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Localized Implicit Time Stepping for the Wave Equation 波方程的局部隐式时间步进
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-07-15 DOI: 10.1137/23m1582618
Dietmar Gallistl, Roland Maier
{"title":"Localized Implicit Time Stepping for the Wave Equation","authors":"Dietmar Gallistl, Roland Maier","doi":"10.1137/23m1582618","DOIUrl":"https://doi.org/10.1137/23m1582618","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1589-1608, August 2024. <br/> Abstract. This work proposes a discretization of the acoustic wave equation with possibly oscillatory coefficients based on a superposition of discrete solutions to spatially localized subproblems computed with an implicit time discretization. Based on exponentially decaying entries of the global system matrices and an appropriate partition of unity, it is proved that the superposition of localized solutions is appropriately close to the solution of the (global) implicit scheme. It is thereby justified that the localized (and especially parallel) computation on multiple overlapping subdomains is reasonable. Moreover, a restart is introduced after a certain number of time steps to maintain a moderate overlap of the subdomains. Overall, the approach may be understood as a domain decomposition strategy in space on successive short time intervals that completely avoids inner iterations. Numerical examples are presented.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"47 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141625086","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Full-Spectrum Dispersion Relation Preserving Summation-by-Parts Operators 全谱色散关系保全逐部求和算子
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-07-11 DOI: 10.1137/23m1586471
Christopher Williams, Kenneth Duru
{"title":"Full-Spectrum Dispersion Relation Preserving Summation-by-Parts Operators","authors":"Christopher Williams, Kenneth Duru","doi":"10.1137/23m1586471","DOIUrl":"https://doi.org/10.1137/23m1586471","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1565-1588, August 2024. <br/> Abstract. The dispersion error is currently the dominant error for computed solutions of wave propagation problems with high-frequency components. In this paper, we define and give explicit examples of interior [math]-dispersion-relation-preserving schemes, of interior order of accuracy 4, 5, 6, and 7, with a complete methodology to construct them. These are dual-pair finite-difference schemes for systems of hyperbolic partial differential equations which satisfy the summation-by-parts principle and preserve the dispersion relation of the continuous problem uniformly to an [math] error tolerance for their interior stencil. We give a general framework to design provably stable finite-difference operators whose interior stencil preserves the dispersion relation for hyperbolic systems such as the elastic wave equation. The operators we derive here can resolve the highest frequency ([math]-mode) present on any equidistant grid at a tolerance of [math] maximum error within the interior stencil, with minimal extra stencil points. As standard finite-difference schemes have a [math] dispersion error for high-frequency components, fine meshes must be used to resolve these components. Our derived schemes may compute solutions with the same accuracy as traditional schemes on far coarser meshes, which in high dimensions significantly improves the computational cost.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"6 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141597537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Randomized Least-Squares with Minimal Oversampling and Interpolation in General Spaces 一般空间中最小过采样和插值的随机最小二乘法
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-07-09 DOI: 10.1137/23m160178x
Matthieu Dolbeault, Moulay Abdellah Chkifa
{"title":"Randomized Least-Squares with Minimal Oversampling and Interpolation in General Spaces","authors":"Matthieu Dolbeault, Moulay Abdellah Chkifa","doi":"10.1137/23m160178x","DOIUrl":"https://doi.org/10.1137/23m160178x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1515-1538, August 2024. <br/> Abstract. In approximation of functions based on point values, least-squares methods provide more stability than interpolation, at the expense of increasing the sampling budget. We show that near-optimal approximation error can nevertheless be achieved, in an expected [math] sense, as soon as the sample size [math] is larger than the dimension [math] of the approximation space by a constant multiplicative ratio. On the other hand, for [math], we obtain an interpolation strategy with a stability factor of order [math]. The proposed sampling algorithms are greedy procedures based on [Batson, Spielman, and Srivastava, Twice-Ramanujan sparsifiers, in Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, 2009, pp. 255–262] and [Lee and Sun, SIAM J. Comput., 47 (2018), pp. 2315–2336], with polynomial computational complexity.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"3 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141566271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Robust Finite Elements for Linearized Magnetohydrodynamics 线性化磁流体力学的稳健有限元
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-07-09 DOI: 10.1137/23m1582783
L. Beirão da Veiga, F. Dassi, G. Vacca
