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page: <a href=/list/math.NA/new?skip=0&show=1000 rel="nofollow"> fewer</a> | <span style="color: #454545">more</span> | <span style="color: #454545">all</span> </div> <dl id='articles'> <h3>New submissions (showing 6 of 6 entries)</h3> <dt> <a name='item1'>[1]</a> <a href ="/abs/2503.13526" title="Abstract" id="2503.13526"> arXiv:2503.13526 </a> [<a href="/pdf/2503.13526" title="Download PDF" id="pdf-2503.13526" aria-labelledby="pdf-2503.13526">pdf</a>, <a href="https://arxiv.org/html/2503.13526v1" title="View HTML" id="html-2503.13526" aria-labelledby="html-2503.13526" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.13526" title="Other formats" id="oth-2503.13526" aria-labelledby="oth-2503.13526">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Time parallelization for hyperbolic and parabolic problems </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Gander,+M+J">Martin J. Gander</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Wu,+S">Shu-Lin Wu</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Zhou,+T">Tao Zhou</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 107 pages; this paper is accepted for publication in Acta Numerica </div> <div class='list-journal-ref'><span class='descriptor'>Journal-ref:</span> Vol. 4, Acta Numerica, pp. 1-107, 2025 </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Numerical Analysis (math.NA)</span> </div> <p class='mathjax'> Time parallelization, also known as PinT (Parallel-in-Time) is a new research direction for the development of algorithms used for solving very large scale evolution problems on highly parallel computing architectures. Despite the fact that interesting theoretical work on PinT appeared as early 1964, it was not until 2004, when processor clock speeds reached their physical limit, that research in PinT took off. A distinctive characteristic of parallelization in time is that information flow only goes forward in time, meaning that time evolution processes seem necessarily to be sequential. Nevertheless, many algorithms have been developed over the last two decades to do PinT computations, and they are often grouped into four basic classes according to how the techniques work and are used: shooting-type methods; waveform relaxation methods based on domain decomposition; multigrid methods in space-time; and direct time parallel methods. However, over the past few years, it has been recognized that highly successful PinT algorithms for parabolic problems struggle when applied to hyperbolic problems. We focus in this survey therefore on this important aspect, by first providing a summary of the fundamental differences between parabolic and hyperbolic problems for time parallelization. We then group PinT algorithms into two basic groups: the first group contains four effective PinT techniques for hyperbolic problems, namely Schwarz Waveform Relaxation with its relation to Tent Pitching; Parallel Integral Deferred Correction; ParaExp; and ParaDiag. While the methods in the first group also work well for parabolic problems, we then present PinT methods especially designed for parabolic problems in the second group: Parareal: the Parallel Full Approximation Scheme in Space-Time; Multigrid Reduction in Time; and Space-Time Multigrid. </p> </div> </dd> <dt> <a name='item2'>[2]</a> <a href ="/abs/2503.13711" title="Abstract" id="2503.13711"> arXiv:2503.13711 </a> [<a href="/pdf/2503.13711" title="Download PDF" id="pdf-2503.13711" aria-labelledby="pdf-2503.13711">pdf</a>, <a href="/format/2503.13711" title="Other formats" id="oth-2503.13711" aria-labelledby="oth-2503.13711">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Constructing Sobolev orthonormal rational functions via an updating procedure </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Faghih,+A">Amin Faghih</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Van+Barel,+M">Marc Van Barel</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Van+Buggenhout,+N">Niel Van Buggenhout</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Vandebril,+R">Raf Vandebril</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Numerical Analysis (math.NA)</span> </div> <p class='mathjax'> In this paper, we generate the recursion coefficients for rational functions with prescribed poles that are orthonormal with respect to a continuous Sobolev inner product. Using a rational Gauss quadrature rule, the inner product can be discretized, thus allowing a linear algebraic approach. The presented approach involves reformulating the problem as an inverse eigenvalue problem involving a Hessenberg pencil, where the pencil will contain the recursion coefficients that generate the sequence of Sobolev orthogonal rational functions. This reformulation is based on the connection between Sobolev orthonormal rational functions and the orthonormal bases for rational Krylov subspaces generated by a Jordan-like matrix. An updating procedure, introducing the nodes of the inner product one after the other, is proposed and the performance is examined through some numerical examples. </p> </div> </dd> <dt> <a name='item3'>[3]</a> <a href ="/abs/2503.13802" title="Abstract" id="2503.13802"> arXiv:2503.13802 </a> [<a href="/pdf/2503.13802" title="Download PDF" id="pdf-2503.13802" aria-labelledby="pdf-2503.13802">pdf</a>, <a href="https://arxiv.org/html/2503.13802v1" title="View HTML" id="html-2503.13802" aria-labelledby="html-2503.13802" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.13802" title="Other formats" id="oth-2503.13802" aria-labelledby="oth-2503.13802">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Multi-Harmonic