SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1949-A1971, June 2024. Abstract. The ensemble Gaussian mixture filter (EnGMF) combines the simplicity and power of Gaussian mixture models with the provable convergence and power of particle filters. The quality of the EnGMF heavily depends on the choice of covariance matrix in each Gaussian mixture. This work extends the EnGMF to an adaptive choice of covariance based on the parameterized estimates of the sample covariance matrix. Through the use of the expectation maximization algorithm, optimal choices of the covariance matrix parameters are computed in an online fashion. Numerical experiments on the Lorenz ’63 equations show that the proposed methodology converges to classical results known in particle filtering. Further numerical results with more advanced choices of covariance parameterization and the medium-size Lorenz ’96 equations show that the proposed approach can perform significantly better than the standard EnGMF and other classical data assimilation algorithms.
{"title":"An Adaptive Covariance Parameterization Technique for the Ensemble Gaussian Mixture Filter","authors":"Andrey A. Popov, Renato Zanetti","doi":"10.1137/22m1544312","DOIUrl":"https://doi.org/10.1137/22m1544312","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1949-A1971, June 2024. <br/> Abstract. The ensemble Gaussian mixture filter (EnGMF) combines the simplicity and power of Gaussian mixture models with the provable convergence and power of particle filters. The quality of the EnGMF heavily depends on the choice of covariance matrix in each Gaussian mixture. This work extends the EnGMF to an adaptive choice of covariance based on the parameterized estimates of the sample covariance matrix. Through the use of the expectation maximization algorithm, optimal choices of the covariance matrix parameters are computed in an online fashion. Numerical experiments on the Lorenz ’63 equations show that the proposed methodology converges to classical results known in particle filtering. Further numerical results with more advanced choices of covariance parameterization and the medium-size Lorenz ’96 equations show that the proposed approach can perform significantly better than the standard EnGMF and other classical data assimilation algorithms.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141517780","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 Scientific Computing, Ahead of Print. Abstract. We present a deep learning–based iterative approach to solve the discrete heterogeneous Helmholtz equation for high wavenumbers. Combining classical iterative multigrid solvers and convolutional neural networks (CNNs) via preconditioning, we obtain a faster, learned neural solver that scales better than a standard multigrid solver. Our approach offers three main contributions over previous neural methods of this kind. First, we construct a multilevel U-Net-like encoder-solver CNN with an implicit layer on the coarsest grid of the U-Net, where convolution kernels are inverted. This alleviates the field of view problem in CNNs and allows better scalability. Second, we improve upon the previous CNN preconditioner in terms of the number of parameters, computation time, and convergence rates. Third, we propose a multiscale training approach that enables the network to scale to problems of previously unseen dimensions while still maintaining a reasonable training procedure. Our encoder-solver architecture can be used to generalize over different slowness models of various difficulties and is efficient at solving for many right-hand sides per slowness model. We demonstrate the benefits of our novel architecture with numerical experiments on various heterogeneous two-dimensional problems at high wavenumbers.
{"title":"Multigrid-Augmented Deep Learning Preconditioners for the Helmholtz Equation Using Compact Implicit Layers","authors":"Bar Lerer, Ido Ben-Yair, Eran Treister","doi":"10.1137/23m1583302","DOIUrl":"https://doi.org/10.1137/23m1583302","url":null,"abstract":"SIAM Journal on Scientific Computing, Ahead of Print. <br/> Abstract. We present a deep learning–based iterative approach to solve the discrete heterogeneous Helmholtz equation for high wavenumbers. Combining classical iterative multigrid solvers and convolutional neural networks (CNNs) via preconditioning, we obtain a faster, learned neural solver that scales better than a standard multigrid solver. Our approach offers three main contributions over previous neural methods of this kind. First, we construct a multilevel U-Net-like encoder-solver CNN with an implicit layer on the coarsest grid of the U-Net, where convolution kernels are inverted. This alleviates the field of view problem in CNNs and allows better scalability. Second, we improve upon the previous CNN preconditioner in terms of the number of parameters, computation time, and convergence rates. Third, we propose a multiscale training approach that enables the network to scale to problems of previously unseen dimensions while still maintaining a reasonable training procedure. Our encoder-solver architecture can be used to generalize over different slowness models of various difficulties and is efficient at solving for many right-hand sides per slowness model. We demonstrate the benefits of our novel architecture with numerical experiments on various heterogeneous two-dimensional problems at high wavenumbers.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141517781","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 Scientific Computing, Volume 46, Issue 3, Page A1903-A1922, June 2024. Abstract. This work considers the convergence of GMRES for nonsingular problems. GMRES is interpreted as the generalized conjugate residual method which allows for simple proofs of the convergence estimates. Preconditioning and weighted norms within GMRES are considered. The objective is to provide a way of choosing the preconditioner and GMRES norm that ensures fast convergence. The main focus of the article is on Hermitian preconditioning (even for non-Hermitian problems). It is proposed to choose a Hermitian preconditioner [math] and to apply GMRES in the inner product induced by [math]. If, moreover, the problem matrix [math] is positive definite, then a new convergence bound is proved that depends only on how well [math] preconditions the Hermitian part of [math], and on how non-Hermitian [math] is. In particular, if a scalable preconditioner is known for the Hermitian part of [math], then the proposed method is also scalable. This result is illustrated numerically.