{"title":"Robust Finite Elements for Linearized Magnetohydrodynamics","authors":"L. Beirão da Veiga, F. Dassi, G. Vacca","doi":"10.1137/23m1582783","DOIUrl":"https://doi.org/10.1137/23m1582783","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1539-1564, August 2024. <br/> Abstract. We introduce a pressure robust finite element method for the linearized magnetohydrodynamics equations in three space dimensions, which is provably quasi-robust also in the presence of high fluid and magnetic Reynolds numbers. The proposed scheme uses a nonconforming BDM approach with suitable DG terms for the fluid part, combined with an [math]-conforming choice for the magnetic fluxes. The method introduces also a specific CIP-type stabilization associated to the coupling terms. Finally, the theoretical result are further validated by numerical experiments.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"54 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141566270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Discrete Weak Duality of Hybrid High-Order Methods for Convex Minimization Problems 凸最小化问题混合高阶方法的离散弱对偶性
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-07-04 DOI: 10.1137/23m1594534
Ngoc Tien Tran
{"title":"Discrete Weak Duality of Hybrid High-Order Methods for Convex Minimization Problems","authors":"Ngoc Tien Tran","doi":"10.1137/23m1594534","DOIUrl":"https://doi.org/10.1137/23m1594534","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1492-1514, August 2024. <br/> Abstract. This paper derives a discrete dual problem for a prototypical hybrid high-order method for convex minimization problems. The discrete primal and dual problem satisfy a weak convex duality that leads to a priori error estimates with convergence rates under additional smoothness assumptions. This duality holds for general polyhedral meshes and arbitrary polynomial degrees of the discretization. A novel postprocessing is proposed and allows for a posteriori error estimates on regular triangulations into simplices using primal-dual techniques. This motivates an adaptive mesh-refining algorithm, which performs better compared to uniform mesh refinements.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"53 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141545843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Uniform Substructuring Preconditioners for High Order FEM on Triangles and the Influence of Nodal Basis Functions 三角形上高阶有限元的均匀子结构预处理以及节点基函数的影响
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-07-01 DOI: 10.1137/23m1561920
Mark Ainsworth, Shuai Jiang
{"title":"Uniform Substructuring Preconditioners for High Order FEM on Triangles and the Influence of Nodal Basis Functions","authors":"Mark Ainsworth, Shuai Jiang","doi":"10.1137/23m1561920","DOIUrl":"https://doi.org/10.1137/23m1561920","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1465-1491, August 2024. <br/> Abstract. A robust substructuring type preconditioner is developed for high order approximation of problem for which the element matrix takes the form [math] where [math] and [math] are the mass and stiffness matrices, respectively. A standard preconditioner for the pure stiffness matrix results in a condition number bounded by [math] where [math] blows up as [math]. It is shown that the best uniform bound in [math] that one can hope for is [math]. More precisely, we show that the upper envelope of the bound [math] is [math]. What, then, can be done to obtain a preconditioner that is robust for all [math]? The solution turns out to be a relatively minor modification of the basic substructuring algorithm of [I. Babuška et al., SIAM J. Numer. Anal., 28 (1991), pp. 624–661]: one can simply augment the preconditioner with a suitable Jacobi smoothener over the coarse grid degrees of freedom. This is shown to result in a condition number bounded by [math] where the constant is independent of [math]. Numerical results are given which shows that the simple expedient of augmentation with nodal smoothening reduces the condition number by a factor of up to two orders of magnitude.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"24 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141489608","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Kernel Machine Learning for Inverse Source and Scattering Problems 用于反源和散射问题的核机器学习
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-06-19 DOI: 10.1137/23m1597381
Shixu Meng, Bo Zhang
{"title":"A Kernel Machine Learning for Inverse Source and Scattering Problems","authors":"Shixu Meng, Bo Zhang","doi":"10.1137/23m1597381","DOIUrl":"https://doi.org/10.1137/23m1597381","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1443-1464, June 2024. <br/> Abstract. In this work we connect machine learning techniques, in particular kernel machine learning, to inverse source and scattering problems. We show the proposed kernel machine learning has demonstrated generalization capability and has a rigorous mathematical foundation. The proposed learning is based on the Mercer kernel, the reproducing kernel Hilbert space, the kernel trick, as well as the mathematical theory of inverse source and scattering theory, and the restricted Fourier integral operator. The kernel machine learns a multilayer neural network which outputs an [math]-neighborhood average of the unknown or its nonlinear transformation. We then apply the general architecture to the multifrequency inverse source problem for a fixed observation direction and the Born inverse medium scattering problem. We establish a mathematically justified kernel machine indicator with demonstrated capability in both shape identification and parameter identification, under very general assumptions on the physical unknowns. More importantly, stability estimates are established in the case of both noiseless and noisy measurement data. Of central importance is the interplay between a restricted Fourier integral operator and a corresponding Sturm–Liouville differential operator. Several numerical examples are presented to demonstrate the capability of the proposed kernel machine learning.