Gridded 3D Deconvolution (MH3D) for Robust and Accurate Image Reconstruction in MPI for Single Axis Drive Field Scanners </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Sanders,+T">Toby Sanders</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Konkle,+J+J">Justin J. Konkle</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Mason,+E+E">Erica E. Mason</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Goodwill,+P+W">Patrick W. Goodwill</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Numerical Analysis (math.NA)</span> </div> <p class='mathjax'> This article presents a new robust model for image reconstruction in magnetic particle imaging (MPI) for single-axis drive field scans, which is based on the deconvolution of gridded harmonic data. Gridded harmonic data, used commonly in MPI, does not map to underlying iron density but rather to the iron density convolved with the harmonic point-spread functions. We refer to the gridded harmonic data as harmonic portraits, since they only represent a portrait-like representation of the iron density, and a deconvolution method is implemented to reconstruct the true underlying density. The advantage of this new method is primarily in the intermediate data analysis that comes in the harmonic portrait domain, where we are able to perform artifact correction, parameter selection, and general data assessment and calibrations efficiently. Furthermore, we show with several examples that our new method closely compares qualitatively with current state-of-the-art image reconstruction models in MPI. While the general concept of gridding harmonic data in MPI is not new, the complete modeling and characterization in order to use the data for image reconstruction has remained an ongoing area of research. We provide detailed analysis, theoretical insights, and many nuanced techniques that make our new methodology and algorithm accurate and robust. </p> </div> </dd> <dt> <a name='item4'>[4]</a> <a href ="/abs/2503.13830" title="Abstract" id="2503.13830"> arXiv:2503.13830 </a> [<a href="/pdf/2503.13830" title="Download PDF" id="pdf-2503.13830" aria-labelledby="pdf-2503.13830">pdf</a>, <a href="https://arxiv.org/html/2503.13830v1" title="View HTML" id="html-2503.13830" aria-labelledby="html-2503.13830" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.13830" title="Other formats" id="oth-2503.13830" aria-labelledby="oth-2503.13830">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Hierarchical Gaussian Random Fields for Multilevel Markov Chain Monte Carlo: Coupling Stochastic Partial Differential Equation and The Karhunen-Lo猫ve Decomposition </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Reddy,+S">Sohail Reddy</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Numerical Analysis (math.NA)</span>; Probability (math.PR) </div> <p class='mathjax'> This work introduces structure preserving hierarchical decompositions for sampling Gaussian random fields (GRF) within the context of multilevel Bayesian inference in high-dimensional space. Existing scalable hierarchical sampling methods, such as those based on stochastic partial differential equation (SPDE), often reduce the dimensionality of the sample space at the cost of accuracy of inference. Other approaches, such that those based on Karhunen-Lo猫ve (KL) expansions, offer sample space dimensionality reduction but sacrifice GRF representation accuracy and ergodicity of the Markov Chain Monte Carlo (MCMC) sampler, and are computationally expensive for high-dimensional problems. The proposed method integrates the dimensionality reduction capabilities of KL expansions with the scalability of stochastic partial differential equation (SPDE)-based sampling, thereby providing a robust, unified framework for high-dimensional uncertainty quantification (UQ) that is scalable, accurate, preserves ergodicity, and offers dimensionality reduction of the sample space. The hierarchy in our multilevel algorithm is derived from the geometric multigrid hierarchy. By constructing a hierarchical decomposition that maintains the covariance structure across the levels in the hierarchy, the approach enables efficient coarse-to-fine sampling while ensuring that all samples are drawn from the desired distribution. The effectiveness of the proposed method is demonstrated on a benchmark subsurface flow problem, demonstrating its effectiveness in improving computational efficiency and statistical accuracy. Our proposed technique is more efficient, accurate, and displays better convergence properties than existing methods for high-dimensional Bayesian inference problems. </p> </div> </dd> <dt> <a name='item5'>[5]</a> <a href ="/abs/2503.13930" title="Abstract" id="2503.13930"> arXiv:2503.13930 </a> [<a href="/pdf/2503.13930" title="Download PDF" id="pdf-2503.13930" aria-labelledby="pdf-2503.13930">pdf</a>, <a href="https://arxiv.org/html/2503.13930v1" title="View HTML" id="html-2503.13930" aria-labelledby="html-2503.13930" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.13930" title="Other formats" id="oth-2503.13930" aria-labelledby="oth-2503.13930">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> SIAC Accuracy Enhancement of Stochastic Galerkin Solutions for Wave Equations with Uncertain Coefficients </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Galindo-Olarte,+A">Andr茅s Galindo-Olarte</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Ryan,+J+K">Jennifer K. Ryan</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 19 pages, 3 figures </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Numerical