{"title":"Hermitian Preconditioning for a Class of Non-Hermitian Linear Systems","authors":"Nicole Spillane","doi":"10.1137/23m1559026","DOIUrl":"https://doi.org/10.1137/23m1559026","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1903-A1922, June 2024. <br/> Abstract. This work considers the convergence of GMRES for nonsingular problems. GMRES is interpreted as the generalized conjugate residual method which allows for simple proofs of the convergence estimates. Preconditioning and weighted norms within GMRES are considered. The objective is to provide a way of choosing the preconditioner and GMRES norm that ensures fast convergence. The main focus of the article is on Hermitian preconditioning (even for non-Hermitian problems). It is proposed to choose a Hermitian preconditioner [math] and to apply GMRES in the inner product induced by [math]. If, moreover, the problem matrix [math] is positive definite, then a new convergence bound is proved that depends only on how well [math] preconditions the Hermitian part of [math], and on how non-Hermitian [math] is. In particular, if a scalable preconditioner is known for the Hermitian part of [math], then the proposed method is also scalable. This result is illustrated numerically.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141188083","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 Scientific Computing, Volume 46, Issue 3, Page A1879-A1902, June 2024. Abstract. While extracting information from data with machine learning plays an increasingly important role, physical laws and other first principles continue to provide critical insights about systems and processes of interest in science and engineering. This work introduces a method that infers models from data with physical insights encoded in the form of structure and that minimizes the model order so that the training data are fitted well while redundant degrees of freedom without conditions and sufficient data to fix them are automatically eliminated. The models are formulated via solution matrices of specific instances of generalized Sylvester equations that enforce interpolation of the training data and relate the model order to the rank of the solution matrices. The proposed method numerically solves the Sylvester equations for minimal-rank solutions and so obtains models of low order. Numerical experiments demonstrate that the combination of structure preservation and rank minimization leads to accurate models with orders of magnitude fewer degrees of freedom than models of comparable prediction quality that are learned with structure preservation alone.
{"title":"Rank-Minimizing and Structured Model Inference","authors":"Pawan Goyal, Benjamin Peherstorfer, Peter Benner","doi":"10.1137/23m1554308","DOIUrl":"https://doi.org/10.1137/23m1554308","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1879-A1902, June 2024. <br/> Abstract. While extracting information from data with machine learning plays an increasingly important role, physical laws and other first principles continue to provide critical insights about systems and processes of interest in science and engineering. This work introduces a method that infers models from data with physical insights encoded in the form of structure and that minimizes the model order so that the training data are fitted well while redundant degrees of freedom without conditions and sufficient data to fix them are automatically eliminated. The models are formulated via solution matrices of specific instances of generalized Sylvester equations that enforce interpolation of the training data and relate the model order to the rank of the solution matrices. The proposed method numerically solves the Sylvester equations for minimal-rank solutions and so obtains models of low order. Numerical experiments demonstrate that the combination of structure preservation and rank minimization leads to accurate models with orders of magnitude fewer degrees of freedom than models of comparable prediction quality that are learned with structure preservation alone.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141168810","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 Scientific Computing, Volume 46, Issue 3, Page A1850-A1878, June 2024. Abstract. For a nonlinear dynamical system that depends on parameters, this paper introduces a novel tensorial reduced-order model (TROM). The reduced model is projection-based, and for systems with no parameters involved, it resembles proper orthogonal decomposition (POD) combined with the discrete empirical interpolation method (DEIM). For parametric systems, TROM employs low-rank tensor approximations in place of truncated SVD, a key dimension-reduction technique in POD with DEIM. Three popular low-rank tensor compression formats are considered for this purpose: canonical polyadic, Tucker, and tensor train. The use of multilinear algebra tools allows the incorporation of information about the parameter dependence of the system into the reduced model and leads to a POD-DEIM type ROM that (i) is parameter-specific (localized) and predicts the system dynamics for out-of-training set (unseen) parameter values, (ii) mitigates the adverse effects of high parameter space dimension, (iii) has online computational costs that depend only on tensor compression ranks but not on the full-order model size, and (iv) achieves lower reduced space dimensions compared to the conventional POD-DEIM ROM. This paper explains the method, analyzes its prediction power, and assesses its performance for two specific parameter-dependent nonlinear dynamical systems.
