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<button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <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/2412.04596">arXiv:2412.04596</a> <span> [<a href="https://arxiv.org/pdf/2412.04596">pdf</a>, <a href="https://arxiv.org/format/2412.04596">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="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Nonlinear Operator Learning Using Energy Minimization and MLPs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Larson%2C+M+G">Mats G. Larson</a>, <a href="/search/math?searchtype=author&query=Lundholm%2C+C">Carl Lundholm</a>, <a href="/search/math?searchtype=author&query=Persson%2C+A">Anna Persson</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="2412.04596v1-abstract-short" style="display: inline;"> We develop and evaluate a method for learning solution operators to nonlinear problems governed by partial differential equations. The approach is based on a finite element discretization and aims at representing the solution operator by an MLP that takes latent variables as input. The latent variables will typically correspond to parameters in a parametrization of input data such as boundary cond… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.04596v1-abstract-full').style.display = 'inline'; document.getElementById('2412.04596v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.04596v1-abstract-full" style="display: none;"> We develop and evaluate a method for learning solution operators to nonlinear problems governed by partial differential equations. The approach is based on a finite element discretization and aims at representing the solution operator by an MLP that takes latent variables as input. The latent variables will typically correspond to parameters in a parametrization of input data such as boundary conditions, coefficients, and right-hand sides. The loss function is most often an energy functional and we formulate efficient parallelizable training algorithms based on assembling the energy locally on each element. For large problems, the learning process can be made more efficient by using only a small fraction of randomly chosen elements in the mesh in each iteration. The approach is evaluated on several relevant test cases, where learning the solution operator turns out to be beneficial compared to classical numerical methods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.04596v1-abstract-full').style.display = 'none'; document.getElementById('2412.04596v1-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, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 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">13 pages, 3 figures (8 subfigures in total)</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 65K10 65N30 65Y20 68T07 <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> G.1.8; I.2.6; J.2 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.22384">arXiv:2410.22384</a> <span> [<a href="https://arxiv.org/pdf/2410.22384">pdf</a>, <a href="https://arxiv.org/format/2410.22384">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="Probability">math.PR</span> </div> </div> <p class="title is-5 mathjax"> On parameter estimation for $N(渭,蟽^2 I_3)$ based on projected data into $\mathbb{S}^2$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Figueras%2C+J">Jordi-Llu铆s Figueras</a>, <a href="/search/math?searchtype=author&query=Persson%2C+A">Aron Persson</a>, <a href="/search/math?searchtype=author&query=Viitasaari%2C+L">Lauri Viitasaari</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="2410.22384v1-abstract-short" style="display: inline;"> We consider the projected normal distribution, with isotropic variance, on the 2-sphere using intrinsic statistics. We show that in this case, the expectation commutes with the projection and that the covariance of the normal variable has a 1-1 correspondence with the intrinsic covariance of the projected normal distribution. This allows to estimate, after model identification, the parameters of t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.22384v1-abstract-full').style.display = 'inline'; document.getElementById('2410.22384v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.22384v1-abstract-full" style="display: none;"> We consider the projected normal distribution, with isotropic variance, on the 2-sphere using intrinsic statistics. We show that in this case, the expectation commutes with the projection and that the covariance of the normal variable has a 1-1 correspondence with the intrinsic covariance of the projected normal distribution. This allows to estimate, after model identification, the parameters of the underlying normal distribution that generates the data. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.22384v1-abstract-full').style.display = 'none'; document.getElementById('2410.22384v1-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> 29 October, 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">MSC Class:</span> 62H11; 62F10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2401.09875">arXiv:2401.09875</a> <span> [<a href="https://arxiv.org/pdf/2401.09875">pdf</a>, <a href="https://arxiv.org/ps/2401.09875">ps</a>, <a href="https://arxiv.org/format/2401.09875">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Symplectic Geometry">math.SG</span> </div> </div> <p class="title is-5 mathjax"> On manifold-like polyfolds as differential geometrical objects with applications in complex geometry </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=%C3%85hag%2C+P">Per 脜hag</a>, <a href="/search/math?searchtype=author&query=Czy%C5%BC%2C+R">Rafa艂 Czy偶</a>, <a href="/search/math?searchtype=author&query=Kalm%2C+H+S">H氓kan Samuelsson Kalm</a>, <a href="/search/math?searchtype=author&query=Persson%2C+A">Aron Persson</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="2401.09875v2-abstract-short" style="display: inline;"> We argue for more widespread use of manifold-like polyfolds (M-polyfolds) as differential geometric objects. M-polyfolds possess a distinct advantage over differentiable