Analysis (math.NA)</span> </div> <p class='mathjax'> This article establishes the usefulness of the Smoothness-Increasing Accuracy-Increasing (SIAC) filter for reducing the errors in the mean and variance for a wave equation with uncertain coefficients solved via generalized polynomial chaos (gPC) whose coefficients are approximated using discontinuous Galerkin (DG-gPC). Theoretical error estimates that utilize information in the negative-order norm are established. While the gPC approximation leads to order of accuracy of $m-1/2$ for a sufficiently smooth solution (smoothness of $m$ in random space), the approximated coefficients solved via DG improves from order $k+1$ to $2k+1$ for a solution of smoothness $2k+2$ in physical space. Our numerical examples verify the performance of the filter for improving the quality of the approximation and reducing the numerical error and significantly eliminating the noise from the spatial approximation of the mean and variance. Further, we illustrate how the errors are effected by both the choice of smoothness of the kernel and number of function translates in the kernel. Hence, this article opens the applicability of SIAC filters to other hyperbolic problems with uncertainty, and other stochastic equations. </p> </div> </dd> <dt> <a name='item6'>[6]</a> <a href ="/abs/2503.13941" title="Abstract" id="2503.13941"> arXiv:2503.13941 </a> [<a href="/pdf/2503.13941" title="Download PDF" id="pdf-2503.13941" aria-labelledby="pdf-2503.13941">pdf</a>, <a href="https://arxiv.org/html/2503.13941v1" title="View HTML" id="html-2503.13941" aria-labelledby="html-2503.13941" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.13941" title="Other formats" id="oth-2503.13941" aria-labelledby="oth-2503.13941">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Randomized block Kaczmarz with volume sampling: Momentum acceleration and efficient implementation </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Xiang,+R">Ruike Xiang</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Xie,+J">Jiaxin Xie</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Zhang,+Q">Qiye Zhang</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Numerical Analysis (math.NA)</span> </div> <p class='mathjax'> The randomized block Kaczmarz (RBK) method is a widely utilized iterative scheme for solving large-scale linear systems. However, the theoretical analysis and practical effectiveness of this method heavily rely on a good row paving of the coefficient matrix. This motivates us to introduce a novel block selection strategy to the RBK method, called volume sampling, in which the probability of selection is proportional to the volume spanned by the rows of the selected submatrix. To further enhance the practical performance, we develop and analyze a momentum variant of the method. Convergence results are established and demonstrate the notable improvements in convergence factor of the RBK method brought by the volume sampling and the momentum acceleration. Furthermore, to efficiently implement the RBK method with volume sampling, we propose an efficient algorithm that enables volume sampling from a sparse matrix with sampling complexity that is only logarithmic in dimension. Numerical experiments confirm our theoretical results. </p> </div> </dd> </dl> <dl id='articles'> <h3>Cross submissions (showing 3 of 3 entries)</h3> <dt> <a name='item7'>[7]</a> <a href ="/abs/2503.13756" title="Abstract" id="2503.13756"> arXiv:2503.13756 </a> (cross-list from cs.CV) [<a href="/pdf/2503.13756" title="Download PDF" id="pdf-2503.13756" aria-labelledby="pdf-2503.13756">pdf</a>, <a href="https://arxiv.org/html/2503.13756v1" title="View HTML" id="html-2503.13756" aria-labelledby="html-2503.13756" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.13756" title="Other formats" id="oth-2503.13756" aria-labelledby="oth-2503.13756">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Fast alignment of heterogeneous images in sliced Wasserstein distance </div> <div class='list-authors'><a href="https://arxiv.org/search/cs?searchtype=author&query=Shi,+Y">Yunpeng Shi</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Singer,+A">Amit Singer</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Verbeke,+E+J">Eric J. Verbeke</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Computer Vision and Pattern Recognition (cs.CV)</span>; Numerical Analysis (math.NA) </div> <p class='mathjax'> Many applications of computer vision rely on the alignment of similar but non-identical images. We present a fast algorithm for aligning heterogeneous images based on optimal transport. Our approach combines the speed of fast Fourier methods with the robustness of sliced probability metrics and allows us to efficiently compute the alignment between two $L \times L$ images using the sliced 2-Wasserstein distance in $O(L^2 \log L)$ operations. We show that our method is robust to translations, rotations and deformations in the images. </p> </div> </dd> <dt> <a name='item8'>[8]</a> <a href ="/abs/2503.13877" title="Abstract" id="2503.13877"> arXiv:2503.13877 </a> (cross-list from cs.LO) [<a href="/pdf/2503.13877" title="Download PDF" id="pdf-2503.13877" aria-labelledby="pdf-2503.13877">pdf</a>, <a href="https://arxiv.org/html/2503.13877v1" title="View HTML" id="html-2503.13877" aria-labelledby="html-2503.13877" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.13877" title="Other formats" id="oth-2503.13877" aria-labelledby="oth-2503.13877">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Shock with Confidence: Formal Proofs of Correctness for Hyperbolic Partial Differential