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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
Bilinear Optimal Control for the Fractional Laplacian: Analysis and Discretization 分数拉普拉奇的双线性最优控制：分析与离散化
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-06-04 DOI: 10.1137/23m154947x
Francisco Bersetche, Francisco Fuica, Enrique Otárola, Daniel Quero
{"title":"Bilinear Optimal Control for the Fractional Laplacian: Analysis and Discretization","authors":"Francisco Bersetche, Francisco Fuica, Enrique Otárola, Daniel Quero","doi":"10.1137/23m154947x","DOIUrl":"https://doi.org/10.1137/23m154947x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1344-1371, June 2024. <br/> Abstract. We adopt the integral definition of the fractional Laplace operator and study an optimal control problem on Lipschitz domains that involves a fractional elliptic PDE as the state equation and a control variable that enters the state equation as a coefficient; pointwise constraints on the control variable are considered as well. We establish the existence of optimal solutions and analyze first- and necessary and sufficient second-order optimality conditions. Regularity estimates for optimal variables are also analyzed. We develop two finite element discretization strategies: a semidiscrete scheme in which the control variable is not discretized and a fully discrete scheme in which the control variable is discretized with piecewise constant functions. For both schemes, we analyze the convergence properties of discretizations and derive error estimates.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"54 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141320015","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
Error Bounds for Discrete Minimizers of the Ginzburg–Landau Energy in the High-[math] Regime 高[数学]区金兹堡-朗道能量离散最小值的误差边界
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-05-30 DOI: 10.1137/23m1560938
Benjamin Dörich, Patrick Henning
{"title":"Error Bounds for Discrete Minimizers of the Ginzburg–Landau Energy in the High-[math] Regime","authors":"Benjamin Dörich, Patrick Henning","doi":"10.1137/23m1560938","DOIUrl":"https://doi.org/10.1137/23m1560938","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1313-1343, June 2024. <br/> Abstract. In this work, we study discrete minimizers of the Ginzburg–Landau energy in finite element spaces. Special focus is given to the influence of the Ginzburg–Landau parameter [math]. This parameter is of physical interest as large values can trigger the appearance of vortex lattices. Since the vortices have to be resolved on sufficiently fine computational meshes, it is important to translate the size of [math] into a mesh resolution condition, which can be done through error estimates that are explicit with respect to [math] and the spatial mesh width [math]. For that, we first work in an abstract framework for a general class of discrete spaces, where we present convergence results in a problem-adapted [math]-weighted norm. Afterward we apply our findings to Lagrangian finite elements and a particular generalized finite element construction. In numerical experiments we confirm that our derived [math]- and [math]-error estimates are indeed optimal in [math] and [math].","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"66 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141165169","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
Solving PDEs with Incomplete Information 用不完全信息求解 PDEs
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-05-30 DOI: 10.1137/23m1546671
Peter Binev, Andrea Bonito, Albert Cohen, Wolfgang Dahmen, Ronald DeVore, Guergana Petrova
{"title":"Solving PDEs with Incomplete Information","authors":"Peter Binev, Andrea Bonito, Albert Cohen, Wolfgang Dahmen, Ronald DeVore, Guergana Petrova","doi":"10.1137/23m1546671","DOIUrl":"https://doi.org/10.1137/23m1546671","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1278-1312, June 2024. <br/> Abstract. We consider the problem of numerically approximating the solutions to a partial differential equation (PDE) when there is insufficient information to determine a unique solution. Our main example is the Poisson boundary value problem, when the boundary data is unknown and instead one observes finitely many linear measurements of the solution. We view this setting as an optimal recovery problem and develop theory and numerical algorithms for its solution. The main vehicle employed is the derivation and approximation of the Riesz representers of these functionals with respect to relevant Hilbert spaces of harmonic functions.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"24 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141182385","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
On Bernoulli’s Method 关于伯努利方法
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-05-24 DOI: 10.1137/22m1528501
Tamás Dózsa, Ferenc Schipp, Alexandros Soumelidis
{"title":"On Bernoulli’s Method","authors":"Tamás Dózsa, Ferenc Schipp, Alexandros Soumelidis","doi":"10.1137/22m1528501","DOIUrl":"https://doi.org/10.1137/22m1528501","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1259-1277, June 2024. <br/> Abstract. We generalize Bernoulli’s classical method for finding poles of rational functions using the rational orthogonal Malmquist–Takenaka system. We show that our approach overcomes the limitations of previous methods, especially their dependence on the existence of a so-called dominant pole, while significantly simplifying the required calculations. A description of the identifiable poles is provided, as well as an iterative algorithm that can be applied to find every pole of a rational function. We discuss automatic parameter choice for the proposed algorithm and demonstrate its effectiveness through numerical examples.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"48 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141096724","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
Contraction and Convergence Rates for Discretized Kinetic Langevin Dynamics 离散动能朗万动力学的收缩与收敛速率
IF 2.9 2区 数学
SIAM Journal on Numerical Analysis Pub Date : 2024-05-22 DOI: 10.1137/23m1556289
Benedict J. Leimkuhler, Daniel Paulin, Peter A. Whalley
{"title":"Contraction and Convergence Rates for Discretized Kinetic Langevin Dynamics","authors":"Benedict J. Leimkuhler, Daniel Paulin, Peter A. Whalley","doi":"10.1137/23m1556289","DOIUrl":"https://doi.org/10.1137/23m1556289","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1226-1258, June 2024. <br/>Abstract. We provide a framework to analyze the convergence of discretized kinetic Langevin dynamics for [math]-[math]Lipschitz, [math]-convex potentials. Our approach gives convergence rates of [math], with explicit step size restrictions, which are of the same order as the stability threshold for Gaussian targets and are valid for a large interval of the friction parameter. We apply this methodology to various integration schemes which are popular in the molecular dynamics and machine learning communities. Further, we introduce the property “[math]-limit convergent” to characterize underdamped Langevin schemes that converge to overdamped dynamics in the high-friction limit and which have step size restrictions that are independent of the friction parameter; we show that this property is not generic by exhibiting methods from both the class and its complement. Finally, we provide asymptotic bias estimates for the BAOAB scheme, which remain accurate in the high-friction limit by comparison to a modified stochastic dynamics which preserves the invariant measure.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"41 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141085304","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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