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<button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Garnier%2C+J&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Garnier%2C+J&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Garnier%2C+J&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> </ul> </nav> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2503.15922">arXiv:2503.15922</a> <span> [<a href="https://arxiv.org/pdf/2503.15922">pdf</a>, <a href="https://arxiv.org/ps/2503.15922">ps</a>, <a href="https://arxiv.org/format/2503.15922">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> General reproducing properties in RKHS with application to derivative and integral operators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=El-Boukkouri%2C+F">Fatima-Zahrae El-Boukkouri</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Roustant%2C+O">Olivier Roustant</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2503.15922v1-abstract-short" style="display: inline;"> In this paper, we generalize the reproducing property in Reproducing Kernel Hilbert Spaces (RKHS). We establish a reproducing property for the closure of the class of combinations of composition operators under minimal conditions. As an application, we improve the existing sufficient conditions for the reproducing property to hold for the derivative operator, as well as for the existence of the me… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.15922v1-abstract-full').style.display = 'inline'; document.getElementById('2503.15922v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2503.15922v1-abstract-full" style="display: none;"> In this paper, we generalize the reproducing property in Reproducing Kernel Hilbert Spaces (RKHS). We establish a reproducing property for the closure of the class of combinations of composition operators under minimal conditions. As an application, we improve the existing sufficient conditions for the reproducing property to hold for the derivative operator, as well as for the existence of the mean embedding function. These results extend the scope of applicability of the representer theorem for regularized learning algorithms that involve data for function values, gradients, or any other operator from the considered class. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2503.15922v1-abstract-full').style.display = 'none'; document.getElementById('2503.15922v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2025. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2502.07028">arXiv:2502.07028</a> <span> [<a href="https://arxiv.org/pdf/2502.07028">pdf</a>, <a href="https://arxiv.org/format/2502.07028">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Fluid Dynamics">physics.flu-dyn</span> </div> </div> <p class="title is-5 mathjax"> Stochastic Dynamics of Incoherent Branched Flows </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Picozzi%2C+A">Antonio Picozzi</a>, <a href="/search/math?searchtype=author&query=Torres%2C+T">Theo Torres</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2502.07028v1-abstract-short" style="display: inline;"> Waves propagating through weakly disordered smooth linear media undergo a universal phenomenon called branched flow. Branched flows have been observed and studied experimentally in various systems by considering coherent waves. Recent experiments have reported the observation of optical branched flows by using an incoherent light source, thus revealing the key role of coherent phase-sensitive effe… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.07028v1-abstract-full').style.display = 'inline'; document.getElementById('2502.07028v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2502.07028v1-abstract-full" style="display: none;"> Waves propagating through weakly disordered smooth linear media undergo a universal phenomenon called branched flow. Branched flows have been observed and studied experimentally in various systems by considering coherent waves. Recent experiments have reported the observation of optical branched flows by using an incoherent light source, thus revealing the key role of coherent phase-sensitive effects in the development of incoherent branched flows. By considering the paraxial wave equation as a generic representative model, we elaborate a stochastic theory of both coherent and incoherent branched flows. We derive closed-form equations that determine the evolution of the intensity correlation function, as well as the value and the propagation distance of the maximum of the scintillation index, which characterize the dynamical formation of incoherent branched flows. We report accurate numerical simulations that are found in quantitative agreement with the theory without free parameters. Our theory elucidates the key role of coherence and interference on branched flows, thereby providing a framework for exploring branched flow in nonlinear media, in relation with the formation of freak waves in oceans. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2502.07028v1-abstract-full').style.display = 'none'; document.getElementById('2502.07028v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2025. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35R60; 78A48 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.00116">arXiv:2410.00116</a> <span> [<a href="https://arxiv.org/pdf/2410.00116">pdf</a>, <a href="https://arxiv.org/format/2410.00116">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Methodology">stat.ME</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Bayesian Calibration for Prediction in a Multi-Output Transposition Context </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Sire%2C+C">Charlie Sire</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Durantin%2C+C">C茅dric Durantin</a>, <a href="/search/math?searchtype=author&query=Kerleguer%2C+B">Baptiste Kerleguer</a>, <a href="/search/math?searchtype=author&query=Defaux%2C+G">Gilles Defaux</a>, <a href="/search/math?searchtype=author&query=Perrin%2C+G">Guillaume Perrin</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.00116v3-abstract-short" style="display: inline;"> Numerical simulations are widely used to predict the behavior of physical systems, with Bayesian approaches being particularly well suited for this purpose. However, experimental observations are necessary to calibrate certain simulator parameters for the prediction. In this work, we use a multi-output simulator to predict all its outputs, including those that have never been experimentally observ… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.00116v3-abstract-full').style.display = 'inline'; document.getElementById('2410.00116v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.00116v3-abstract-full" style="display: none;"> Numerical simulations are widely used to predict the behavior of physical systems, with Bayesian approaches being particularly well suited for this purpose. However, experimental observations are necessary to calibrate certain simulator parameters for the prediction. In this work, we use a multi-output simulator to predict all its outputs, including those that have never been experimentally observed. This situation is referred to as the transposition context. To accurately quantify the discrepancy between model outputs and real data in this context, conventional methods cannot be applied, and the Bayesian calibration must be augmented by incorporating a joint model error across all outputs. To achieve this, the proposed method is to consider additional numerical input parameters within a hierarchical Bayesian model, which includes hyperparameters for the prior distribution of the calibration variables. This approach is applied on a computer code with three outputs that models the Taylor cylinder impact test with a small number of observations. The outputs are considered as the observed variables one at a time, to work with three different transposition situations. The proposed method is compared with other approaches that embed model errors to demonstrate the significance of the hierarchical formulation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.00116v3-abstract-full').style.display = 'none'; document.getElementById('2410.00116v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 February, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 30 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Submitted to International Journal for Uncertainty Quantification</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.05056">arXiv:2407.05056</a> <span> [<a href="https://arxiv.org/pdf/2407.05056">pdf</a>, <a href="https://arxiv.org/format/2407.05056">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Classical Analysis and ODEs">math.CA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Surface waves in randomly perturbed discrete models </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Sharma%2C+B+L">Basant Lal Sharma</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.05056v1-abstract-short" style="display: inline;"> We study the propagation of surface waves across structured surfaces with random, localized inhomogeneities. A discrete analogue of Gurtin-Murdoch model is employed and surface elasticity, in contrast to bulk elasticity, is captured by distinct point masses and elastic constants for nearest-neighbour interactions parallel to the surface. Expressions for the surface wave reflectance and transmittan… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.05056v1-abstract-full').style.display = 'inline'; document.getElementById('2407.05056v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.05056v1-abstract-full" style="display: none;"> We study the propagation of surface waves across structured surfaces with random, localized inhomogeneities. A discrete analogue of Gurtin-Murdoch model is employed and surface elasticity, in contrast to bulk elasticity, is captured by distinct point masses and elastic constants for nearest-neighbour interactions parallel to the surface. Expressions for the surface wave reflectance and transmittance, as well as the radiative loss, are provided for every localized patch of point mass perturbation on the surface. As the main result in the article, we provide the statistics of surface wave reflectance and transmittance and the radiative loss for an ensemble of random mass perturbations, independent and identically distributed with mean zero, on the surface. In the weakly scattering regime, the mean radiative loss is found to be proportional to the size of the perturbed patch, to the variance of the mass perturbations, and to an effective parameter that depends on the continuous spectrum of the unperturbed system. In the strongly scattering regime, the mean radiative loss is found to depend on another effective parameter that depends on the continuous spectrum, but not on the variance of the mass perturbations. Numerical simulations are found in quantitative agreement with the theoretical predictions for several illustrative values of the surface structure parameters. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.05056v1-abstract-full').style.display = 'none'; document.getElementById('2407.05056v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">27 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.11454">arXiv:2406.11454</a> <span> [<a href="https://arxiv.org/pdf/2406.11454">pdf</a>, <a href="https://arxiv.org/format/2406.11454">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Sensitivity analysis of colored noise-driven interacting particle systems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Ip%2C+H+L+F">Harry L. F. Ip</a>, <a href="/search/math?searchtype=author&query=Mertz%2C+L">Laurent Mertz</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2406.11454v1-abstract-short" style="display: inline;"> We propose an efficient sensitivity analysis method for a wide class of colored noise-driven interacting particle systems (IPS). Our method is based on unperturbed simulations and significantly extends the Malliavin weight sampling method proposed by Szamel (EPL, 117 (2017) 50010) for evaluating sensitivities such as linear response functions of IPS driven by simple Ornstein-Uhlenbeck processes. W… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.11454v1-abstract-full').style.display = 'inline'; document.getElementById('2406.11454v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.11454v1-abstract-full" style="display: none;"> We propose an efficient sensitivity analysis method for a wide class of colored noise-driven interacting particle systems (IPS). Our method is based on unperturbed simulations and significantly extends the Malliavin weight sampling method proposed by Szamel (EPL, 117 (2017) 50010) for evaluating sensitivities such as linear response functions of IPS driven by simple Ornstein-Uhlenbeck processes. We show that the sensitivity index depends not only on two effective parameters that characterize the variance and correlation time of the noise, but also on the noise spectrum. In the case of a single particle in a harmonic potential, we obtain exact analytical formulas for two types of linear response functions. By applying our method to a system of many particles interacting via a repulsive screened Coulomb potential, we compute the mobility and effective temperature of the system. Our results show that the system dynamics depend, in a nontrivial way, on the noise spectrum. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.11454v1-abstract-full').style.display = 'none'; document.getElementById('2406.11454v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2405.15676">arXiv:2405.15676</a> <span> [<a href="https://arxiv.org/pdf/2405.15676">pdf</a>, <a href="https://arxiv.org/format/2405.15676">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Taming Score-Based Diffusion Priors for Infinite-Dimensional Nonlinear Inverse Problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Baldassari%2C+L">Lorenzo Baldassari</a>, <a href="/search/math?searchtype=author&query=Siahkoohi%2C+A">Ali Siahkoohi</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Solna%2C+K">Knut Solna</a>, <a href="/search/math?searchtype=author&query=de+Hoop%2C+M+V">Maarten V. de Hoop</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2405.15676v1-abstract-short" style="display: inline;"> This work introduces a sampling method capable of solving Bayesian inverse problems in function space. It does not assume the log-concavity of the likelihood, meaning that it is compatible with nonlinear inverse problems. The method leverages the recently defined infinite-dimensional score-based diffusion models as a learning-based prior, while enabling provable posterior sampling through a Langev… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.15676v1-abstract-full').style.display = 'inline'; document.getElementById('2405.15676v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2405.15676v1-abstract-full" style="display: none;"> This work introduces a sampling method capable of solving Bayesian inverse problems in function space. It does not assume the log-concavity of the likelihood, meaning that it is compatible with nonlinear inverse problems. The method leverages the recently defined infinite-dimensional score-based diffusion models as a learning-based prior, while enabling provable posterior sampling through a Langevin-type MCMC algorithm defined on function spaces. A novel convergence analysis is conducted, inspired by the fixed-point methods established for traditional regularization-by-denoising algorithms and compatible with weighted annealing. The obtained convergence bound explicitly depends on the approximation error of the score; a well-approximated score is essential to obtain a well-approximated posterior. Stylized and PDE-based examples are provided, demonstrating the validity of our convergence analysis. We conclude by presenting a discussion of the method's challenges related to learning the score and computational complexity. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2405.15676v1-abstract-full').style.display = 'none'; document.getElementById('2405.15676v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 May, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 62F15; 65N21; 68Q32; 60Hxx; 60Jxx; 65C05; 82C31 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.19482">arXiv:2403.19482</a> <span> [<a href="https://arxiv.org/pdf/2403.19482">pdf</a>, <a href="https://arxiv.org/format/2403.19482">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> The linear sampling method for data generated by small random scatterers </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">J. Garnier</a>, <a href="/search/math?searchtype=author&query=Haddar%2C+H">H. Haddar</a>, <a href="/search/math?searchtype=author&query=Montanelli%2C+H">H. Montanelli</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2403.19482v2-abstract-short" style="display: inline;"> We present an extension of the linear sampling method for solving the sound-soft inverse scattering problem in two dimensions with data generated by randomly distributed small scatterers. The theoretical justification of our novel sampling method is based on a rigorous asymptotic model, a modified Helmholtz--Kirchhoff identity, and our previous work on the linear sampling method for random sources… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.19482v2-abstract-full').style.display = 'inline'; document.getElementById('2403.19482v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.19482v2-abstract-full" style="display: none;"> We present an extension of the linear sampling method for solving the sound-soft inverse scattering problem in two dimensions with data generated by randomly distributed small scatterers. The theoretical justification of our novel sampling method is based on a rigorous asymptotic model, a modified Helmholtz--Kirchhoff identity, and our previous work on the linear sampling method for random sources. Our numerical implementation incorporates boundary elements, Singular Value Decomposition, Tikhonov regularization, and Morozov's discrepancy principle. We showcase the robustness and accuracy of our algorithms with a series of numerical experiments. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.19482v2-abstract-full').style.display = 'none'; document.getElementById('2403.19482v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.02575">arXiv:2312.02575</a> <span> [<a href="https://arxiv.org/pdf/2312.02575">pdf</a>, <a href="https://arxiv.org/format/2312.02575">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1615/Int.J.UncertaintyQuantification.2023044584">10.1615/Int.J.UncertaintyQuantification.2023044584 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A Bayesian neural network approach to Multi-fidelity surrogate modelling </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Kerleguer%2C+B">Baptiste Kerleguer</a>, <a href="/search/math?searchtype=author&query=Cannamela%2C+C">Claire Cannamela</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2312.02575v1-abstract-short" style="display: inline;"> This paper deals with surrogate modelling of a computer code output in a hierarchical multi-fidelity context, i.e., when the output can be evaluated at different levels of accuracy and computational cost. Using observations of the output at low- and high-fidelity levels, we propose a method that combines Gaussian process (GP) regression and Bayesian neural network (BNN), in a method called GPBNN.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.02575v1-abstract-full').style.display = 'inline'; document.getElementById('2312.02575v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.02575v1-abstract-full" style="display: none;"> This paper deals with surrogate modelling of a computer code output in a hierarchical multi-fidelity context, i.e., when the output can be evaluated at different levels of accuracy and computational cost. Using observations of the output at low- and high-fidelity levels, we propose a method that combines Gaussian process (GP) regression and Bayesian neural network (BNN), in a method called GPBNN. The low-fidelity output is treated as a single-fidelity code using classical GP regression. The high-fidelity output is approximated by a BNN that incorporates, in addition to the high-fidelity observations, well-chosen realisations of the low-fidelity output emulator. The predictive uncertainty of the final surrogate model is then quantified by a complete characterisation of the uncertainties of the different models and their interaction. GPBNN is compared with most of the multi-fidelity regression methods allowing to quantify the prediction uncertainty. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.02575v1-abstract-full').style.display = 'none'; document.getElementById('2312.02575v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> International Journal for Uncertainty Quantification, 2024, 14 (1), pp.43-60 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2309.07777">arXiv:2309.07777</a> <span> [<a href="https://arxiv.org/pdf/2309.07777">pdf</a>, <a href="https://arxiv.org/format/2309.07777">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Scattered wavefield in the stochastic homogenization regime </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Giovangigli%2C+L">Laure Giovangigli</a>, <a href="/search/math?searchtype=author&query=Goepfert%2C+Q">Quentin Goepfert</a>, <a href="/search/math?searchtype=author&query=Millien%2C+P">Pierre Millien</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2309.07777v1-abstract-short" style="display: inline;"> In the context of providing a mathematical framework for the propagation of ultrasound waves in a random multiscale medium, we consider the scattering of classical waves (modeled by a divergence form scalar Helmholtz equation) by a bounded object with a random composite micro-structure embedded in an unbounded homogeneous background medium. Using quantitative stochastic homogenization techniques,… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.07777v1-abstract-full').style.display = 'inline'; document.getElementById('2309.07777v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2309.07777v1-abstract-full" style="display: none;"> In the context of providing a mathematical framework for the propagation of ultrasound waves in a random multiscale medium, we consider the scattering of classical waves (modeled by a divergence form scalar Helmholtz equation) by a bounded object with a random composite micro-structure embedded in an unbounded homogeneous background medium. Using quantitative stochastic homogenization techniques, we provide asymptotic expansions of the scattered field in the background medium with respect to a scaling parameter describing the spatial random oscillations of the micro-structure. Introducing a boundary layer corrector to compensate the breakdown of stationarity assumptions at the boundary of the scattering medium, we prove quantitative $L^2$- and $H^1$- error estimates for the asymptotic first-order expansion. The theoretical results are supported by numerical experiments. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.07777v1-abstract-full').style.display = 'none'; document.getElementById('2309.07777v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2309.03090">arXiv:2309.03090</a> <span> [<a href="https://arxiv.org/pdf/2309.03090">pdf</a>, <a href="https://arxiv.org/format/2309.03090">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Classical Analysis and ODEs">math.CA</span> </div> </div> <p class="title is-5 mathjax"> Effective dynamics in lattices with random mass perturbations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Sharma%2C+B+L">Basant Lal Sharma</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2309.03090v2-abstract-short" style="display: inline;"> We consider a one-dimensional mono-atomic lattice with random perturbations of masses spread over a finite number of particles. Assuming Newtonian dynamics and linear nearest-neighbour interactions and allowing for a provision of pinning due to substrate interaction, we discuss a transient dynamics problem and a time-harmonic transmission problem. By a stochastic, multiscale analysis we provide as… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.03090v2-abstract-full').style.display = 'inline'; document.getElementById('2309.03090v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2309.03090v2-abstract-full" style="display: none;"> We consider a one-dimensional mono-atomic lattice with random perturbations of masses spread over a finite number of particles. Assuming Newtonian dynamics and linear nearest-neighbour interactions and allowing for a provision of pinning due to substrate interaction, we discuss a transient dynamics problem and a time-harmonic transmission problem. By a stochastic, multiscale analysis we provide asymptotic expressions for the displacement field that propagates through the random perturbations and for the time-harmonic transmission coefficients. These theoretical predictions are supported by illustrations of their agreements with numerical simulations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.03090v2-abstract-full').style.display = 'none'; document.getElementById('2309.03090v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">6 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2305.19147">arXiv:2305.19147</a> <span> [<a href="https://arxiv.org/pdf/2305.19147">pdf</a>, <a href="https://arxiv.org/format/2305.19147">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Conditional score-based diffusion models for Bayesian inference in infinite dimensions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Baldassari%2C+L">Lorenzo Baldassari</a>, <a href="/search/math?searchtype=author&query=Siahkoohi%2C+A">Ali Siahkoohi</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Solna%2C+K">Knut Solna</a>, <a href="/search/math?searchtype=author&query=de+Hoop%2C+M+V">Maarten V. de Hoop</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2305.19147v2-abstract-short" style="display: inline;"> Since their initial introduction, score-based diffusion models (SDMs) have been successfully applied to solve a variety of linear inverse problems in finite-dimensional vector spaces due to their ability to efficiently approximate the posterior distribution. However, using SDMs for inverse problems in infinite-dimensional function spaces has only been addressed recently, primarily through methods… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.19147v2-abstract-full').style.display = 'inline'; document.getElementById('2305.19147v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2305.19147v2-abstract-full" style="display: none;"> Since their initial introduction, score-based diffusion models (SDMs) have been successfully applied to solve a variety of linear inverse problems in finite-dimensional vector spaces due to their ability to efficiently approximate the posterior distribution. However, using SDMs for inverse problems in infinite-dimensional function spaces has only been addressed recently, primarily through methods that learn the unconditional score. While this approach is advantageous for some inverse problems, it is mostly heuristic and involves numerous computationally costly forward operator evaluations during posterior sampling. To address these limitations, we propose a theoretically grounded method for sampling from the posterior of infinite-dimensional Bayesian linear inverse problems based on amortized conditional SDMs. In particular, we prove that one of the most successful approaches for estimating the conditional score in finite dimensions - the conditional denoising estimator - can also be applied in infinite dimensions. A significant part of our analysis is dedicated to demonstrating that extending infinite-dimensional SDMs to the conditional setting requires careful consideration, as the conditional score typically blows up for small times, contrarily to the unconditional score. We conclude by presenting stylized and large-scale numerical examples that validate our approach, offer additional insights, and demonstrate that our method enables large-scale, discretization-invariant Bayesian inference. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.19147v2-abstract-full').style.display = 'none'; document.getElementById('2305.19147v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 October, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">NeurIPS 2023 (Spotlight)</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 62F15; 65N21; 68Q32; 60Hxx; 60Jxx </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2305.12426">arXiv:2305.12426</a> <span> [<a href="https://arxiv.org/pdf/2305.12426">pdf</a>, <a href="https://arxiv.org/format/2305.12426">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistical Mechanics">cond-mat.stat-mech</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Computing the diffusivity of a particle subject to dry friction with colored noise </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Mertz%2C+L">Laurent Mertz</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2305.12426v2-abstract-short" style="display: inline;"> This paper considers the motion of an object subjected to dry friction and an external random force. The objective is to characterize the role of the correlation time of the external random force. We develop efficient stochastic simulation methods for computing the diffusivity (the linear growth rate of the variance of the displacement) and other related quantities of interest when the external ra… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.12426v2-abstract-full').style.display = 'inline'; document.getElementById('2305.12426v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2305.12426v2-abstract-full" style="display: none;"> This paper considers the motion of an object subjected to dry friction and an external random force. The objective is to characterize the role of the correlation time of the external random force. We develop efficient stochastic simulation methods for computing the diffusivity (the linear growth rate of the variance of the displacement) and other related quantities of interest when the external random force is white or colored. These methods are based on original representation formulas for the quantities of interest which make it possible to build unbiased and consistent estimators. The numerical results obtained with these original methods are in perfect agreement with known closed-form formulas valid in the white noise regime. In the colored noise regime the numerical results show that the predictions obtained from the white-noise approximation are reasonable for quantities such as the histograms of the stationary velocity but can be wrong for the diffusivity unless the correlation time is extremely small. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.12426v2-abstract-full').style.display = 'none'; document.getElementById('2305.12426v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2302.08173">arXiv:2302.08173</a> <span> [<a href="https://arxiv.org/pdf/2302.08173">pdf</a>, <a href="https://arxiv.org/format/2302.08173">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/1361-6420/ad2781">10.1088/1361-6420/ad2781 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Inverse problem for Love waves in a layered, elastic half-space </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=de+Hoop%2C+M+V">Maarten V. de Hoop</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Iantchenko%2C+A">Alexei Iantchenko</a>, <a href="/search/math?searchtype=author&query=Ricaud%2C+J">Julien Ricaud</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2302.08173v3-abstract-short" style="display: inline;"> In this paper we study Love waves in a layered, elastic half-space. We first address the direct problem and we characterize the existence of Love waves through the dispersion relation. We then address the inverse problem and we show how to recover the parameters of the elastic medium from the empirical knowledge of the frequency--wavenumber couples of the Love waves. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2302.08173v3-abstract-full" style="display: none;"> In this paper we study Love waves in a layered, elastic half-space. We first address the direct problem and we characterize the existence of Love waves through the dispersion relation. We then address the inverse problem and we show how to recover the parameters of the elastic medium from the empirical knowledge of the frequency--wavenumber couples of the Love waves. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.08173v3-abstract-full').style.display = 'none'; document.getElementById('2302.08173v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 February, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">37 pages, 7 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 74J25; 74J15; 86A15; 86A22; 35R30 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Inverse Problems 40 (4), 045013 (2024) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2302.05988">arXiv:2302.05988</a> <span> [<a href="https://arxiv.org/pdf/2302.05988">pdf</a>, <a href="https://arxiv.org/format/2302.05988">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geophysics">physics.geo-ph</span> </div> </div> <p class="title is-5 mathjax"> When data driven reduced order modeling meets full waveform inversion </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Borcea%2C+L">Liliana Borcea</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Mamonov%2C+A+V">Alexander V. Mamonov</a>, <a href="/search/math?searchtype=author&query=Zimmerling%2C+J">J枚rn Zimmerling</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2302.05988v3-abstract-short" style="display: inline;"> Waveform inversion is concerned with estimating a heterogeneous medium, modeled by variable coefficients of wave equations, using sources that emit probing signals and receivers that record the generated waves. It is an old and intensively studied inverse problem with a wide range of applications, but the existing inversion methodologies are still far from satisfactory. The typical mathematical fo… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.05988v3-abstract-full').style.display = 'inline'; document.getElementById('2302.05988v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2302.05988v3-abstract-full" style="display: none;"> Waveform inversion is concerned with estimating a heterogeneous medium, modeled by variable coefficients of wave equations, using sources that emit probing signals and receivers that record the generated waves. It is an old and intensively studied inverse problem with a wide range of applications, but the existing inversion methodologies are still far from satisfactory. The typical mathematical formulation is a nonlinear least squares data fit optimization and the difficulty stems from the non-convexity of the objective function that displays numerous local minima at which local optimization approaches stagnate. This pathological behavior has at least three unavoidable causes: (1) The mapping from the unknown coefficients to the wave field is nonlinear and complicated. (2) The sources and receivers typically lie on a single side of the medium, so only backscattered waves are measured. (3) The probing signals are band limited and with high frequency content. There is a lot of activity in the computational science and engineering communities that seeks to mitigate the difficulty of estimating the medium by data fitting. In this paper we present a different point of view, based on reduced order models (ROMs) of two operators that control the wave propagation. The ROMs are called data driven because they are computed directly from the measurements, without any knowledge of the wave field inside the inaccessible medium. This computation is non-iterative and uses standard numerical linear algebra methods. The resulting ROMs capture features of the physics of wave propagation in a complementary way and have surprisingly good approximation properties that facilitate waveform inversion. In this arxiv version two important typos are corrected when compared to the published version. The typo was in the second equation in Theorem 3 and carried over into Corollary 1. The proofs are correct. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2302.05988v3-abstract-full').style.display = 'none'; document.getElementById('2302.05988v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 February, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 65M32; 41A20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2210.16185">arXiv:2210.16185</a> <span> [<a href="https://arxiv.org/pdf/2210.16185">pdf</a>, <a href="https://arxiv.org/format/2210.16185">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computation">stat.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Applications">stat.AP</span> </div> </div> <p class="title is-5 mathjax"> Adaptive importance sampling based on fault tree analysis for piecewise deterministic Markov process </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chennetier%2C+G">Guillaume Chennetier</a>, <a href="/search/math?searchtype=author&query=Chraibi%2C+H">Hassane Chraibi</a>, <a href="/search/math?searchtype=author&query=Dutfoy%2C+A">Anne Dutfoy</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2210.16185v2-abstract-short" style="display: inline;"> Piecewise deterministic Markov processes (PDMPs) can be used to model complex dynamical industrial systems. The counterpart of this modeling capability is their simulation cost, which makes reliability assessment untractable with standard Monte Carlo methods. A significant variance reduction can be obtained with an adaptive importance sampling (AIS) method based on a cross-entropy (CE) procedure.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2210.16185v2-abstract-full').style.display = 'inline'; document.getElementById('2210.16185v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2210.16185v2-abstract-full" style="display: none;"> Piecewise deterministic Markov processes (PDMPs) can be used to model complex dynamical industrial systems. The counterpart of this modeling capability is their simulation cost, which makes reliability assessment untractable with standard Monte Carlo methods. A significant variance reduction can be obtained with an adaptive importance sampling (AIS) method based on a cross-entropy (CE) procedure. The success of this method relies on the selection of a good family of approximations of the committor function of the PDMP. In this paper original families are proposed. They are well adapted to high-dimensional industrial systems. Their forms are based on reliability concepts related to fault tree analysis: minimal path sets and minimal cut sets. The proposed method is discussed in detail and applied to academic systems and to a realistic system from the nuclear industry. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2210.16185v2-abstract-full').style.display = 'none'; document.getElementById('2210.16185v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 June, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 September, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 65C05; 62L12; 65C60; 60J25 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2210.15560">arXiv:2210.15560</a> <span> [<a href="https://arxiv.org/pdf/2210.15560">pdf</a>, <a href="https://arxiv.org/format/2210.15560">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> The linear sampling method for random sources </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Haddar%2C+H">Houssem Haddar</a>, <a href="/search/math?searchtype=author&query=Montanelli%2C+H">Hadrien Montanelli</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2210.15560v2-abstract-short" style="display: inline;"> We present an extension of the linear sampling method for solving the sound-soft inverse acoustic scattering problem with randomly distributed point sources. The theoretical justification of our sampling method is based on the Helmholtz--Kirchhoff identity, the cross-correlation between measurements, and the volume and imaginary near-field operators, which we introduce and analyze. Implementations… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2210.15560v2-abstract-full').style.display = 'inline'; document.getElementById('2210.15560v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2210.15560v2-abstract-full" style="display: none;"> We present an extension of the linear sampling method for solving the sound-soft inverse acoustic scattering problem with randomly distributed point sources. The theoretical justification of our sampling method is based on the Helmholtz--Kirchhoff identity, the cross-correlation between measurements, and the volume and imaginary near-field operators, which we introduce and analyze. Implementations in MATLAB using boundary elements, the SVD, Tikhonov regularization, and Morozov's discrepancy principle are also discussed. We demonstrate the robustness and accuracy of our algorithms with several numerical experiments in two dimensions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2210.15560v2-abstract-full').style.display = 'none'; document.getElementById('2210.15560v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 March, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 October, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2208.11051">arXiv:2208.11051</a> <span> [<a href="https://arxiv.org/pdf/2208.11051">pdf</a>, <a href="https://arxiv.org/format/2208.11051">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Waveform inversion with a data driven estimate of the internal wave </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Borcea%2C+L">Liliana Borcea</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Mamonov%2C+A+V">Alexander V. Mamonov</a>, <a href="/search/math?searchtype=author&query=Zimmerling%2C+J">J枚rn Zimmerling</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2208.11051v1-abstract-short" style="display: inline;"> We study an inverse problem for the wave equation, concerned with estimating the wave speed, aka velocity, from data gathered by an array of sources and receivers that emit probing signals and measure the resulting waves. The typical mathematical formulation of velocity estimation is a nonlinear least squares minimization of the data misfit, over a search velocity space. There are two main impedim… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.11051v1-abstract-full').style.display = 'inline'; document.getElementById('2208.11051v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2208.11051v1-abstract-full" style="display: none;"> We study an inverse problem for the wave equation, concerned with estimating the wave speed, aka velocity, from data gathered by an array of sources and receivers that emit probing signals and measure the resulting waves. The typical mathematical formulation of velocity estimation is a nonlinear least squares minimization of the data misfit, over a search velocity space. There are two main impediments to this approach, which manifest as multiple local minima of the objective function: The nonlinearity of the mapping from the velocity to the data, which accounts for multiple scattering effects, and poor knowledge of the kinematics (smooth part of the wave speed) which causes cycle-skipping. We show that the nonlinearity can be mitigated using a data driven estimate of the internal wave field. This leads to improved performance of the inversion for a reasonable initial guess of the kinematics. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.11051v1-abstract-full').style.display = 'none'; document.getElementById('2208.11051v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 August, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2208.01209">arXiv:2208.01209</a> <span> [<a href="https://arxiv.org/pdf/2208.01209">pdf</a>, <a href="https://arxiv.org/format/2208.01209">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Velocity estimation via model order reduction </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Mamonov%2C+A+V">Alexander V. Mamonov</a>, <a href="/search/math?searchtype=author&query=Borcea%2C+L">Liliana Borcea</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Zimmerling%2C+J">J枚rn Zimmerling</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2208.01209v1-abstract-short" style="display: inline;"> A novel approach to full waveform inversion (FWI), based on a data driven reduced order model (ROM) of the wave equation operator is introduced. The unknown medium is probed with pulses and the time domain pressure waveform data is recorded on an active array of sensors. The ROM, a projection of the wave equation operator is constructed from the data via a nonlinear process and is used for efficie… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.01209v1-abstract-full').style.display = 'inline'; document.getElementById('2208.01209v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2208.01209v1-abstract-full" style="display: none;"> A novel approach to full waveform inversion (FWI), based on a data driven reduced order model (ROM) of the wave equation operator is introduced. The unknown medium is probed with pulses and the time domain pressure waveform data is recorded on an active array of sensors. The ROM, a projection of the wave equation operator is constructed from the data via a nonlinear process and is used for efficient velocity estimation. While the conventional FWI via nonlinear least-squares data fitting is challenging without low frequency information, and prone to getting stuck in local minima (cycle skipping), minimization of ROM misfit is behaved much better, even for a poor initial guess. For low-dimensional parametrizations of the unknown velocity the ROM misfit function is close to convex. The proposed approach consistently outperforms conventional FWI in standard synthetic tests. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.01209v1-abstract-full').style.display = 'none'; document.getElementById('2208.01209v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 August, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">5 pages, 3 figures, to be presented at IMAGE 2022</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 65M32; 41A20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2206.13248">arXiv:2206.13248</a> <span> [<a href="https://arxiv.org/pdf/2206.13248">pdf</a>, <a href="https://arxiv.org/ps/2206.13248">ps</a>, <a href="https://arxiv.org/format/2206.13248">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Populations and Evolution">q-bio.PE</span> </div> </div> <p class="title is-5 mathjax"> Adaptation to a changing environment: what me Normal? </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">Jimmy Garnier</a>, <a href="/search/math?searchtype=author&query=Cotto%2C+O">O Cotto</a>, <a href="/search/math?searchtype=author&query=Bourgeron%2C+T">T Bourgeron</a>, <a href="/search/math?searchtype=author&query=Bouin%2C+E">E Bouin</a>, <a href="/search/math?searchtype=author&query=Lepoutre%2C+T">T Lepoutre</a>, <a href="/search/math?searchtype=author&query=Ronce%2C+O">O Ronce</a>, <a href="/search/math?searchtype=author&query=Calvez%2C+V">V Calvez</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2206.13248v1-abstract-short" style="display: inline;"> Predicting the adaptation of populations to a changing environment is crucial to assess the impact of human activities on biodiversity. Many theoretical studies have tackled this issue by modeling the evolution of quantitative traits subject to stabilizing selection around an optimum phenotype, whose value is shifted continuously through time. In this context, the population fate results from the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2206.13248v1-abstract-full').style.display = 'inline'; document.getElementById('2206.13248v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2206.13248v1-abstract-full" style="display: none;"> Predicting the adaptation of populations to a changing environment is crucial to assess the impact of human activities on biodiversity. Many theoretical studies have tackled this issue by modeling the evolution of quantitative traits subject to stabilizing selection around an optimum phenotype, whose value is shifted continuously through time. In this context, the population fate results from the equilibrium distribution of the trait, relative to the moving optimum. Such a distribution may vary with the shape of selection, the system of reproduction, the number of loci, the mutation kernel or their interactions. Here, we develop a methodology that provides quantitative measures of population maladaptation and potential of survival directly from the entire profile of the phenotypic distribution, without any a priori on its shape.We investigate two different models of reproduction (asexual and infinitesimal sexual models of inheritance), with general forms of selection.In particular, we recover that fitness functions such that selection weakens away from the optimum lead to evolutionary tipping points, with an abrupt collapse of the population when the speed of environmental change is too high. Our unified framework furthermore allows highlighting the underlying mechanisms that lead to this phenomenon. More generally, it allows discussing similarities and discrepancies between the two reproduction models, the latter being ultimately explained by different constraints on the evolution of the phenotypic variance. We demonstrate that the mean fitness in the population crucially depends on the shape of the selection function in the sexual infinitesimal model, in contrast with the asexual model.In the asexual model, we also investigate the effect of the mutation kernel and we show that kernels with higher kurtosis tend to reduce maladaptation and improve fitness, especially in fast changing environments. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2206.13248v1-abstract-full').style.display = 'none'; document.getElementById('2206.13248v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 June, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2205.04451">arXiv:2205.04451</a> <span> [<a href="https://arxiv.org/pdf/2205.04451">pdf</a>, <a href="https://arxiv.org/format/2205.04451">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optics">physics.optics</span> </div> </div> <p class="title is-5 mathjax"> Paraxial wave propagation in random media with long-range correlations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Borcea%2C+L">Liliana Borcea</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Solna%2C+K">Knut Solna</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2205.04451v1-abstract-short" style="display: inline;"> We study the paraxial wave equation with a randomly perturbed index of refraction, which can model the propagation of a wave beam in a turbulent medium. The random perturbation is a stationary and isotropic process with a general form of the covariance that may be integrable or not. We focus attention mostly on the non-integrable case, which corresponds to a random perturbation with long-range cor… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.04451v1-abstract-full').style.display = 'inline'; document.getElementById('2205.04451v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2205.04451v1-abstract-full" style="display: none;"> We study the paraxial wave equation with a randomly perturbed index of refraction, which can model the propagation of a wave beam in a turbulent medium. The random perturbation is a stationary and isotropic process with a general form of the covariance that may be integrable or not. We focus attention mostly on the non-integrable case, which corresponds to a random perturbation with long-range correlations, that is relevant for propagation through a cloudy turbulent atmosphere. The analysis is carried out in a high-frequency regime where the forward scattering approximation holds. It reveals that the randomization of the wave field is multiscale: The travel time of the wave front is randomized at short distances of propagation and it can be described by a fractional Brownian motion. The wave field observed in the random travel time frame is affected by the random perturbations at long distances, and it is described by a Schroedinger-type equation driven by a standard Brownian field. We use these results to quantify how scattering leads to decorrelation of the spatial and spectral components of the wave field and to a deformation of the pulse emitted by the source. These are important questions for applications like imaging and free space communications with pulsed laser beams through a turbulent atmosphere. We also compare the results with those used in the optics literature, which are based on the Kolmogorov model of turbulence. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.04451v1-abstract-full').style.display = 'none'; document.getElementById('2205.04451v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 May, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">26 pages, 1 figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 76B15; 35Q94; 60F05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2202.12552">arXiv:2202.12552</a> <span> [<a href="https://arxiv.org/pdf/2202.12552">pdf</a>, <a href="https://arxiv.org/ps/2202.12552">ps</a>, <a href="https://arxiv.org/format/2202.12552">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> A piecewise deterministic Markov process approach modeling a dry friction problem with noise </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Lu%2C+Z">Ziyu Lu</a>, <a href="/search/math?searchtype=author&query=Mertz%2C+L">Laurent Mertz</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2202.12552v2-abstract-short" style="display: inline;"> Understanding and predicting the dynamical properties of systems involving dry friction is a major concern in physics and engineering. It abounds in many mechanical processes, from the sound produced by a violin to the screeching of chalk on a blackboard to human infant crawling dynamics and friction-based locomotion of a multitude of living organisms (snakes, bacteria, scallops..) to the displace… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.12552v2-abstract-full').style.display = 'inline'; document.getElementById('2202.12552v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2202.12552v2-abstract-full" style="display: none;"> Understanding and predicting the dynamical properties of systems involving dry friction is a major concern in physics and engineering. It abounds in many mechanical processes, from the sound produced by a violin to the screeching of chalk on a blackboard to human infant crawling dynamics and friction-based locomotion of a multitude of living organisms (snakes, bacteria, scallops..) to the displacement of mechanical structures (building, bridges, nuclear plants, massive industrial infrastructures) under earthquakes and beyond. Surprisingly, even for low-dimensional systems, the modeling of dry friction in the presence of random forcing has not been elucidated. In this paper, we propose a piecewise deterministic Markov process approach modeling a system with dry friction including different coefficients for the static and dynamic forces. In this mathematical framework, we derive the corresponding Kolmogorov equations and related tools to compute statistical quantities of interest related to the distributions of the static (sticked) and dynamic phases. We show ergodicity and provide a representation formula of the stationary measure using independent identically distributed portions of the trajectory (excursions). We also obtain deterministic characterizations of the Laplace transforms of the probability density functions of the durations of the static and dynamic phases. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.12552v2-abstract-full').style.display = 'none'; document.getElementById('2202.12552v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 25 February, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2202.01824">arXiv:2202.01824</a> <span> [<a href="https://arxiv.org/pdf/2202.01824">pdf</a>, <a href="https://arxiv.org/format/2202.01824">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geophysics">physics.geo-ph</span> </div> </div> <p class="title is-5 mathjax"> Waveform inversion via reduced order modeling </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Borcea%2C+L">Liliana Borcea</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Mamonov%2C+A+V">Alexander V. Mamonov</a>, <a href="/search/math?searchtype=author&query=Zimmerling%2C+J">J枚rn Zimmerling</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2202.01824v5-abstract-short" style="display: inline;"> We introduce a novel approach to waveform inversion, based on a data driven reduced order model (ROM) of the wave operator. The presentation is for the acoustic wave equation, but the approach can be extended to elastic or electromagnetic waves. The data are time resolved measurements of the pressure wave gathered by an acquisition system which probes the unknown medium with pulses and measures th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.01824v5-abstract-full').style.display = 'inline'; document.getElementById('2202.01824v5-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2202.01824v5-abstract-full" style="display: none;"> We introduce a novel approach to waveform inversion, based on a data driven reduced order model (ROM) of the wave operator. The presentation is for the acoustic wave equation, but the approach can be extended to elastic or electromagnetic waves. The data are time resolved measurements of the pressure wave gathered by an acquisition system which probes the unknown medium with pulses and measures the generated waves. We propose to solve the inverse problem of velocity estimation by minimizing the square misfit between the ROM computed from the recorded data and the ROM computed from the modeled data, at the current guess of the velocity. We give the step by step computation of the ROM, which depends nonlinearly on the data and yet can be obtained from them in a non-iterative fashion, using efficient methods from linear algebra. We also explain how to make the ROM robust to data inaccuracy. The ROM computation requires the full array response matrix gathered with collocated sources and receivers. However, we show that the computation can deal with an approximation of this matrix, obtained from towed-streamer data using interpolation and reciprocity on-the-fly. While the full-waveform inversion approach of nonlinear least-squares data fitting is challenging without low frequency information, due to multiple minima of the data fit objective function, we show that the ROM misfit objective function has a better behavior, even for a poor initial guess. We also show by an explicit computation of the objective functions in a simple setting that the ROM misfit objective function has convexity properties, whereas the least squares data fit objective function displays multiple local minima. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.01824v5-abstract-full').style.display = 'none'; document.getElementById('2202.01824v5-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 December, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 3 February, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 65M32; 41A20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2201.06671">arXiv:2201.06671</a> <span> [<a href="https://arxiv.org/pdf/2201.06671">pdf</a>, <a href="https://arxiv.org/format/2201.06671">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optics">physics.optics</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1364/JOSAA.453358">10.1364/JOSAA.453358 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Scintillation of Partially Coherent Light in Time Varying Complex Media </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Solna%2C+K">Knut Solna</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2201.06671v1-abstract-short" style="display: inline;"> We present a theory for wave scintillation in the situation with a time-dependent partially coherent source and a time-dependent randomly heterogeneous medium. Our objective is to understand how the scintillation index of the measured intensity depends on the source and medium parameters. We deduce from an asymptotic analysis of the random wave equation a general form of the scintillation index an… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.06671v1-abstract-full').style.display = 'inline'; document.getElementById('2201.06671v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2201.06671v1-abstract-full" style="display: none;"> We present a theory for wave scintillation in the situation with a time-dependent partially coherent source and a time-dependent randomly heterogeneous medium. Our objective is to understand how the scintillation index of the measured intensity depends on the source and medium parameters. We deduce from an asymptotic analysis of the random wave equation a general form of the scintillation index and we evaluate this in various scaling regimes. The scintillation index is a fundamental quantity that is used to analyze and optimize imaging and communication schemes. Our results are useful to quantify the scintillation index under realistic propagation scenarios and to address such optimization challenges. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.06671v1-abstract-full').style.display = 'none'; document.getElementById('2201.06671v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 January, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">28 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 60H15; 35R60; 74J20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2201.06649">arXiv:2201.06649</a> <span> [<a href="https://arxiv.org/pdf/2201.06649">pdf</a>, <a href="https://arxiv.org/format/2201.06649">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optics">physics.optics</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Partially Coherent Electromagnetic Beam Propagation in Random Media </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=S%C3%B8lna%2C+K">Knut S酶lna</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2201.06649v1-abstract-short" style="display: inline;"> A theory for the characterization of the fourth moment of electromagnetic wave beams is presented in the case when the source is partially coherent. A Gaussian-Schell model is used for the partially coherent random source. The white-noise paraxial regime is considered, which holds when the wavelength is much smaller than the correlation radius of the source, the beam radius of the source, and the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.06649v1-abstract-full').style.display = 'inline'; document.getElementById('2201.06649v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2201.06649v1-abstract-full" style="display: none;"> A theory for the characterization of the fourth moment of electromagnetic wave beams is presented in the case when the source is partially coherent. A Gaussian-Schell model is used for the partially coherent random source. The white-noise paraxial regime is considered, which holds when the wavelength is much smaller than the correlation radius of the source, the beam radius of the source, and the correlation length of the medium, which are themselves much smaller than the propagation distance. The complex wave amplitude field can then be described by the It么-Schr枚dinger equation. This equation gives closed evolution equations for the wave field moments at all orders and here the fourth order equations are considered. The general fourth moment equations are solved explicitly in the scintillation regime (when the correlation radius of the source is of the same order as the correlation radius of the medium, but the beam radius is much larger) and the result gives a characterization of the intensity covariance function. The form of the intensity covariance function derives from the solution of the transport equation for the Wigner distribution associated with the second wave moment. The fourth moment results for polarized waves is used in an application to imaging of partially coherent sources. