SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1660-1686, August 2024. Abstract. We propose a finite element discretization for the steady, generalized Navier–Stokes equations for fluids with shear-dependent viscosity, completed with inhomogeneous Dirichlet boundary conditions and an inhomogeneous divergence constraint. We establish (weak) convergence of discrete solutions as well as a priori error estimates for the velocity vector field and the scalar kinematic pressure. Numerical experiments complement the theoretical findings.
{"title":"Finite Element Discretization of the Steady, Generalized Navier–Stokes Equations with Inhomogeneous Dirichlet Boundary Conditions","authors":"Julius Jeßberger, Alex Kaltenbach","doi":"10.1137/23m1607398","DOIUrl":"https://doi.org/10.1137/23m1607398","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1660-1686, August 2024. <br/> Abstract. We propose a finite element discretization for the steady, generalized Navier–Stokes equations for fluids with shear-dependent viscosity, completed with inhomogeneous Dirichlet boundary conditions and an inhomogeneous divergence constraint. We establish (weak) convergence of discrete solutions as well as a priori error estimates for the velocity vector field and the scalar kinematic pressure. Numerical experiments complement the theoretical findings.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"21 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141755090","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1638-1659, August 2024. Abstract. Maximal regularity is a kind of a priori estimate for parabolic-type equations, and it plays an important role in the theory of nonlinear differential equations. The aim of this paper is to investigate the temporally discrete counterpart of maximal regularity for the discontinuous Galerkin (DG) time-stepping method. We will establish such an estimate without logarithmic factor over a quasi-uniform temporal mesh. To show the main result, we introduce the temporally regularized Green’s function and then reduce the discrete maximal regularity to a weighted error estimate for its DG approximation. Our results would be useful for investigation of DG approximation of nonlinear parabolic problems.
{"title":"Discrete Maximal Regularity for the Discontinuous Galerkin Time-Stepping Method without Logarithmic Factor","authors":"Takahito Kashiwabara, Tomoya Kemmochi","doi":"10.1137/23m1580802","DOIUrl":"https://doi.org/10.1137/23m1580802","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1638-1659, August 2024. <br/> Abstract. Maximal regularity is a kind of a priori estimate for parabolic-type equations, and it plays an important role in the theory of nonlinear differential equations. The aim of this paper is to investigate the temporally discrete counterpart of maximal regularity for the discontinuous Galerkin (DG) time-stepping method. We will establish such an estimate without logarithmic factor over a quasi-uniform temporal mesh. To show the main result, we introduce the temporally regularized Green’s function and then reduce the discrete maximal regularity to a weighted error estimate for its DG approximation. Our results would be useful for investigation of DG approximation of nonlinear parabolic problems.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"27 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737012","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1609-1637, August 2024. Abstract. When applying the classical multistep schemes for solving differential equations, one often faces the dilemma that smaller time steps are needed with higher-order schemes, making it impractical to use high-order schemes for stiff problems. We construct in this paper a new class of BDF and implicit-explicit schemes for parabolic type equations based on the Taylor expansions at time [math] with [math] being a tunable parameter. These new schemes, with a suitable [math], allow larger time steps at higher order for stiff problems than that which is allowed with a usual higher-order scheme. For parabolic type equations, we identify an explicit uniform multiplier for the new second- to fourth-order schemes and conduct rigorously stability and error analysis by using the energy argument. We also present ample numerical examples to validate our findings.
{"title":"On a New Class of BDF and IMEX Schemes for Parabolic Type Equations","authors":"Fukeng Huang, Jie Shen","doi":"10.1137/23m1612986","DOIUrl":"https://doi.org/10.1137/23m1612986","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1609-1637, August 2024. <br/> Abstract. When applying the classical multistep schemes for solving differential equations, one often faces the dilemma that smaller time steps are needed with higher-order schemes, making it impractical to use high-order schemes for stiff problems. We construct in this paper a new class of BDF and implicit-explicit schemes for parabolic type equations based on the Taylor expansions at time [math] with [math] being a tunable parameter. These new schemes, with a suitable [math], allow larger time steps at higher order for stiff problems than that which is allowed with a usual higher-order scheme. For parabolic type equations, we identify an explicit uniform multiplier for the new second- to fourth-order schemes and conduct rigorously stability and error analysis by using the energy argument. We also present ample numerical examples to validate our findings.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"18 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141631349","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1589-1608, August 2024. 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.
{"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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1565-1588, August 2024. 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.
{"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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1515-1538, August 2024. 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.
SIAM 数值分析期刊》第 62 卷第 4 期第 1515-1538 页，2024 年 8 月。 摘要。在基于点值的函数逼近中，最小二乘法比插值法更稳定，但代价是增加了采样预算。我们的研究表明，只要样本量[math]比近似空间的维数[math]大一个恒定的乘法比，就能在预期[math]意义上实现近似误差接近最优。另一方面，对于 [math]，我们得到的插值策略的稳定系数为 [math]。所提出的采样算法是基于 [Batson, Spielman, and Srivastava, Twice-Ramanujan sparsifiers, in Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, 2009, pp.
{"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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1539-1564, August 2024. 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.
{"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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1492-1514, August 2024. 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.
{"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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1465-1491, August 2024. 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.
{"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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1443-1464, June 2024. 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.
{"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}
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