Equation Solvers </div> <div class='list-authors'><a href="https://arxiv.org/search/cs?searchtype=author&query=Gorard,+J">Jonathan Gorard</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Hakim,+A">Ammar Hakim</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 13 pages, prepared for submission to ACM </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Logic in Computer Science (cs.LO)</span>; Numerical Analysis (math.NA); Computational Physics (physics.comp-ph) </div> <p class='mathjax'> First-order systems of hyperbolic partial differential equations (PDEs) occur ubiquitously throughout computational physics, commonly used in simulations of fluid turbulence, shock waves, electromagnetic interactions, and even general relativistic phenomena. Such equations are often challenging to solve numerically in the non-linear case, due to their tendency to form discontinuities even for smooth initial data, which can cause numerical algorithms to become unstable, violate conservation laws, or converge to physically incorrect solutions. In this paper, we introduce a new formal verification pipeline for such algorithms in Racket, which allows a user to construct a bespoke hyperbolic PDE solver for a specified equation system, generate low-level C code which verifiably implements that solver, and then produce formal proofs of various mathematical and physical correctness properties of the resulting implementation, including L^2 stability, flux conservation, and physical validity. We outline how these correctness proofs are generated, using a custom-built theorem-proving and automatic differentiation framework that fully respects the algebraic structure of floating-point arithmetic, and show how the resulting C code may either be used to run standalone simulations, or integrated into a larger computational multiphysics framework such as Gkeyll. </p> </div> </dd> <dt> <a name='item9'>[9]</a> <a href ="/abs/2503.14011" title="Abstract" id="2503.14011"> arXiv:2503.14011 </a> (cross-list from eess.SP) [<a href="/pdf/2503.14011" title="Download PDF" id="pdf-2503.14011" aria-labelledby="pdf-2503.14011">pdf</a>, <a href="/format/2503.14011" title="Other formats" id="oth-2503.14011" aria-labelledby="oth-2503.14011">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Multitaper-Based Post-Processing of Compact Antenna Responses Obtained in Non-Anechoic Conditions </div> <div class='list-authors'><a href="https://arxiv.org/search/eess?searchtype=author&query=Dzwonkowski,+M">Mariusz Dzwonkowski</a>, <a href="https://arxiv.org/search/eess?searchtype=author&query=Bekasiewicz,+A">Adrian Bekasiewicz</a>, <a href="https://arxiv.org/search/eess?searchtype=author&query=Koziel,+S">Slawomir Koziel</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Signal Processing (eess.SP)</span>; Numerical Analysis (math.NA) </div> <p class='mathjax'> The process of developing antenna structures typically involves prototype measurements. While accurate validation of far-field performance can be performed in dedicated facilities like anechoic chambers, high cost of construction and maintenance might not justify their use for teaching, or low-budget research scenarios. Non-anechoic experiments provide a cost-effective alternative, however the performance metrics obtained in such conditions require appropriate correction. In this paper, we consider a multitaper approach for post-processing antenna far-field characteristics measured in challenging, non-anechoic environments. The discussed algorithm enhances one-shot measurements to enable extraction of line-of-sight responses while attenuating interferences from multi-path propagation and the noise from external sources of electromagnetic radiation. The performance of the considered method has been demonstrated in uncontrolled conditions using a compact spline-based monopole. Furthermore, the approach has been favorably validated against the state-of-the-art techniques from the literature. </p> </div> </dd> </dl> <dl id='articles'> <h3>Replacement submissions (showing 10 of 10 entries)</h3> <dt> <a name='item10'>[10]</a> <a href ="/abs/2404.16699" title="Abstract" id="2404.16699"> arXiv:2404.16699 </a> (replaced) [<a href="/pdf/2404.16699" title="Download PDF" id="pdf-2404.16699" aria-labelledby="pdf-2404.16699">pdf</a>, <a href="/format/2404.16699" title="Other formats" id="oth-2404.16699" aria-labelledby="oth-2404.16699">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Generalized cyclic symmetric decompositions for the matrix multiplication tensor </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Vermeylen,+C">Charlotte Vermeylen</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Van+Barel,+M">Marc Van Barel</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Numerical Analysis (math.NA)</span>; Optimization and Control (math.OC) </div> <p class='mathjax'> A new generalized cyclic symmetric structure in the factor matrices of polyadic decompositions of matrix multiplication tensors for non-square matrix multiplication is proposed to reduce the number of variables in the optimization problem and in this way improve the convergence. The structure is implemented in an existing numerical optimization algorithm. Extensive numerical experiments are given that the proposed structure indeed finds more (practical) decompositions. </p> </div> </dd> <dt> <a name='item11'>[11]</a> <a href ="/abs/2406.03885" title="Abstract" id="2406.03885"> arXiv:2406.03885 </a> (replaced) [<a href="/pdf/2406.03885" title="Download PDF" id="pdf-2406.03885" aria-labelledby="pdf-2406.03885">pdf</a>, <a href="/format/2406.03885" title="Other