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.06649v1-abstract-full').style.display = 'none'; document.getElementById('2201.06649v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 January, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">23 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 60H15; 35R60; 74J20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2201.05558">arXiv:2201.05558</a> <span> [<a href="https://arxiv.org/pdf/2201.05558">pdf</a>, <a href="https://arxiv.org/ps/2201.05558">ps</a>, <a href="https://arxiv.org/format/2201.05558">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optics">physics.optics</span> </div> </div> <p class="title is-5 mathjax"> Speckle Memory Effect in the Frequency Domain and Stability in Time-Reversal Experiments </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Solna%2C+K">Knut Solna</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2201.05558v1-abstract-short" style="display: inline;"> When waves propagate through a complex medium like the turbulent atmosphere the wave field becomes incoherent and the wave intensity forms a complex speckle pattern. In this paper we study a speckle memory effect in the frequency domain and some of its consequences. This effect means that certain properties of the speckle pattern produced by wave transmission through a randomly scattering medium i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.05558v1-abstract-full').style.display = 'inline'; document.getElementById('2201.05558v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2201.05558v1-abstract-full" style="display: none;"> When waves propagate through a complex medium like the turbulent atmosphere the wave field becomes incoherent and the wave intensity forms a complex speckle pattern. In this paper we study a speckle memory effect in the frequency domain and some of its consequences. This effect means that certain properties of the speckle pattern produced by wave transmission through a randomly scattering medium is preserved when shifting the frequency of the illumination. The speckle memory effect is characterized via a detailed novel analysis of the fourth-order moment of the random paraxial Green's function at four different frequencies. We arrive at a precise characterization of the frequency memory effect and what governs the strength of the memory. As an application we quantify the statistical stability of time-reversal wave refocusing through a randomly scattering medium in the paraxial or beam regime. Time reversal refers to the situation when a transmitted wave field is recorded on a time-reversal mirror then time reversed and sent back into the complex medium. The reemitted wave field then refocuses at the original source point. We compute the mean of the refocused wave and identify a novel quantitative description of its variance in terms of the radius of the time-reversal mirror, the size of its elements, the source bandwidth and the statistics of the random medium fluctuations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.05558v1-abstract-full').style.display = 'none'; document.getElementById('2201.05558v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 January, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 60H15; 35R60; 35L05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2108.01609">arXiv:2108.01609</a> <span> [<a href="https://arxiv.org/pdf/2108.01609">pdf</a>, <a href="https://arxiv.org/format/2108.01609">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/1361-6420/ac41d0">10.1088/1361-6420/ac41d0 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Reduced order model approach for imaging with waves </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Borcea%2C+L">Liliana Borcea</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Mamonov%2C+A+V">Alexander V. Mamonov</a>, <a href="/search/math?searchtype=author&query=Zimmerling%2C+J">J枚rn Zimmerling</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2108.01609v1-abstract-short" style="display: inline;"> We introduce a novel, computationally inexpensive approach for imaging with an active array of sensors, which probe an unknown medium with a pulse and measure the resulting waves. The imaging function uses a data driven estimate of the "internal wave" originating from the vicinity of the imaging point and propagating to the sensors through the unknown medium. We explain how this estimate can be ob… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2108.01609v1-abstract-full').style.display = 'inline'; document.getElementById('2108.01609v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2108.01609v1-abstract-full" style="display: none;"> We introduce a novel, computationally inexpensive approach for imaging with an active array of sensors, which probe an unknown medium with a pulse and measure the resulting waves. The imaging function uses a data driven estimate of the "internal wave" originating from the vicinity of the imaging point and propagating to the sensors through the unknown medium. We explain how this estimate can be obtained using a reduced order model (ROM) for the wave propagation. We analyze the imaging function, connect it to the time reversal process and describe how its resolution depends on the aperture of the array, the bandwidth of the probing pulse and the medium through which the waves propagate. We also show how the internal wave can be used for selective focusing of waves at points in the imaging region. This can be implemented experimentally and can be used for pixel scanning imaging. We assess the performance of the imaging methods with numerical simulations and compare them to the conventional reverse-time migration method and the "backprojection" method introduced recently as an application of the same ROM. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2108.01609v1-abstract-full').style.display = 'none'; document.getElementById('2108.01609v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 August, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 65M32; 41A20 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2105.12174">arXiv:2105.12174</a> <span> [<a href="https://arxiv.org/pdf/2105.12174">pdf</a>, <a href="https://arxiv.org/format/2105.12174">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Image and Video Processing">eess.IV</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Imaging in random media by two-point coherent interferometry </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Borcea%2C+L">Liliana Borcea</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2105.12174v1-abstract-short" style="display: inline;"> This paper considers wave-based imaging through a heterogeneous (random) scattering medium. The goal is to estimate the support of the reflectivity function of a remote scene from measurements of the backscattered wave field. The proposed imaging methodology is based on the coherent interferometric (CINT) approach that exploits the local empirical cross correlations of the measurements of the wave… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.12174v1-abstract-full').style.display = 'inline'; document.getElementById('2105.12174v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2105.12174v1-abstract-full" style="display: none;"> This paper considers wave-based imaging through a heterogeneous (random) scattering medium. The goal is to estimate the support of the reflectivity function of a remote scene from measurements of the backscattered wave field. The proposed imaging methodology is based on the coherent interferometric (CINT) approach that exploits the local empirical cross correlations of the measurements of the wave field. The standard CINT images are known to be robust (statistically stable) with respect to the random medium, but the stability comes at the expense of a loss of resolution. This paper shows that a two-point CINT function contains the information needed to obtain statistically stable and high-resolution images. Different methods to build such images are presented, theoretically analyzed and compared with the standard imaging approaches using numerical simulations. The first method involves a phase-retrieval step to extract the reflectivity function from the modulus of its Fourier transform. The second method involves the evaluation of the leading eigenvector of the two-point CINT imaging function seen as the kernel of a linear operator. The third method uses an optimization step to extract the reflectivity function from some cross products of its Fourier transform. The presentation is for the synthetic aperture radar data acquisition setup, where a moving sensor probes the scene with signals emitted periodically and records the resulting backscattered wave. The generalization to other imaging setups, with passive or active arrays of sensors, is discussed briefly. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.12174v1-abstract-full').style.display = 'none'; document.getElementById('2105.12174v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 May, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">31 pages, 5 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35Q93; 45Q05; 49R99; 60H30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2011.05192">arXiv:2011.05192</a> <span> [<a href="https://arxiv.org/pdf/2011.05192">pdf</a>, <a href="https://arxiv.org/format/2011.05192">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Ancestral lineages in mutation-selection equilibria with moving optimum </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Forien%2C+R">Rapha毛l Forien</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Jimmy Garnier</a>, <a href="/search/math?searchtype=author&query=Patout%2C+F">Florian Patout</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2011.05192v2-abstract-short" style="display: inline;"> We investigate the evolutionary dynamics of a population structured in phenotype, subjected to trait dependent selection with a linearly moving optimum and an asexual mode of reproduction. Our model consists of a non-local and non-linear parabolic PDE. Our main goal is to measure the history of traits when the population stays around an equilibrium. We define an ancestral process based on the idea… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.05192v2-abstract-full').style.display = 'inline'; document.getElementById('2011.05192v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2011.05192v2-abstract-full" style="display: none;"> We investigate the evolutionary dynamics of a population structured in phenotype, subjected to trait dependent selection with a linearly moving optimum and an asexual mode of reproduction. Our model consists of a non-local and non-linear parabolic PDE. Our main goal is to measure the history of traits when the population stays around an equilibrium. We define an ancestral process based on the idea of neutral fractions. It allows us to derive quantitative information upon the evolution of diversity in the population along time. First, we study the long-time asymptotics of the ancestral process. We show that the very few fittest individuals drive adaptation. We then tackle the adaptive dynamics regime, where the effect of mutations is asymptotically small. In this limit, we provide an interpretation for the minimizer of some related optimization problem, an Hamilton Jacobi equation, as the typical ancestral lineage. We check the theoretical results against individual based simulations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.05192v2-abstract-full').style.display = 'none'; document.getElementById('2011.05192v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 December, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 10 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2020. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2009.06618">arXiv:2009.06618</a> <span> [<a href="https://arxiv.org/pdf/2009.06618">pdf</a>, <a href="https://arxiv.org/format/2009.06618">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Signal Processing">eess.SP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Passive communication with ambient noise </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2009.06618v2-abstract-short" style="display: inline;"> Motivated by applications to wireless communications, this paper addresses the propagation of waves transmitted by ambient noise sources and interacting with metamaterials. We discuss a generalized Helmholtz-Kirchhoff identity that is valid in dispersive media and we characterize the statistical properties of the empirical cross spectral density of the wave field. We can then introduce and analyze… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2009.06618v2-abstract-full').style.display = 'inline'; document.getElementById('2009.06618v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2009.06618v2-abstract-full" style="display: none;"> Motivated by applications to wireless communications, this paper addresses the propagation of waves transmitted by ambient noise sources and interacting with metamaterials. We discuss a generalized Helmholtz-Kirchhoff identity that is valid in dispersive media and we characterize the statistical properties of the empirical cross spectral density of the wave field. We can then introduce and analyze an original communication scheme between two passive arrays that uses only ambient noise illumination. The passive transmitter array does not transmit anything but it is a tunable metamaterial surface that can modulate its dispersive properties and encode a message in the modulation. The passive receiver array made of two receivers that are half-a-wavelength apart from each other can decode the message from the empirical cross spectral density of the wave field. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2009.06618v2-abstract-full').style.display = 'none'; document.getElementById('2009.06618v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 February, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 September, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 78A48; 35R60; 35Q61; 78A55 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2008.02022">arXiv:2008.02022</a> <span> [<a href="https://arxiv.org/pdf/2008.02022">pdf</a>, <a href="https://arxiv.org/format/2008.02022">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/1361-6420/abae10">10.1088/1361-6420/abae10 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Low-frequency source imaging in an acoustic waveguide </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2008.02022v1-abstract-short" style="display: inline;"> Time-harmonic far-field source array imaging in a two-dimensional waveguide is analyzed. A low-frequency situation is considered in which the diameter of the waveguide is slightly larger than the wavelength, so that the waveguide supports a limited number of guided modes, and the diameter of the antenna array is smaller than the wavelength, so that the standard resolution formulas in open media pr… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.02022v1-abstract-full').style.display = 'inline'; document.getElementById('2008.02022v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2008.02022v1-abstract-full" style="display: none;"> Time-harmonic far-field source array imaging in a two-dimensional waveguide is analyzed. A low-frequency situation is considered in which the diameter of the waveguide is slightly larger than the wavelength, so that the waveguide supports a limited number of guided modes, and the diameter of the antenna array is smaller than the wavelength, so that the standard resolution formulas in open media predict very poor imaging resolution. A general framework to analyze the resolution and stability performances of such antenna arrays is introduced. It is shown that planar antenna arrays perform better (in terms of resolution and stability with respect to measurement noise) than linear (horizontal or vertical) arrays and that vertical linear arrays perform better than horizontal arrays, for a given diameter. However a fundamental limitation to imaging in waveguides is identified that is due to the form of the dispersion relation. It is intrinsic to scalar waves, whatever the complexity of the medium and the array geometry. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.02022v1-abstract-full').style.display = 'none'; document.getElementById('2008.02022v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35R30; 35L05; 65M32 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1906.06735">arXiv:1906.06735</a> <span> [<a href="https://arxiv.org/pdf/1906.06735">pdf</a>, <a href="https://arxiv.org/ps/1906.06735">ps</a>, <a href="https://arxiv.org/format/1906.06735">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Intensity fluctuations in random waveguides </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1906.06735v2-abstract-short" style="display: inline;"> An asymptotic analysis of wave propagation in randomly perturbed waveguides is carried out in order to identify the effective Markovian dynamics of the guided mode powers. The main result consists in a quantification of the fluctuations of the mode powers and wave intensities that increase exponentially with the propagation distance. The exponential growth rate is studied in detail so as to determ… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.06735v2-abstract-full').style.display = 'inline'; document.getElementById('1906.06735v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1906.06735v2-abstract-full" style="display: none;"> An asymptotic analysis of wave propagation in randomly perturbed waveguides is carried out in order to identify the effective Markovian dynamics of the guided mode powers. The main result consists in a quantification of the fluctuations of the mode powers and wave intensities that increase exponentially with the propagation distance. The exponential growth rate is studied in detail so as to determine its dependence with respect to the waveguide geometry, the statistics of the random perturbations, and the operating wavelength. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.06735v2-abstract-full').style.display = 'none'; document.getElementById('1906.06735v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 January, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 June, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35R60; 35L05; 60F05; 35Q60; 35Q99 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1906.01225">arXiv:1906.01225</a> <span> [<a href="https://arxiv.org/pdf/1906.01225">pdf</a>, <a href="https://arxiv.org/ps/1906.01225">ps</a>, <a href="https://arxiv.org/format/1906.01225">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> A control variate method driven by diffusion approximation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Mertz%2C+L">Laurent Mertz</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1906.01225v2-abstract-short" style="display: inline;"> In this paper we examine a control variate estimator for a quantity that can be expressed as the expectation of a functional of a random process, that is itself the solution of a differential equation driven by fast mean-reverting ergodic forces. The control variate is the expectation of the same functional for the limit diffusion process that approximates the original process when the mean-revers… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.01225v2-abstract-full').style.display = 'inline'; document.getElementById('1906.01225v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1906.01225v2-abstract-full" style="display: none;"> In this paper we examine a control variate estimator for a quantity that can be expressed as the expectation of a functional of a random process, that is itself the solution of a differential equation driven by fast mean-reverting ergodic forces. The control variate is the expectation of the same functional for the limit diffusion process that approximates the original process when the mean-reversion time goes to zero. To get an efficient control variate estimator, we propose a coupling method to build the original process and the limit diffusion process. We show that the correlation between the two processes indeed goes to one when the mean reversion time goes to zero and we quantify the convergence rate, which makes it possible to characterize the variance reduction of the proposed control variate method. The efficiency of the method is illustrated on a few examples. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.01225v2-abstract-full').style.display = 'none'; document.getElementById('1906.01225v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 June, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2019. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1905.09044">arXiv:1905.09044</a> <span> [<a href="https://arxiv.org/pdf/1905.09044">pdf</a>, <a href="https://arxiv.org/format/1905.09044">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computation">stat.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1063/1.5081446">10.1063/1.5081446 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Application of the interacting particle system method to piecewise deterministic Markov processes used in reliability </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chraibi%2C+H">H. Chraibi</a>, <a href="/search/math?searchtype=author&query=Dutfoy%2C+A">A. Dutfoy</a>, <a href="/search/math?searchtype=author&query=Galtier%2C+T">T. Galtier</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">J. Garnier</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1905.09044v1-abstract-short" style="display: inline;"> Variance reduction methods are often needed for the reliability assessment of complex industrial systems, we focus on one variance reduction method in a given context, that is the interacting particle system method (IPS) used on piecewise deterministic Markov processes (PDMP) for reliability assessment . The PDMPs are a very large class of processes which benefit from high modeling capacities, the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1905.09044v1-abstract-full').style.display = 'inline'; document.getElementById('1905.09044v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1905.09044v1-abstract-full" style="display: none;"> Variance reduction methods are often needed for the reliability assessment of complex industrial systems, we focus on one variance reduction method in a given context, that is the interacting particle system method (IPS) used on piecewise deterministic Markov processes (PDMP) for reliability assessment . The PDMPs are a very large class of processes which benefit from high modeling capacities, they can model almost any Markovian phenomenon that does not include diffusion. In reliability assessment, the PDMPs modeling industrial systems generally involve low jump rates and jump kernels favoring one safe arrival, we call such model a "concentrated PDMP". Used on such concentrated PDMPs, the IPS is inefficient and does not always provide a variance reduction. Indeed, the efficiency of the IPS method relies on simulating many different trajectories during its propagation steps, but unfortunately concentrated PDMPs are likely to generate the same deterministic trajectories over and over. We propose an adaptation of the IPS method called IPS+M that reduces this phenomenon. The IPS+M consists in modifying the propagation steps of the IPS, by conditioning the propagation to avoid generating the same trajectories multiple times. We prove that, compared to the IPS, the IPS+M method always provides an estimator with a lower variance. We also carry out a quick simulation study on a two-components system that confirms this result. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1905.09044v1-abstract-full').style.display = 'none'; document.getElementById('1905.09044v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 May, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2019. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1901.06575">arXiv:1901.06575</a> <span> [<a href="https://arxiv.org/pdf/1901.06575">pdf</a>, <a href="https://arxiv.org/format/1901.06575">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> How a moving passive observer can perceive its environment ? The Unruh effect revisited </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Fink%2C+M">Mathias Fink</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1901.06575v2-abstract-short" style="display: inline;"> We consider a point-like observer that moves in a medium illuminated by noise sources with Lorentz-invariant spectrum. We show that the autocorrelation function of the signal recorded by the observer allows it to perceive its environment. More precisely, we consider an observer with constant acceleration (along a Rindler trajectory) and we exploit the recent work on the emergence of the Green's fu… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1901.06575v2-abstract-full').style.display = 'inline'; document.getElementById('1901.06575v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1901.06575v2-abstract-full" style="display: none;"> We consider a point-like observer that moves in a medium illuminated by noise sources with Lorentz-invariant spectrum. We show that the autocorrelation function of the signal recorded by the observer allows it to perceive its environment. More precisely, we consider an observer with constant acceleration (along a Rindler trajectory) and we exploit the recent work on the emergence of the Green's function from the cross correlation of signals transmitted by noise sources. First we recover the result that the signal recorded by the observer has a constant Wigner transform, i.e. a constant local spectrum, when the medium is homogeneous (this is the classical analogue of the Unruh effect). We complete that result by showing that the Rindler trajectory is the only straight-line trajectory that satisfies this property. We also show that, in the presence of an obstacle in the form of an infinite perfect mirror, the Wigner transform is perturbed when the observer comes into the neighborhood of the obstacle. The perturbation makes it possible for the observer to determine its position relative to the obstacle once the entire trajectory has been traversed. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1901.06575v2-abstract-full').style.display = 'none'; document.getElementById('1901.06575v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 November, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 January, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35R30; 35R60 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1812.10141">arXiv:1812.10141</a> <span> [<a href="https://arxiv.org/pdf/1812.10141">pdf</a>, <a href="https://arxiv.org/format/1812.10141">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geophysics">physics.geo-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1121/1.5116569">10.1121/1.5116569 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Acoustic and geoacoustic inverse problems in randomly perturbed shallow-water environments </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Dumaz%2C+L">Laure Dumaz</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Lepoultier%2C+G">Guilhem Lepoultier</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1812.10141v1-abstract-short" style="display: inline;"> The main goal of this paper is to estimate the regional acoustic and geoacoustic shallow-water environment from data collected by a vertical hydrophone array and transmitted by distant time-harmonic point sources. We aim at estimating the statistical properties of the random fluctuations of the index of refraction in the water column and the characteristics of the sea bottom. We first explain from… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1812.10141v1-abstract-full').style.display = 'inline'; document.getElementById('1812.10141v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1812.10141v1-abstract-full" style="display: none;"> The main goal of this paper is to estimate the regional acoustic and geoacoustic shallow-water environment from data collected by a vertical hydrophone array and transmitted by distant time-harmonic point sources. We aim at estimating the statistical properties of the random fluctuations of the index of refraction in the water column and the characteristics of the sea bottom. We first explain from first principles how acoustic wave propagation can be expressed as Markovian dynamics for the complex mode amplitudes of the sound pressure, which makes it possible to express the cross moments of the sound pressure in terms of the parameters to be estimated. We then show how the estimation problem can be formulated as a nonlinear inverse problem using this formulation, that can be solved by minimization of a misfit function. We apply this method to experimental data collected by the ALMA system (Acoustic Laboratory for Marine Applications). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1812.10141v1-abstract-full').style.display = 'none'; document.getElementById('1812.10141v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 December, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35Q60; 35R30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1812.01131">arXiv:1812.01131</a> <span> [<a href="https://arxiv.org/pdf/1812.01131">pdf</a>, <a href="https://arxiv.org/format/1812.01131">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Wave propagation in randomly perturbed weakly coupled waveguides </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Borcea%2C+L">Liliana Borcea</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1812.01131v1-abstract-short" style="display: inline;"> We present an analysis of wave propagation in a two step-index, parallel waveguide system. The goal is to quantify the effect of scattering at randomly perturbed interfaces between the guiding layers of high index of refraction and the host medium. The analysis is based on the expansion of the solution of the wave equation in a complete set of guided, radiation and evanescent modes with amplitudes… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1812.01131v1-abstract-full').style.display = 'inline'; document.getElementById('1812.01131v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1812.01131v1-abstract-full" style="display: none;"> We present an analysis of wave propagation in a two step-index, parallel waveguide system. The goal is to quantify the effect of scattering at randomly perturbed interfaces between the guiding layers of high index of refraction and the host medium. The analysis is based on the expansion of the solution of the wave equation in a complete set of guided, radiation and evanescent modes with amplitudes that are random fields, due to scattering. We obtain a detailed characterization of these amplitudes and thus quantify the transfer of power between the two waveguides in terms of their separation distance. The results show that, no matter how small the fluctuations of the interfaces are, they have significant effect at sufficiently large distance of propagation, which manifests in two ways: The first effect is well known and consists of power leakage from the guided modes to the radiation ones. The second effect consists of blurring of the periodic transfer of power between the waveguides and the eventual equipartition of power. Its quantification is the main practical result of the paper. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1812.01131v1-abstract-full').style.display = 'none'; document.getElementById('1812.01131v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 December, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">34 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35Q60; 35R60 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1811.10450">arXiv:1811.10450</a> <span> [<a href="https://arxiv.org/pdf/1811.10450">pdf</a>, <a href="https://arxiv.org/ps/1811.10450">ps</a>, <a href="https://arxiv.org/format/1811.10450">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Statistics Theory">math.ST</span> </div> </div> <p class="title is-5 mathjax"> Optimal input potential functions in the interacting particle system method </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Chraibi%2C+H">H. Chraibi</a>, <a href="/search/math?searchtype=author&query=Dutfoy%2C+A">A. Dutfoy</a>, <a href="/search/math?searchtype=author&query=Galtier%2C+T">T. Galtier</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">J. Garnier</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1811.10450v2-abstract-short" style="display: inline;"> The assessment of the probability of a rare event with a naive Monte-Carlo method is computationally intensive, so faster estimation or variance reduction methods are needed. We focus on one of these methods which is the interacting particle system (IPS) method. The method is not intrusive in the sense that the random Markov system under consideration is simulated with its original distribution, b… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.10450v2-abstract-full').style.display = 'inline'; document.getElementById('1811.10450v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1811.10450v2-abstract-full" style="display: none;"> The assessment of the probability of a rare event with a naive Monte-Carlo method is computationally intensive, so faster estimation or variance reduction methods are needed. We focus on one of these methods which is the interacting particle system (IPS) method. The method is not intrusive in the sense that the random Markov system under consideration is simulated with its original distribution, but selection steps are introduced that favor trajectories (particles) with high potential values. An unbiased estimator with reduced variance can then be proposed. The method requires to specify a set of potential functions. The choice of these functions is crucial, because it determines the magnitude of the variance reduction. So far, little information was available on how to choose the potential functions. This paper provides the expressions of the optimal potential functions minimizing the asymptotic variance of the estimator of the IPS method and it proposes recommendations for the practical design of the potential functions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.10450v2-abstract-full').style.display = 'none'; document.getElementById('1811.10450v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 26 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 2010 : 65C05; 65C35; 60K35 65C40; 60J22 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1811.01779">arXiv:1811.01779</a> <span> [<a href="https://arxiv.org/pdf/1811.01779">pdf</a>, <a href="https://arxiv.org/ps/1811.01779">ps</a>, <a href="https://arxiv.org/format/1811.01779">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Asymptotic analysis of a quantitative genetics model withnonlinear integral operator </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Calvez%2C+V">Vincent Calvez</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Jimmy Garnier</a>, <a href="/search/math?searchtype=author&query=Patout%2C+F">Florian Patout</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1811.01779v3-abstract-short" style="display: inline;"> We study the asymptotic behavior of stationary solutions to a quantitative genetics model with trait-dependent mortality and sexual reproduction. The infinitesimal model accounts for the mixing of parental phenotypes at birth.Our asymptotic analysis encompasses the case when deviations between the offspring and the mean parental trait are typically small. Under suitable regularity and growth condi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.01779v3-abstract-full').style.display = 'inline'; document.getElementById('1811.01779v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1811.01779v3-abstract-full" style="display: none;"> We study the asymptotic behavior of stationary solutions to a quantitative genetics model with trait-dependent mortality and sexual reproduction. The infinitesimal model accounts for the mixing of parental phenotypes at birth.Our asymptotic analysis encompasses the case when deviations between the offspring and the mean parental trait are typically small. Under suitable regularity and growth conditions on the mortality rate, we prove existence and local uniqueness of a stationary profile that get concentrated around a local optimum of mortality, with a Gaussian shape having small variance. Our approach is based on perturbative analysis techniques that require to describe accurately the correction to the Gaussian leading order profile. Our result extends previous results obtained with an asexual mode of reproduction, but using an alternative methodology. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.01779v3-abstract-full').style.display = 'none'; document.getElementById('1811.01779v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 August, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 5 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal de l'{脡}cole polytechnique - Math{茅}matiques, {脡}cole polytechnique, 2019, 6, pp.537-579. \&\#x27E8;10.5802/jep.100\&\#x27E9 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1810.08337">arXiv:1810.08337</a> <span> [<a href="https://arxiv.org/pdf/1810.08337">pdf</a>, <a href="https://arxiv.org/ps/1810.08337">ps</a>, <a href="https://arxiv.org/format/1810.08337">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Pricing of Securities">q-fin.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Optimal hedging under fast-varying stochastic volatility </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Solna%2C+K">Knut Solna</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1810.08337v3-abstract-short" style="display: inline;"> In a market with a rough or Markovian mean-reverting stochastic volatility there is no perfect hedge. Here it is shown how various delta-type hedging strategies perform and can be evaluated in such markets in the case of European options. A precise characterization of the hedging cost, the replication cost caused by the volatility fluctuations, is presented in an asymptotic regime of rapid mean re… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.08337v3-abstract-full').style.display = 'inline'; document.getElementById('1810.08337v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1810.08337v3-abstract-full" style="display: none;"> In a market with a rough or Markovian mean-reverting stochastic volatility there is no perfect hedge. Here it is shown how various delta-type hedging strategies perform and can be evaluated in such markets in the case of European options. A precise characterization of the hedging cost, the replication cost caused by the volatility fluctuations, is presented in an asymptotic regime of rapid mean reversion for the volatility fluctuations. The optimal dynamic asset based hedging strategy in the considered regime is identified as the so-called `practitioners' delta hedging scheme. It is moreover shown that the performances of the delta-type hedging schemes are essentially independent of the regularity of the volatility paths in the considered regime and that the hedging costs are related to a vega risk martingale whose magnitude is proportional to a new market risk parameter. It is also shown via numerical simulations that the proposed hedging schemes which derive from option price approximations in the regime of rapid mean reversion, are robust: the `practitioners' delta hedging scheme that is identified as being optimal by our asymptotic analysis when the mean reversion time is small seems to be optimal with arbitrary mean reversion times. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.08337v3-abstract-full').style.display = 'none'; document.getElementById('1810.08337v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 March, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 October, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 91G80; 60H10; 60G22; 60K37 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1810.01240">arXiv:1810.01240</a> <span> [<a href="https://arxiv.org/pdf/1810.01240">pdf</a>, <a href="https://arxiv.org/format/1810.01240">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Classical Analysis and ODEs">math.CA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> </div> <p class="title is-5 mathjax"> Efficient Seismic fragility curve estimation by Active Learning on Support Vector Machines </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Sainct%2C+R">R茅mi Sainct</a>, <a href="/search/math?searchtype=author&query=Feau%2C+C">Cyril Feau</a>, <a href="/search/math?searchtype=author&query=Martinez%2C+J">Jean-Marc Martinez</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1810.01240v1-abstract-short" style="display: inline;"> Fragility curves which express the failure probability of a structure, or critical components, as function of a loading intensity measure are nowadays widely used (i) in Seismic Probabilistic Risk Assessment studies, (ii) to evaluate impact of construction details on the structural performance of installations under seismic excitations or under other loading sources such as wind. To avoid the use… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.01240v1-abstract-full').style.display = 'inline'; document.getElementById('1810.01240v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1810.01240v1-abstract-full" style="display: none;"> Fragility curves which express the failure probability of a structure, or critical components, as function of a loading intensity measure are nowadays widely used (i) in Seismic Probabilistic Risk Assessment studies, (ii) to evaluate impact of construction details on the structural performance of installations under seismic excitations or under other loading sources such as wind. To avoid the use of parametric models such as lognormal model to estimate fragility curves from a reduced number of numerical calculations, a methodology based on Support Vector Machines coupled with an active learning algorithm is proposed in this paper. In practice, input excitation is reduced to some relevant parameters and, given these parameters, SVMs are used for a binary classification of the structural responses relative to a limit threshold of exceedance. Since the output is not only binary, this is a score, a probabilistic interpretation of the output is exploited to estimate very efficiently fragility curves as score functions or as functions of classical seismic intensity measures. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.01240v1-abstract-full').style.display = 'none'; document.getElementById('1810.01240v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 September, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">24 pages, 14 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1806.06217">arXiv:1806.06217</a> <span> [<a href="https://arxiv.org/pdf/1806.06217">pdf</a>, <a href="https://arxiv.org/format/1806.06217">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Wave propagation and imaging in moving random media </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Borcea%2C+L">Liliana Borcea</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Solna%2C+K">Knut Solna</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1806.06217v1-abstract-short" style="display: inline;"> We present a study of sound wave propagation in a time dependent random medium and an application to imaging. The medium is modeled by small temporal and spatial random fluctuations in the wave speed and density, and it moves due to an ambient flow. We develop a transport theory for the energy density of the waves, in a forward scattering regime, within a cone (beam) of propagation with small open… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1806.06217v1-abstract-full').style.display = 'inline'; document.getElementById('1806.06217v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1806.06217v1-abstract-full" style="display: none;"> We present a study of sound wave propagation in a time dependent random medium and an application to imaging. The medium is modeled by small temporal and spatial random fluctuations in the wave speed and density, and it moves due to an ambient flow. We develop a transport theory for the energy density of the waves, in a forward scattering regime, within a cone (beam) of propagation with small opening angle. We apply the transport theory to the inverse problem of estimating a stationary wave source from measurements at a remote array of receivers. The estimation requires knowledge of the mean velocity of the ambient flow and the second-order statistics of the random medium. If these are not known, we show how they may be estimated from additional measurements gathered at the array, using a few known sources. We also show how the transport theory can be used to estimate the mean velocity of the medium. If the array has large aperture and the scattering in the random medium is strong, this estimate does not depend on the knowledge of the statistics of the random medium. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1806.06217v1-abstract-full').style.display = 'none'; document.getElementById('1806.06217v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 June, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">36 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1804.00552">arXiv:1804.00552</a> <span> [<a href="https://arxiv.org/pdf/1804.00552">pdf</a>, <a href="https://arxiv.org/ps/1804.00552">ps</a>, <a href="https://arxiv.org/format/1804.00552">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/1361-6420/aacfb0">10.1088/1361-6420/aacfb0 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Imaging through a scattering medium by speckle intensity correlations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Solna%2C+K">Knut Solna</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1804.00552v2-abstract-short" style="display: inline;"> In this paper we analyze an imaging technique based on intensity speckle correlations over incident field position proposed in [J. A. Newmann and K. J. Webb, Phys. Rev. Lett. 113, 263903 (2014)]. Its purpose is to reconstruct a field incident on a strongly scattering random medium. The thickness of the complex medium is much larger than the scattering mean free path so that the wave emerging from… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1804.00552v2-abstract-full').style.display = 'inline'; document.getElementById('1804.00552v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1804.00552v2-abstract-full" style="display: none;"> In this paper we analyze an imaging technique based on intensity speckle correlations over incident field position proposed in [J. A. Newmann and K. J. Webb, Phys. Rev. Lett. 113, 263903 (2014)]. Its purpose is to reconstruct a field incident on a strongly scattering random medium. The thickness of the complex medium is much larger than the scattering mean free path so that the wave emerging from the random section forms an incoherent speckle pattern. Our analysis clarifies the conditions under which the method can give a good reconstruction and characterizes its performance. The analysis is carried out in the white-noise paraxial regime, which is relevant for the applications in optics that motivated the original paper. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1804.00552v2-abstract-full').style.display = 'none'; document.getElementById('1804.00552v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 June, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 April, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">23 pages, 1 figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35R60; 78A48 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1804.00549">arXiv:1804.00549</a> <span> [<a href="https://arxiv.org/pdf/1804.00549">pdf</a>, <a href="https://arxiv.org/format/1804.00549">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/1361-6420/aaea37">10.1088/1361-6420/aaea37 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A ghost imaging modality in a random waveguide </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Borcea%2C+L">Liliana Borcea</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1804.00549v2-abstract-short" style="display: inline;"> We study the imaging of a penetrable scatterer, aka target, in a waveguide with randomly perturbed boundary. The target is located between a partially coherent source which transmits the wave, and a detector which measures the spatially integrated energy flux of the wave. The imaging is impeded by random boundary scattering effects that accumulate as the wave propagates. We consider a very large d… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1804.00549v2-abstract-full').style.display = 'inline'; document.getElementById('1804.00549v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1804.00549v2-abstract-full" style="display: none;"> We study the imaging of a penetrable scatterer, aka target, in a waveguide with randomly perturbed boundary. The target is located between a partially coherent source which transmits the wave, and a detector which measures the spatially integrated energy flux of the wave. The imaging is impeded by random boundary scattering effects that accumulate as the wave propagates. We consider a very large distance (range) between the target and the detector, where that cumulative scattering is so strong that it distributes the energy evenly among the components (modes) of the wave. Conventional imaging is impossible in this equipartition regime. Nevertheless, we show that the target can be located with a ghost imaging modality. This forms an image using the cross-correlation of the measured energy flux, integrated over the aperture of the detector, with the time and space resolved energy flux in a reference waveguide, at the search range. We consider two reference waveguides: The waveguide with unperturbed boundary, in which we can calculate the energy flux, and the actual random waveguide, before the presence of the target, in which the energy flux should be measured. We analyze the ghost imaging modality from first principles and show that it can be efficient in a random waveguide geometry in which there is both strong modal dispersion and mode coupling induced by scattering, provided that the standard ghost imaging function is modified and integrated over a suitable time offset window in order to compensate for dispersion and diffusion. The analysis quantifies the resolution of the image in terms of the source and detector aperture, the range offset between the source and the target, and the duration of the measurements. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1804.00549v2-abstract-full').style.display = 'none'; document.getElementById('1804.00549v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 July, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 April, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">34 pages, 4 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35R60; 35R30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1802.07883">arXiv:1802.07883</a> <span> [<a href="https://arxiv.org/pdf/1802.07883">pdf</a>, <a href="https://arxiv.org/format/1802.07883">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optics">physics.optics</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Non-invasive imaging through random media </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Solna%2C+K">Knut Solna</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1802.07883v1-abstract-short" style="display: inline;"> When waves propagate through a strongly scattering medium the energy is transferred to the incoherent wave part by scattering. The wave intensity then forms a random speckle pattern seemingly without much useful information. However, a number of recent physical experiments show how one can extract useful information from this speckle pattern. Here we present the mathematical analysis that explains… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1802.07883v1-abstract-full').style.display = 'inline'; document.getElementById('1802.07883v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1802.07883v1-abstract-full" style="display: none;"> When waves propagate through a strongly scattering medium the energy is transferred to the incoherent wave part by scattering. The wave intensity then forms a random speckle pattern seemingly without much useful information. However, a number of recent physical experiments show how one can extract useful information from this speckle pattern. Here we present the mathematical analysis that explains the quite stunning performance of such a scheme for speckle imaging. Our analysis identifies a scaling regime where the scheme works well. This regime is the white-noise paraxial regime, which leads to the Ito-Schrodinger model for the wave amplitude. The results presented in this paper conform with the sophisticated physical intuition that has motivated these schemes, but give a more detailed characterization of the performance. The analysis gives a description of (i) the information that can be extracted and with what resolution (ii) the statistical stability or signal-to-noise ratio with which the information can be extracted. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1802.07883v1-abstract-full').style.display = 'none'; document.getElementById('1802.07883v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 February, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2018. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1707.08452">arXiv:1707.08452</a> <span> [<a href="https://arxiv.org/pdf/1707.08452">pdf</a>, <a href="https://arxiv.org/format/1707.08452">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Analysis, Statistics and Probability">physics.data-an</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optics">physics.optics</span> </div> </div> <p class="title is-5 mathjax"> Laser beam imaging from the speckle pattern of the off-axis scattered intensity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Borcea%2C+L">Liliana Borcea</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1707.08452v1-abstract-short" style="display: inline;"> We study the inverse problem of localization (imaging) of a laser beam from measurements of the intensity of light scattered off-axis by a Poisson cloud of small particles. Starting from the wave equation, we analyze the microscopic coherence of the scattered intensity and show that it is possible to determine the laser beam from the speckle pattern captured by a group of CCD cameras. Two groups o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1707.08452v1-abstract-full').style.display = 'inline'; document.getElementById('1707.08452v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1707.08452v1-abstract-full" style="display: none;"> We study the inverse problem of localization (imaging) of a laser beam from measurements of the intensity of light scattered off-axis by a Poisson cloud of small particles. Starting from the wave equation, we analyze the microscopic coherence of the scattered intensity and show that it is possible to determine the laser beam from the speckle pattern captured by a group of CCD cameras. Two groups of cameras are sufficient when the particles are either small or large with respect to the wavelength. For general particle sizes the accuracy of the laser localization with two groups of cameras is subject to knowing the scattering properties of the cloud. However, three or more groups of cameras allow accurate localization that is robust to uncertainty of the type, size, shape and concentration of the particles in the cloud. We introduce a novel laser beam localization algorithm and give some numerical illustrations in a regime relevant to the application of imaging high energy lasers in a maritime atmosphere. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1707.08452v1-abstract-full').style.display = 'none'; document.getElementById('1707.08452v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 July, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">27 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1707.00610">arXiv:1707.00610</a> <span> [<a href="https://arxiv.org/pdf/1707.00610">pdf</a>, <a href="https://arxiv.org/ps/1707.00610">ps</a>, <a href="https://arxiv.org/format/1707.00610">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Pricing of Securities">q-fin.PR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> Option Pricing under Fast-varying and Rough Stochastic Volatility </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Solna%2C+K">Knut Solna</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1707.00610v2-abstract-short" style="display: inline;"> Recent empirical studies suggest that the volatilities associated with financial time series exhibit short-range correlations. This entails that the volatility process is very rough and its autocorrelation exhibits sharp decay at the origin. Another classic stylistic feature often assumed for the volatility is that it is mean reverting. In this paper it is shown that the price impact of a rapidly… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1707.00610v2-abstract-full').style.display = 'inline'; document.getElementById('1707.00610v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1707.00610v2-abstract-full" style="display: none;"> Recent empirical studies suggest that the volatilities associated with financial time series exhibit short-range correlations. This entails that the volatility process is very rough and its autocorrelation exhibits sharp decay at the origin. Another classic stylistic feature often assumed for the volatility is that it is mean reverting. In this paper it is shown that the price impact of a rapidly mean reverting rough volatility model coincides with that associated with fast mean reverting Markov stochastic volatility models. This reconciles the empirical observation of rough volatility paths with the good fit of the implied volatility surface to models of fast mean reverting Markov volatilities. Moreover, the result conforms with recent numerical results regarding rough stochastic volatility models. It extends the scope of models for which the asymptotic results of fast mean reverting Markov volatilities are valid. The paper concludes with a general discussion of fractional volatility asymptotics and their interrelation. The regimes discussed there include fast and slow volatility factors with strong or small volatility fluctuations and with the limits not commuting in general. The notion of a characteristic term structure exponent is introduced, this exponent governs the implied volatility term structure in the various asymptotic regimes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1707.00610v2-abstract-full').style.display = 'none'; document.getElementById('1707.00610v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 April, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 June, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">arXiv admin note: text overlap with arXiv:1604.00105</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 91G80; 60H10; 60G22; 60K37 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1705.10997">arXiv:1705.10997</a> <span> [<a href="https://arxiv.org/pdf/1705.10997">pdf</a>, <a href="https://arxiv.org/format/1705.10997">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Thin front limit of an integro--differential Fisher--KPP equation with fat--tailed kernels </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bouin%2C+E">Emeric Bouin</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Jimmy Garnier</a>, <a href="/search/math?searchtype=author&query=Henderson%2C+C">Christopher Henderson</a>, <a href="/search/math?searchtype=author&query=Patout%2C+F">Florian Patout</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1705.10997v2-abstract-short" style="display: inline;"> We study the asymptotic behavior of solutions to a monostable integro-differential Fisher-KPP equation , that is where the standard Laplacian is replaced by a convolution term, when the dispersal kernel is fat-tailed. We focus on two different regimes. Firstly, we study the long time/long range scaling limit by introducing a relevant rescaling in space and time and prove a sharp bound on the (supe… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1705.10997v2-abstract-full').style.display = 'inline'; document.getElementById('1705.10997v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1705.10997v2-abstract-full" style="display: none;"> We study the asymptotic behavior of solutions to a monostable integro-differential Fisher-KPP equation , that is where the standard Laplacian is replaced by a convolution term, when the dispersal kernel is fat-tailed. We focus on two different regimes. Firstly, we study the long time/long range scaling limit by introducing a relevant rescaling in space and time and prove a sharp bound on the (super-linear) spreading rate in the Hamilton-Jacobi sense by means of sub-and super-solutions. Secondly, we investigate a long time/small mutation regime for which, after identifying a relevant rescaling for the size of mutations, we derive a Hamilton-Jacobi limit. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1705.10997v2-abstract-full').style.display = 'none'; document.getElementById('1705.10997v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 April, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 31 May, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2017. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1703.09078">arXiv:1703.09078</a> <span> [<a href="https://arxiv.org/pdf/1703.09078">pdf</a>, <a href="https://arxiv.org/format/1703.09078">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Populations and Evolution">q-bio.PE</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> </div> <p class="title is-5 mathjax"> Existence of recombination-selection equilibria for sexual populations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Bourgeron%2C+T">Thibault Bourgeron</a>, <a href="/search/math?searchtype=author&query=Calvez%2C+V">Vincent Calvez</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Jimmy Garnier</a>, <a href="/search/math?searchtype=author&query=Lepoutre%2C+T">Thomas Lepoutre</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1703.09078v1-abstract-short" style="display: inline;"> We study a birth and death model for the adapatation of a sexual population to an environment. The population is structured by a phenotypical trait, and, possibly, an age variable. Recombination is modeled by Fisher's infinitesimal operator. We prove the existence of principal eigenelements for the corresponding eigenproblem. As the infinitesimal operator is 1-homogeneous but nor linear nor monoto… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1703.09078v1-abstract-full').style.display = 'inline'; document.getElementById('1703.09078v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1703.09078v1-abstract-full" style="display: none;"> We study a birth and death model for the adapatation of a sexual population to an environment. The population is structured by a phenotypical trait, and, possibly, an age variable. Recombination is modeled by Fisher's infinitesimal operator. We prove the existence of principal eigenelements for the corresponding eigenproblem. As the infinitesimal operator is 1-homogeneous but nor linear nor monotone, the general Krein-Rutman theory cannot be applied to this problem. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1703.09078v1-abstract-full').style.display = 'none'; document.getElementById('1703.09078v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 March, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2017. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1702.03921">arXiv:1702.03921</a> <span> [<a href="https://arxiv.org/pdf/1702.03921">pdf</a>, <a href="https://arxiv.org/format/1702.03921">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Transport of power in random waveguides with turning points </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Borcea%2C+L">Liliana Borcea</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Wood%2C+D">Derek Wood</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1702.03921v1-abstract-short" style="display: inline;"> We present a mathematical theory of time-harmonic wave propagation and reflection in a two-dimensional random acoustic waveguide with sound soft boundary and turning points. The boundary has small fluctuations on the scale of the wavelength, modeled as random. The waveguide supports multiple propagating modes. The number of these modes changes due to slow variations of the waveguide cross-section.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.03921v1-abstract-full').style.display = 'inline'; document.getElementById('1702.03921v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1702.03921v1-abstract-full" style="display: none;"> We present a mathematical theory of time-harmonic wave propagation and reflection in a two-dimensional random acoustic waveguide with sound soft boundary and turning points. The boundary has small fluctuations on the scale of the wavelength, modeled as random. The waveguide supports multiple propagating modes. The number of these modes changes due to slow variations of the waveguide cross-section. The changes occur at turning points, where waves transition from propagating to evanescent or the other way around. We consider a regime where scattering at the random boundary has significant effect on the wave traveling from one turning point to another. This effect is described by the coupling of its components, the modes. We derive the mode coupling theory from first principles, and quantify the randomization of the wave and the transport and reflection of power in the waveguide. We show in particular that scattering at the random boundary may increase or decrease the net power transmitted through the waveguide depending on the source. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.03921v1-abstract-full').style.display = 'none'; document.getElementById('1702.03921v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 February, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">45 pages, 6 figure</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1702.01384">arXiv:1702.01384</a> <span> [<a href="https://arxiv.org/pdf/1702.01384">pdf</a>, <a href="https://arxiv.org/ps/1702.01384">ps</a>, <a href="https://arxiv.org/format/1702.01384">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/1361-6420/aa9ad7">10.1088/1361-6420/aa9ad7 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Global acoustic daylight imaging in a stratified Earth-like model </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=de+Hoop%2C+M+V">Maarten V. de Hoop</a>, <a href="/search/math?searchtype=author&query=Garnier%2C+J">Josselin Garnier</a>, <a href="/search/math?searchtype=author&query=Solna%2C+K">Knut Solna</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1702.01384v2-abstract-short" style="display: inline;"> We present an analysis of acoustic daylight imaging in an Earth-like model assuming a random distribution of noise sources spatially supported in an annulus located away from the surface. We assume a situation with scalar wave propagation and that the measurements are of the wave field at the surface. Then, we obtain a relation between the autocorrelation function of the measurements and the trace… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.01384v2-abstract-full').style.display = 'inline'; document.getElementById('1702.01384v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1702.01384v2-abstract-full" style="display: none;"> We present an analysis of acoustic daylight imaging in an Earth-like model assuming a random distribution of noise sources spatially supported in an annulus located away from the surface. We assume a situation with scalar wave propagation and that the measurements are of the wave field at the surface. Then, we obtain a relation between the autocorrelation function of the measurements and the trace of the scattered field generated by an impulsive source localized just below the surface. From this relation it is, for example, clear that the eigenfrequencies can be recovered from the autocorrelation. Moreover, the complete scattering operator can be extracted under the additional assumption that the annulus is close to the surface and has a thickness smaller than the typical wavelength. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.01384v2-abstract-full').style.display = 'none'; document.getElementById('1702.01384v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 October, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 5 February, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35R30; 35R60; 86A15 </p> </li> </ol> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> 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