manifolds, enabling a smooth and local change of dimension. To establish their utility, we introduce tensors and prove the existence of Riemannian metrics, symplectic structures, and almost complex structures within the M-polyfold… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.09875v2-abstract-full').style.display = 'inline'; document.getElementById('2401.09875v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2401.09875v2-abstract-full" style="display: none;"> We argue for more widespread use of manifold-like polyfolds (M-polyfolds) as differential geometric objects. M-polyfolds possess a distinct advantage over differentiable manifolds, enabling a smooth and local change of dimension. To establish their utility, we introduce tensors and prove the existence of Riemannian metrics, symplectic structures, and almost complex structures within the M-polyfold framework. Drawing inspiration from a series of highly acclaimed articles by L谩szl贸 Lempert, we lay the foundation for advancing geometry and function theory in complex M-polyfolds. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2401.09875v2-abstract-full').style.display = 'none'; document.getElementById('2401.09875v2-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 March, 2025; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary 53C56; 32Q15; 46G20; 58B99; Secondary 32Q15; 32Q60; 53C15; 58C99 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2112.08485">arXiv:2112.08485</a> <span> [<a href="https://arxiv.org/pdf/2112.08485">pdf</a>, <a href="https://arxiv.org/format/2112.08485">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"> On optimal convergence rates for discrete minimizers of the Gross-Pitaevskii energy in LOD spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Henning%2C+P">Patrick Henning</a>, <a href="/search/math?searchtype=author&query=Persson%2C+A">Anna Persson</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="2112.08485v2-abstract-short" style="display: inline;"> In this paper we revisit a two-level discretization based on the Localized Orthogonal Decomposition (LOD). It was originally proposed in [P.Henning, A.M氓lqvist, D.Peterseim. SIAM J. Numer. Anal.52-4:1525-1550, 2014] to compute ground states of Bose-Einstein condensates by finding discrete minimizers of the Gross-Pitaevskii energy functional. The established convergence rates for the method appeare… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2112.08485v2-abstract-full').style.display = 'inline'; document.getElementById('2112.08485v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2112.08485v2-abstract-full" style="display: none;"> In this paper we revisit a two-level discretization based on the Localized Orthogonal Decomposition (LOD). It was originally proposed in [P.Henning, A.M氓lqvist, D.Peterseim. SIAM J. Numer. Anal.52-4:1525-1550, 2014] to compute ground states of Bose-Einstein condensates by finding discrete minimizers of the Gross-Pitaevskii energy functional. The established convergence rates for the method appeared however suboptimal compared to numerical observations and a proof of optimal rates in this setting remained open. In this paper we shall close this gap by proving optimal order error estimates for the $L^2$- and $H^1$-error between the exact ground state and discrete minimizers, as well as error estimates for the ground state energy and the ground state eigenvalue. In particular, the achieved convergence rates for the energy and the eigenvalue are of $6$th order with respect to the mesh size on which the discrete LOD space is based, without making any additional regularity assumptions. These high rates justify the use of very coarse meshes, which significantly reduces the computational effort for finding accurate approximations of ground states. In addition, we include numerical experiments that confirm the optimality of the new theoretical convergence rates, both for smooth and discontinuous potentials. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2112.08485v2-abstract-full').style.display = 'none'; document.getElementById('2112.08485v2-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 April, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 December, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2011.03311">arXiv:2011.03311</a> <span> [<a href="https://arxiv.org/pdf/2011.03311">pdf</a>, <a href="https://arxiv.org/ps/2011.03311">ps</a>, <a href="https://arxiv.org/format/2011.03311">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"> A generalized finite element method for the strongly damped wave equation with rapidly varying data </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ljung%2C+P">Per Ljung</a>, <a href="/search/math?searchtype=author&query=M%C3%A5lqvist%2C+A">Axel M氓lqvist</a>, <a href="/search/math?searchtype=author&query=Persson%2C+A">Anna Persson</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="2011.03311v1-abstract-short" style="display: inline;"> We propose a generalized finite element method for the strongly damped wave equation with highly varying coefficients. The proposed method is based on the localized orthogonal decomposition introduced and is designed to handle independent variations in both the damping and the wave propagation speed respectively. The method does so by automatically correcting for the damping in the transient phase… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.03311v1-abstract-full').style.display = 'inline'; document.getElementById('2011.03311v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2011.03311v1-abstract-full" style="display: none;"> We propose a generalized finite element method for the strongly damped wave equation with highly varying coefficients. The proposed method is based on the localized orthogonal decomposition introduced and is designed to handle independent variations in both the damping and the wave propagation speed respectively. The method does so by automatically correcting for the damping in the transient phase and for the propagation speed in the steady state phase. Convergence of optimal order is proven in $L_2(H^1)$-norm, independent of the derivatives of the coefficients. We present numerical examples that confirm the theoretical findings. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.03311v1-abstract-full').style.display = 'none'; document.getElementById('2011.03311v1-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 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/1910.07390">arXiv:1910.07390</a> <span> [<a href="https://arxiv.org/pdf/1910.07390">pdf</a>, <a href="https://arxiv.org/format/1910.07390">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"> Computational homogenization of time-harmonic Maxwell's equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Henning%2C+P">Patrick Henning</a>, <a href="/search/math?searchtype=author&query=Persson%2C+A">Anna Persson</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="1910.07390v2-abstract-short" style="display: inline;"> In this paper we consider a numerical homogenization technique for curl-curl-problems that is based on the framework of the Localized Orthogonal Decomposition and which was proposed in [D. Gallistl, P. Henning, B. Verf眉rth. SIAM J. Numer. Anal. 56-3:1570-1596, 2018] for problems with essential boundary conditions. The findings of the aforementioned work establish quantitative homogenization result… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1910.07390v2-abstract-full').style.display = 'inline'; document.getElementById('1910.07390v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1910.07390v2-abstract-full" style="display: none;"> In this paper we consider a numerical homogenization technique for curl-curl-problems that is based on the framework of the Localized Orthogonal Decomposition and which was proposed in [D. Gallistl, P. Henning, B. Verf眉rth. SIAM J. Numer. Anal. 56-3:1570-1596, 2018] for problems with essential boundary conditions. The findings of the aforementioned work establish quantitative homogenization results for the time-harmonic Maxwell's equations that hold beyond assumptions of periodicity, however, a practical realization of the approach was left open. In this paper, we transfer the findings from essential boundary conditions to natural boundary conditions and we demonstrate that the approach yields a computable numerical method. We also investigate how boundary values of the source term can effect the computational complexity and accuracy. Our findings will be supported by various numerical experiments, both in $2D$ and $3D$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1910.07390v2-abstract-full').style.display = 'none'; document.getElementById('1910.07390v2-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 March, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 October, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 35Q61; 65N30; 65N12; 78M10 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1801.10105">arXiv:1801.10105</a> <span> [<a href="https://arxiv.org/pdf/1801.10105">pdf</a>, <a href="https://arxiv.org/format/1801.10105">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"> Finite element convergence for the time-dependent Joule heating problem with mixed boundary conditions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Jensen%2C+M">Max Jensen</a>, <a href="/search/math?searchtype=author&query=M%C3%A5lqvist%2C+A">Axel M氓lqvist</a>, <a href="/search/math?searchtype=author&query=Persson%2C+A">Anna Persson</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="1801.10105v1-abstract-short" style="display: inline;"> We prove strong convergence for a large class of finite element methods for the time-dependent Joule heating problem in three spatial dimensions with mixed boundary conditions on Lipschitz domains. We consider conforming subspaces for the spatial discretization and the backward Euler scheme for the temporal discretization. Furthermore, we prove uniqueness and higher regularity of the solution on c… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1801.10105v1-abstract-full').style.display = 'inline'; document.getElementById('1801.10105v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1801.10105v1-abstract-full" style="display: none;"> We prove strong convergence for a large class of finite element methods for the time-dependent Joule heating problem in three spatial dimensions with mixed boundary conditions on Lipschitz domains. We consider conforming subspaces for the spatial discretization and the backward Euler scheme for the temporal discretization. Furthermore, we prove uniqueness and higher regularity of the solution on creased domains and additional regularity in the interior of the domain. Due to a variational formulation with a cut-off functional the convergence analysis does not require a discrete maximum principle, permitting approximation spaces suitable for adaptive mesh refinement, responding to the the difference in regularity within the domain. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1801.10105v1-abstract-full').style.display = 'none'; document.getElementById('1801.10105v1-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> 30 January, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2018. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1706.04380">arXiv:1706.04380</a> <span> [<a href="https://arxiv.org/pdf/1706.04380">pdf</a>, <a href="https://arxiv.org/format/1706.04380">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/10.1137/17M1134500">10.1137/17M1134500 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Multiscale differential Riccati equations for linear quadratic regulator problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=M%C3%A5lqvist%2C+A">Axel M氓lqvist</a>, <a href="/search/math?searchtype=author&query=Persson%2C+A">Anna Persson</a>, <a href="/search/math?searchtype=author&query=Stillfjord%2C+T">Tony Stillfjord</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="1706.04380v3-abstract-short" style="display: inline;"> We consider approximations to the solutions of differential Riccati equations in the context of linear quadratic regulator problems, where the state equation is governed by