formats" id="oth-2406.03885" aria-labelledby="oth-2406.03885">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Convergence of a Riemannian gradient method for the Gross-Pitaevskii energy functional in a rotating frame </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Henning,+P">Patrick Henning</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Yadav,+M">Mahima Yadav</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Numerical Analysis (math.NA)</span> </div> <p class='mathjax'> This paper investigates the numerical approximation of ground states of rotating Bose-Einstein condensates. This problem requires the minimization of the Gross-Pitaevskii energy $E$ on a Hilbert manifold $\mathbb{S}$. To find a corresponding minimizer $u$, we use a generalized Riemannian gradient method that is based on the concept of Sobolev gradients in combination with an adaptively changing metric on the manifold. By a suitable choice of the metric, global energy dissipation for the arising gradient method can be proved. The energy dissipation property in turn implies global convergence to the density $|u|^2$ of a critical point $u$ of $E$ on $\mathbb{S}$. Furthermore, we present a precise characterization of the local convergence rates in a neighborhood of each ground state $u$ and how these rates depend on the first spectral gap of $E^{\prime\prime}(u)$ restricted to the $L^2$-orthogonal complement of $u$. With this we establish the first convergence results for a Riemannian gradient method to minimize the Gross-Pitaevskii energy functional in a rotating frame. At the same, we refine previous results obtained in the case without rotation. The major complication in our new analysis is the missing isolation of minimizers, which are at most unique up to complex phase shifts. For that, we introduce an auxiliary iteration in the tangent space $T_{\mathrm{i} u} \mathbb{S}$ and apply the Ostrowski theorem to characterize the asymptotic convergence rates through a weighted eigenvalue problem. Afterwards, we link the auxiliary iteration to the original Riemannian gradient method and bound the spectrum of the weighted eigenvalue problem to obtain quantitative convergence rates. Our findings are validated in numerical experiments. </p> </div> </dd> <dt> <a name='item12'>[12]</a> <a href ="/abs/2408.03455" title="Abstract" id="2408.03455"> arXiv:2408.03455 </a> (replaced) [<a href="/pdf/2408.03455" title="Download PDF" id="pdf-2408.03455" aria-labelledby="pdf-2408.03455">pdf</a>, <a href="https://arxiv.org/html/2408.03455v3" title="View HTML" id="html-2408.03455" aria-labelledby="html-2408.03455" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2408.03455" title="Other formats" id="oth-2408.03455" aria-labelledby="oth-2408.03455">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Bayesian learning with Gaussian processes for low-dimensional representations of time-dependent nonlinear systems </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=McQuarrie,+S+A">Shane A. McQuarrie</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Chaudhuri,+A">Anirban Chaudhuri</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Willcox,+K+E">Karen E. Willcox</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Guo,+M">Mengwu Guo</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> <a href="https://github.com/Sandialabs/GP-BayesOpInf" rel="external noopener nofollow" class="link-external link-https">this https URL</a> </div> <div class='list-journal-ref'><span class='descriptor'>Journal-ref:</span> Physica D: Nonlinear Phenomena, 475 (2025), 134572 </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Numerical Analysis (math.NA)</span> </div> <p class='mathjax'> This work presents a data-driven method for learning low-dimensional time-dependent physics-based surrogate models whose predictions are endowed with uncertainty estimates. We use the operator inference approach to model reduction that poses the problem of learning low-dimensional model terms as a regression of state space data and corresponding time derivatives by minimizing the residual of reduced system equations. Standard operator inference models perform well with accurate training data that are dense in time, but producing stable and accurate models when the state data are noisy and/or sparse in time remains a challenge. Another challenge is the lack of uncertainty estimation for the predictions from the operator inference models. Our approach addresses these challenges by incorporating Gaussian process surrogates into the operator inference framework to (1) probabilistically describe uncertainties in the state predictions and (2) procure analytical time derivative estimates with quantified uncertainties. The formulation leads to a generalized least-squares regression and, ultimately, reduced-order models that are described probabilistically with a closed-form expression for the posterior distribution of the operators. The resulting probabilistic surrogate model propagates uncertainties from the observed state data to reduced-order predictions. We demonstrate the method is effective for constructing low-dimensional models of two nonlinear partial differential equations representing a compressible flow and a nonlinear diffusion-reaction process, as well as for estimating the parameters of a low-dimensional system of nonlinear ordinary differential equations representing compartmental models in epidemiology. </p> </div> </dd> <dt> <a name='item13'>[13]</a> <a href ="/abs/2410.01467" title="Abstract" id="2410.01467"> arXiv:2410.01467 </a> (replaced) [<a href="/pdf/2410.01467" title="Download