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"44 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141430415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Finite Element Method for Hyperbolic Metamaterials with Applications for Hyperlens 双曲超材料有限元方法及其在超透镜中的应用
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-06-17 DOI: 10.1137/23m1591207
Fuhao Liu, Wei Yang, Jichun Li
{"title":"A Finite Element Method for Hyperbolic Metamaterials with Applications for Hyperlens","authors":"Fuhao Liu, Wei Yang, Jichun Li","doi":"10.1137/23m1591207","DOIUrl":"https://doi.org/10.1137/23m1591207","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1420-1442, June 2024. <br/> Abstract. In this paper, we first derive a time-dependent Maxwell’s equation model for simulating wave propagation in anisotropic dispersive media and hyperbolic metamaterials. The modeling equations are obtained by using the Drude–Lorentz model to approximate both the permittivity and permeability. Then we develop a time-domain finite element method and prove its discrete stability and optimal error estimate. This mathematical model and the proposed numerical method can be used to design effective hyperbolic superlenses by the dielectric-metal multilayer metamaterials in different frequency ranges. Extensive two-dimensional (2D) and 3D numerical results are presented to demonstrate the good performance of many 2D and 3D hyperbolic superlenses in different frequency ranges. This is the first finite element paper on solving the hyperbolic metamaterials in a time domain.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"63 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141333698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The ([math], [math])-HDG Method for the Helmholtz Equation with Large Wave Number 大波数亥姆霍兹方程的（[math], [math]）-HDG 方法
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-06-12 DOI: 10.1137/23m1562639
Bingxin Zhu, Haijun Wu
{"title":"The ([math], [math])-HDG Method for the Helmholtz Equation with Large Wave Number","authors":"Bingxin Zhu, Haijun Wu","doi":"10.1137/23m1562639","DOIUrl":"https://doi.org/10.1137/23m1562639","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1394-1419, June 2024. <br/> Abstract. In this paper, we analyze a hybridizable discontinuous Galerkin method for the Helmholtz equation with large wave number, which uses piecewise polynomials of degree of [math] to approximate the potential [math] and its traces and piecewise polynomials of degree of [math] for the flux [math]. It is proved that [math] and [math] hold under the conditions that [math] is sufficiently small and that the penalty parameter [math], where [math] is the mesh size. Numerical experiments are proposed to verify our theoretical findings and to show that the pollution error may be greatly reduced by tuning the penalty parameter.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"6 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141309134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Inverse Wave-Number-Dependent Source Problems for the Helmholtz Equation 亥姆霍兹方程的反波数依赖源问题
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-06-06 DOI: 10.1137/23m1572696
Hongxia Guo, Guanghui Hu
{"title":"Inverse Wave-Number-Dependent Source Problems for the Helmholtz Equation","authors":"Hongxia Guo, Guanghui Hu","doi":"10.1137/23m1572696","DOIUrl":"https://doi.org/10.1137/23m1572696","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1372-1393, June 2024. <br/> Abstract. This paper is concerned with the multi-frequency factorization method for imaging the support of a wave-number-dependent source function. It is supposed that the source function is given by the inverse Fourier transform of some time-dependent source with a priori given radiating period. Using the multi-frequency far-field data at a fixed observation direction, we provide a computational criterion for characterizing the smallest strip containing the support and perpendicular to the observation direction. The far-field data from sparse observation directions can be used to recover a [math]-convex polygon of the support. The inversion algorithm is proven valid even with multi-frequency near-field data in three dimensions. The connections to time-dependent inverse source problems are discussed in the near-field case. Numerical tests in both two and three dimensions are implemented to show effectiveness and feasibility of the approach. This paper provides numerical analysis for a frequency-domain approach to recover the support of an admissible class of time-dependent sources.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"431 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141287149","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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