a multiscale operator. Similarly to elliptic and parabolic problems, standard finite element discretizations perform poorly in this setting unless the grid resolves the fine-scale features of the problem. This results in unfeas… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1706.04380v3-abstract-full').style.display = 'inline'; document.getElementById('1706.04380v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1706.04380v3-abstract-full" style="display: none;"> We consider approximations to the solutions of differential Riccati equations in the context of linear quadratic regulator problems, where the state equation is governed by a multiscale operator. Similarly to elliptic and parabolic problems, standard finite element discretizations perform poorly in this setting unless the grid resolves the fine-scale features of the problem. This results in unfeasible amounts of computation and high memory requirements. In this paper, we demonstrate how the localized orthogonal decomposition method may be used to acquire accurate results also for coarse discretizations, at the low cost of solving a series of small, localized elliptic problems. We prove second-order convergence (except for a logarithmic factor) in the $L^2$ operator norm, and first-order convergence in the corresponding energy norm. These results are both independent of the multiscale variations in the state equation. In addition, we provide a detailed derivation of the fully discrete matrix-valued equations, and show how they can be handled in a low-rank setting for large-scale computations. In connection to this, we also show how to efficiently compute the relevant operator-norm errors. Finally, our theoretical results are validated by several numerical experiments. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1706.04380v3-abstract-full').style.display = 'none'; document.getElementById('1706.04380v3-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> 18 June, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 June, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 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">Accepted for publication in SIAM J. Sci. Comput. This version differs from the previous one only by the addition of Remark 7.2 and minor changes in formatting. 21 pages, 12 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 49N10; 65N12; 65N30; 93C20 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> SIAM J. Sci. Comput. 40(4) (2018), pp. A2406--A2426 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1702.06171">arXiv:1702.06171</a> <span> [<a href="https://arxiv.org/pdf/1702.06171">pdf</a>, <a href="https://arxiv.org/ps/1702.06171">ps</a>, <a href="https://arxiv.org/format/1702.06171">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> </div> <p class="title is-5 mathjax"> Continuous deformations of harmonic maps and their unitons </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Aleman%2C+A">Alexandru Aleman</a>, <a href="/search/math?searchtype=author&query=Mart%C3%ADn%2C+M+J">Mar铆a J. Mart铆n</a>, <a href="/search/math?searchtype=author&query=Persson%2C+A">Anna-Maria Persson</a>, <a href="/search/math?searchtype=author&query=Svensson%2C+M">Martin Svensson</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="1702.06171v1-abstract-short" style="display: inline;"> We consider harmonic maps on simply connected Riemann surfaces into the group $\mathrm{U}(n)$ of unitary matrices of order $n$. It is known that a harmonic map with an associated algebraic extended solution can be deformed into a new harmonic map that has an $S^1$-invariant associated extended solution. We study this deformation in detail and show that the corresponding unitons are smooth function… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.06171v1-abstract-full').style.display = 'inline'; document.getElementById('1702.06171v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1702.06171v1-abstract-full" style="display: none;"> We consider harmonic maps on simply connected Riemann surfaces into the group $\mathrm{U}(n)$ of unitary matrices of order $n$. It is known that a harmonic map with an associated algebraic extended solution can be deformed into a new harmonic map that has an $S^1$-invariant associated extended solution. We study this deformation in detail and show that the corresponding unitons are smooth functions of the deformation parameter and real analytic along any line through the origin. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.06171v1-abstract-full').style.display = 'none'; document.getElementById('1702.06171v1-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, 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">9 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 58E20; 47A56 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1604.00262">arXiv:1604.00262</a> <span> [<a href="https://arxiv.org/pdf/1604.00262">pdf</a>, <a href="https://arxiv.org/format/1604.00262">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"> A generalized finite element method for linear thermoelasticity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=M%C3%A5lqvist%2C+A">Axel M氓lqvist</a>, <a href="/search/math?searchtype=author&query=Persson%2C+A">Anna Persson</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="1604.00262v1-abstract-short" style="display: inline;"> We propose and analyze a generalized finite element method designed for linear quasistatic thermoelastic systems with spatial multiscale coefficients. The method is based on the local orthogonal decomposition technique introduced by M氓lqvist and Peterseim (Math. Comp., 83(290): 2583--2603, 2014). We prove convergence of optimal order, independent of the derivatives of the coefficients, in the spat… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1604.00262v1-abstract-full').style.display = 'inline'; document.getElementById('1604.00262v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1604.00262v1-abstract-full" style="display: none;"> We propose and analyze a generalized finite element method designed for linear quasistatic thermoelastic systems with spatial multiscale coefficients. The method is based on the local orthogonal decomposition technique introduced by M氓lqvist and Peterseim (Math. Comp., 83(290): 2583--2603, 2014). We prove convergence of optimal order, independent of the derivatives of the coefficients, in the spatial $H^1$-norm. The theoretical results are confirmed by numerical examples. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1604.00262v1-abstract-full').style.display = 'none'; document.getElementById('1604.00262v1-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 April, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2016. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1603.09523">arXiv:1603.09523</a> <span> [<a href="https://arxiv.org/pdf/1603.09523">pdf</a>, <a href="https://arxiv.org/format/1603.09523">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/10.1016/j.cma.2016.06.034">10.1016/j.cma.2016.06.034 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A multiscale method for linear elasticity reducing Poisson locking </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Henning%2C+P">Patrick Henning</a>, <a href="/search/math?searchtype=author&query=Persson%2C+A">Anna Persson</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="1603.09523v1-abstract-short" style="display: inline;"> We propose a generalized finite element method for linear elasticity equations with highly varying and oscillating coefficients. The method is formulated in the framework of localized orthogonal decomposition techniques introduced by M氓lqvist and Peterseim (Math. Comp., 83(290): 2583--2603, 2014). Assuming only $L_\infty$-coefficients we prove linear convergence in the $H^1$-norm, also for materia… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1603.09523v1-abstract-full').style.display = 'inline'; document.getElementById('1603.09523v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1603.09523v1-abstract-full" style="display: none;"> We propose a generalized finite element method for linear elasticity equations with highly varying and oscillating coefficients. The method is formulated in the framework of localized orthogonal decomposition techniques introduced by M氓lqvist and Peterseim (Math. Comp., 83(290): 2583--2603, 2014). Assuming only $L_\infty$-coefficients we prove linear convergence in the $H^1$-norm, also for materials with large Lam茅 parameter $位$. The theoretical a priori error estimate is confirmed by numerical examples. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1603.09523v1-abstract-full').style.display = 'none'; document.getElementById('1603.09523v1-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> 31 March, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2016. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1509.01401">arXiv:1509.01401</a> <span> [<a href="https://arxiv.org/pdf/1509.01401">pdf</a>, <a href="https://arxiv.org/ps/1509.01401">ps</a>, <a href="https://arxiv.org/format/1509.01401">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Complex Variables">math.CV</span> </div> </div> <p class="title is-5 mathjax"> The Spectrum of Volterra-type integration operators on generalized Fock spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Constantin%2C+O">Olivia Constantin</a>, <a href="/search/math?searchtype=author&query=Persson%2C+A">Anna-Maria Persson</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="1509.01401v1-abstract-short" style="display: inline;"> We describe the spectrum of certain integration operators acting on general- ized Fock spaces. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1509.01401v1-abstract-full" style="display: none;"> We describe the spectrum of certain integration operators acting on general- ized Fock spaces. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1509.01401v1-abstract-full').style.display = 'none'; document.getElementById('1509.01401v1-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> 4 September, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2015. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1504.08140">arXiv:1504.08140</a> <span> [<a href="https://arxiv.org/pdf/1504.08140">pdf</a>, <a href="https://arxiv.org/format/1504.08140">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"> Multiscale techniques for parabolic equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=M%C3%A5lqvist%2C+A">Axel M氓lqvist</a>, <a href="/search/math?searchtype=author&query=Persson%2C+A">Anna Persson</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="1504.08140v1-abstract-short" style="display: inline;"> We use the local orthogonal decomposition technique to derive a generalized finite element method for linear and semilinear parabolic equations with spatial multiscale diffusion coefficient. We consider nonsmooth initial data and a backward Euler scheme for the temporal discretization. Optimal order convergence rate, depending only on the contrast, but not on the variations in the diffusion coeffi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1504.08140v1-abstract-full').style.display = 'inline'; document.getElementById('1504.08140v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1504.08140v1-abstract-full" style="display: none;"> We use the local orthogonal decomposition technique to derive a generalized finite element method for linear and semilinear parabolic equations with spatial multiscale diffusion coefficient. We consider nonsmooth initial data and a backward Euler scheme for the temporal discretization. Optimal order convergence rate, depending only on the contrast, but not on the variations in the diffusion coefficient, is proven in the $L_\infty(L_2)$-norm. We present numerical examples, which confirm our theoretical findings. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1504.08140v1-abstract-full').style.display = 'none'; document.getElementById('1504.08140v1-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> 30 April, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2015. </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a 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