PDF" id="pdf-2410.01467" aria-labelledby="pdf-2410.01467">pdf</a>, <a href="https://arxiv.org/html/2410.01467v4" title="View HTML" id="html-2410.01467" aria-labelledby="html-2410.01467" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2410.01467" title="Other formats" id="oth-2410.01467" aria-labelledby="oth-2410.01467">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> A fast numerical scheme for fractional viscoelastic models of wave propagation </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Yuan,+H">Hao Yuan</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Xie,+X">Xiaoping Xie</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Numerical Analysis (math.NA)</span> </div> <p class='mathjax'> Due to the nonlocal feature of fractional differential operators, the numerical solution to fractional partial differential equations usually requires expensive memory and computation costs. This paper develops a fast scheme for fractional viscoelastic models of wave propagation. <br>We first apply the Laplace transform to convert the time-fractional constitutive equation into an integro-differential form that involves the Mittag-Leffler function as a convolution kernel. Then we construct an efficient sum-of-exponentials (SOE) approximation for the Mittag-Leffler function. We use mixed finite elements for the spatial discretization and the Newmark scheme for the temporal discretization of the second time-derivative of the displacement variable in the kinematical equation and finally obtain the fast algorithm. Compared with the traditional L1 scheme for time fractional derivative, our fast scheme reduces the memory complexity from $\mathcal O(N_sN) $ to $\mathcal O(N_sN_{exp})$ and the computation complexity from $\mathcal O(N_sN^2)$ to $\mathcal O(N_sN_{exp}N)$, where $N$ denotes the total number of temporal grid points, $N_{exp}$ is the number of exponentials in SOE, and $N_s$ represents the complexity of memory and computation related to the spatial discretization. Numerical experiments confirm the theoretical results. </p> </div> </dd> <dt> <a name='item14'>[14]</a> <a href ="/abs/2503.10199" title="Abstract" id="2503.10199"> arXiv:2503.10199 </a> (replaced) [<a href="/pdf/2503.10199" title="Download PDF" id="pdf-2503.10199" aria-labelledby="pdf-2503.10199">pdf</a>, <a href="https://arxiv.org/html/2503.10199v3" title="View HTML" id="html-2503.10199" aria-labelledby="html-2503.10199" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.10199" title="Other formats" id="oth-2503.10199" aria-labelledby="oth-2503.10199">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Optimal Estimation and Uncertainty Quantification for Stochastic Inverse Problems via Variational Bayesian Methods </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Song,+R">Ruibiao Song</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Zhang,+L">Liying Zhang</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Numerical Analysis (math.NA)</span> </div> <p class='mathjax'> The Bayesian inversion method demonstrates significant potential for solving inverse problems, enabling both point estimation and uncertainty quantification. However, Bayesian maximum a posteriori (MAP) estimation may become unstable when handling data from diverse distributions (e.g., solutions of stochastic partial differential equations (SPDEs)). Additionally, Monte Carlo sampling methods are computationally expensive. To address these challenges, we propose a novel two-stage optimization method based on optimal control theory and variational Bayesian methods. This method not only achieves stable solutions for stochastic inverse problems but also efficiently quantifies the uncertainty of the solutions. In the first stage, we introduce a new weighting formulation to ensure the stability of the Bayesian MAP estimation. In the second stage, we derive the necessary condition to efficiently quantify the uncertainty of the solutions, by combining the new weighting formula with variational inference. Furthermore, we establish an error estimation theorem that relates the exact solution to the optimally estimated solution under different amounts of observed data. Finally, the efficiency of the proposed method is demonstrated through numerical examples. </p> </div> </dd> <dt> <a name='item15'>[15]</a> <a href ="/abs/2503.12805" title="Abstract" id="2503.12805"> arXiv:2503.12805 </a> (replaced) [<a href="/pdf/2503.12805" title="Download PDF" id="pdf-2503.12805" aria-labelledby="pdf-2503.12805">pdf</a>, <a href="https://arxiv.org/html/2503.12805v2" title="View HTML" id="html-2503.12805" aria-labelledby="html-2503.12805" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.12805" title="Other formats" id="oth-2503.12805" aria-labelledby="oth-2503.12805">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> A fast Fourier spectral method for wave kinetic equation </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Qi,+K">Kunlun Qi</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Shen,+L">Lian Shen</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Wang,+L">Li Wang</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Numerical Analysis (math.NA)</span> </div> <p class='mathjax'> The central object in wave turbulence theory is the wave kinetic equation (WKE), which is an evolution equation for wave action density and acts as the wave analog of the Boltzmann kinetic equations for particle interactions. Despite recent exciting progress in the theoretical aspects of the WKE, numerical developments have lagged behind. In this paper, we introduce a fast Fourier spectral method for solving the WKE. The key idea lies in reformulating the high-dimensional nonlinear wave kinetic operator as a spherical integral, analogous to classical Boltzmann collision operator. The conservation of mass and momentum leads to a double convolution structure in Fourier space, which can be efficiently handled using the fast Fourier transform. We demonstrate the accuracy and efficiency of the proposed method through several numerical tests in both 2D and 3D, revealing some interesting and unique features of this equation. </p> </div> </dd> <dt> <a name='item16'>[16]</a> <a href ="/abs/2310.16202" title="Abstract" id="2310.16202"> arXiv:2310.16202 </a> (replaced) [<a href="/pdf/2310.16202" title="Download PDF" id="pdf-2310.16202" aria-labelledby="pdf-2310.16202">pdf</a>, <a href="https://arxiv.org/html/2310.16202v2" title="View HTML" id="html-2310.16202" aria-labelledby="html-2310.16202" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2310.16202" title="Other formats" id="oth-2310.16202" aria-labelledby="oth-2310.16202">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Existence of solution to a system of PDEs modeling the crystal growth inside lithium batteries </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Lakkis,+O">Omar Lakkis</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Skouras,+A">Alexandros Skouras</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Styles,+V">Vanessa Styles</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 23 pages, 4 figures (25 pictures), free software and open source code available </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Analysis of PDEs (math.AP)</span>; Materials Science (cond-mat.mtrl-sci); Numerical Analysis (math.NA) </div> <p class='mathjax'> We study a model for lithium (Li) electrodeposition on Li-metal electrodes that leads to dendritic pattern formation. The model comprises of a system of three coupled PDEs, taking the form of an Allen--Cahn equation, a Nernst--Planck equation and a Poisson equation. We prove existence of a weak solution and stability results for this system and present numerical simulations resulting from a finite element approximation of the system, which illustrate the dendritic nature of solutions to the model. </p> </div> </dd> <dt> <a name='item17'>[17]</a> <a href ="/abs/2405.14099" title="Abstract" id="2405.14099"> arXiv:2405.14099 </a> (replaced) [<a href="/pdf/2405.14099" title="Download PDF" id="pdf-2405.14099" aria-labelledby="pdf-2405.14099">pdf</a>, <a href="https://arxiv.org/html/2405.14099v4" title="View HTML" id="html-2405.14099" aria-labelledby="html-2405.14099" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2405.14099" title="Other formats" id="oth-2405.14099" aria-labelledby="oth-2405.14099">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Automatic Differentiation is Essential in Training Neural Networks for Solving Differential Equations </div> <div class='list-authors'><a href="https://arxiv.org/search/cs?searchtype=author&query=Chen,+C">Chuqi Chen</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Yang,+Y">Yahong Yang</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Xiang,+Y">Yang Xiang</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Hao,+W">Wenrui Hao</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Machine Learning (cs.LG)</span>; Numerical Analysis (math.NA) </div> <p class='mathjax'> Neural network-based approaches have recently shown significant promise in solving partial differential equations (PDEs) in science and engineering, especially in scenarios featuring complex domains or incorporation of empirical data. One advantage of the neural network methods for PDEs lies in its automatic differentiation (AD), which necessitates only the sample points themselves, unlike traditional finite difference (FD) approximations that require nearby local points to compute derivatives. In this paper, we quantitatively demonstrate the advantage of AD in training neural networks. The concept of truncated entropy is introduced to characterize the training property. Specifically, through comprehensive experimental and theoretical analyses conducted on random feature models and two-layer neural networks, we discover that the defined truncated entropy serves as a reliable metric for quantifying the residual loss of random feature models and the training speed of neural networks for both AD and FD methods. Our experimental and theoretical analyses demonstrate that, from a training perspective, AD outperforms FD in solving PDEs. </p> </div> </dd> <dt> <a name='item18'>[18]</a> <a href ="/abs/2406.10511" title="Abstract" id="2406.10511"> arXiv:2406.10511 </a> (replaced) [<a href="/pdf/2406.10511" title="Download PDF" id="pdf-2406.10511" aria-labelledby="pdf-2406.10511">pdf</a>, <a href="https://arxiv.org/html/2406.10511v3" title="View HTML" id="html-2406.10511" aria-labelledby="html-2406.10511" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2406.10511" title="Other formats" id="oth-2406.10511" aria-labelledby="oth-2406.10511">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Efficient Hardware Accelerator Based on Medium Granularity Dataflow for SpTRSV </div> <div class='list-authors'><a href="https://arxiv.org/search/cs?searchtype=author&query=Chen,+Q">Qian Chen</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Yang,+X">Xiaofeng Yang</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Lu,+S">Shengli Lu</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> Published in IEEE Transactions on Very Large Scale Integration (VLSI) Systems. DOI: <a href="https://doi.org/10.1109/TVLSI.2024.3497166" data-doi="10.1109/TVLSI.2024.3497166" class="link-https link-external" rel="external noopener nofollow">https://doi.org/10.1109/TVLSI.2024.3497166</a> </div> <div class='list-journal-ref'><span class='descriptor'>Journal-ref:</span> Q. Chen, X. Yang, and S. Lu, "Efficient Hardware Accelerator Based on Medium Granularity Dataflow for SpTRSV," IEEE Trans. Very Large Scale Integr. (VLSI) Syst. 33 (2025) 807-820 </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Distributed, Parallel, and Cluster Computing (cs.DC)</span>; Hardware Architecture (cs.AR); Performance (cs.PF); Numerical Analysis (math.NA) </div> <p class='mathjax'> Sparse triangular solve (SpTRSV) is widely used in various domains. Numerous studies have been conducted using CPUs, GPUs, and specific hardware accelerators, where dataflows can be categorized into coarse and fine granularity. Coarse dataflows offer good spatial locality but suffer from low parallelism, while fine dataflows provide high parallelism but disrupt the spatial structure, leading to increased nodes and poor data reuse. This paper proposes a novel hardware accelerator for SpTRSV or SpTRSV-like DAGs. The accelerator implements a medium granularity dataflow through hardware-software codesign and achieves both excellent spatial locality and high parallelism. Additionally, a partial sum caching mechanism is introduced to reduce the blocking frequency of processing elements (PEs), and a reordering algorithm of intra-node edges computation is developed to enhance data reuse. Experimental results on 245 benchmarks with node counts reaching up to 85,392 demonstrate that this work achieves average performance improvements of 7.0$\times$ (up to 27.8$\times$) over CPUs and 5.8$\times$ (up to 98.8$\times$) over GPUs. Compared to the state-of-the-art technique (DPU-v2), this work shows a 2.5$\times$ (up to 5.9$\times$) average performance improvement and 1.7$\times$ (up to 4.1$\times$) average energy efficiency enhancement. </p> </div> </dd> <dt> <a name='item19'>[19]</a> <a href ="/abs/2503.13388" title="Abstract" id="2503.13388"> arXiv:2503.13388 </a> (replaced) [<a href="/pdf/2503.13388" title="Download PDF" id="pdf-2503.13388" aria-labelledby="pdf-2503.13388">pdf</a>, <a href="https://arxiv.org/html/2503.13388v2" title="View HTML" id="html-2503.13388" aria-labelledby="html-2503.13388" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2503.13388" title="Other formats" id="oth-2503.13388" aria-labelledby="oth-2503.13388">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> A mathematical model for a universal digital quantum computer with an application to the Grover-Rudolph algorithm </div> <div class='list-authors'><a href="https://arxiv.org/search/quant-ph?searchtype=author&query=Falc%C3%B3,+A">Antonio Falc贸</a>, <a href="https://arxiv.org/search/quant-ph?searchtype=author&query=Falc%C3%B3--Pomares,+D">Daniela Falc贸--Pomares</a>, <a href="https://arxiv.org/search/quant-ph?searchtype=author&query=Matthies,+H+G">Hermann G. Matthies</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Quantum Physics (quant-ph)</span>; Numerical Analysis (math.NA) </div> <p class='mathjax'> In this work, we develop a novel mathematical framework for universal digital quantum computation using algebraic probability theory. We rigorously define quantum circuits as finite sequences of elementary quantum gates and establish their role in implementing unitary transformations. A key result demonstrates that every unitary matrix in $\mathrm{U}(N)$ can be expressed as a product of elementary quantum gates, leading to the concept of a universal dictionary for quantum computation. We apply this framework to the construction of quantum circuits that encode probability distributions, focusing on the Grover-Rudolph algorithm. By leveraging controlled quantum gates and rotation matrices, we design a quantum circuit that approximates a given probability density function. Numerical simulations, conducted using Qiskit, confirm the theoretical predictions and validate the effectiveness of our approach. These results provide a rigorous foundation for quantum circuit synthesis within an algebraic probability framework and offer new insights into the encoding of probability distributions in quantum algorithms. Potential applications include quantum machine learning, circuit optimization, and experimental implementations on real quantum hardware. </p> </div> </dd> </dl> <div class='paging'>Total of 19 entries </div> <div class='morefewer'>Showing up to 2000 entries per page: <a href=/list/math.NA/new?skip=0&show=1000 rel="nofollow"> fewer</a> | <span style="color: #454545">more</span> | <span style="color: #454545">all</span> </div> </div> </div> </div> </main> <footer style="clear: both;"> <div class="columns is-desktop" role="navigation" aria-label="Secondary" style="margin: -0.75em -0.75em 0.75em -0.75em">  <div class="column" style="padding: 0;"> <div class="columns"> <div class="column"> <ul style="list-style: none; line-height: 2;"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul style="list-style: none; line-height: 2;"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg> <a href="https://info.arxiv.org/help/contact.html"> Contact</a> </li> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>subscribe to arXiv mailings</title><desc>Click here to subscribe</desc><path d="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z"/></svg> <a href="https://info.arxiv.org/help/subscribe"> Subscribe</a> </li> </ul> </div> </div> </div>   <div class="column" style="padding: 0;"> <div class="columns"> <div class="column"> <ul style="list-style: none; 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