{"title":"Tensorial Parametric Model Order Reduction of Nonlinear Dynamical Systems","authors":"Alexander V. Mamonov, Maxim A. Olshanskii","doi":"10.1137/23m1553789","DOIUrl":"https://doi.org/10.1137/23m1553789","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1850-A1878, June 2024. <br/> Abstract. For a nonlinear dynamical system that depends on parameters, this paper introduces a novel tensorial reduced-order model (TROM). The reduced model is projection-based, and for systems with no parameters involved, it resembles proper orthogonal decomposition (POD) combined with the discrete empirical interpolation method (DEIM). For parametric systems, TROM employs low-rank tensor approximations in place of truncated SVD, a key dimension-reduction technique in POD with DEIM. Three popular low-rank tensor compression formats are considered for this purpose: canonical polyadic, Tucker, and tensor train. The use of multilinear algebra tools allows the incorporation of information about the parameter dependence of the system into the reduced model and leads to a POD-DEIM type ROM that (i) is parameter-specific (localized) and predicts the system dynamics for out-of-training set (unseen) parameter values, (ii) mitigates the adverse effects of high parameter space dimension, (iii) has online computational costs that depend only on tensor compression ranks but not on the full-order model size, and (iv) achieves lower reduced space dimensions compared to the conventional POD-DEIM ROM. This paper explains the method, analyzes its prediction power, and assesses its performance for two specific parameter-dependent nonlinear dynamical systems.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141187801","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}
Thomas Frachon, Peter Hansbo, Erik Nilsson, Sara Zahedi
SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1793-A1820, June 2024. Abstract. We study cut finite element discretizations of a Darcy interface problem based on the mixed finite element pairs [math], [math]. Here [math] is the space of discontinuous polynomial functions of degree less than or equal to [math] and [math] is the Raviart–Thomas space. We show that the standard ghost penalty stabilization, often added in the weak forms of cut finite element methods for stability and control of the condition number of the resulting linear system matrix, destroys the divergence-free property of the considered element pairs. Therefore, we propose new stabilization terms for the pressure and show that we recover the optimal approximation of the divergence without losing control of the condition number of the linear system matrix. We prove that with the new stabilization term the proposed cut finite element discretization results in pointwise divergence-free approximations of solenoidal velocity fields. We derive a priori error estimates for the proposed unfitted finite element discretization based on [math], [math]. In addition, by decomposing the computational mesh into macroelements and applying ghost penalty terms only on interior edges of macroelements, stabilization is applied very restrictively and active only where needed. Numerical experiments with element pairs [math], [math], and [math] (where [math] is the Brezzi–Douglas–Marini space) indicate that with the new method we have (1) optimal rates of convergence of the approximate velocity and pressure; (2) well-posed linear systems where the condition number of the system matrix scales as it does for fitted finite element discretizations; (3) optimal rates of convergence of the approximate divergence with pointwise divergence-free approximations of solenoidal velocity fields. All three properties hold independently of how the interface is positioned relative to the computational mesh. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/CutFEM/CutFEM-Library and in the supplementary materials (CutFEM-Library-master.zip [30.5MB]).
{"title":"A Divergence Preserving Cut Finite Element Method for Darcy Flow","authors":"Thomas Frachon, Peter Hansbo, Erik Nilsson, Sara Zahedi","doi":"10.1137/22m149702x","DOIUrl":"https://doi.org/10.1137/22m149702x","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1793-A1820, June 2024. <br/>Abstract. We study cut finite element discretizations of a Darcy interface problem based on the mixed finite element pairs [math], [math]. Here [math] is the space of discontinuous polynomial functions of degree less than or equal to [math] and [math] is the Raviart–Thomas space. We show that the standard ghost penalty stabilization, often added in the weak forms of cut finite element methods for stability and control of the condition number of the resulting linear system matrix, destroys the divergence-free property of the considered element pairs. Therefore, we propose new stabilization terms for the pressure and show that we recover the optimal approximation of the divergence without losing control of the condition number of the linear system matrix. We prove that with the new stabilization term the proposed cut finite element discretization results in pointwise divergence-free approximations of solenoidal velocity fields. We derive a priori error estimates for the proposed unfitted finite element discretization based on [math], [math]. In addition, by decomposing the computational mesh into macroelements and applying ghost penalty terms only on interior edges of macroelements, stabilization is applied very restrictively and active only where needed. Numerical experiments with element pairs [math], [math], and [math] (where [math] is the Brezzi–Douglas–Marini space) indicate that with the new method we have (1) optimal rates of convergence of the approximate velocity and pressure; (2) well-posed linear systems where the condition number of the system matrix scales as it does for fitted finite element discretizations; (3) optimal rates of convergence of the approximate divergence with pointwise divergence-free approximations of solenoidal velocity fields. All three properties hold independently of how the interface is positioned relative to the computational mesh. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/CutFEM/CutFEM-Library and in the supplementary materials (CutFEM-Library-master.zip [30.5MB]).","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141150370","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}
Golo A. Wimmer, Ben S. Southworth, Thomas J. Gregory, Xian-Zhu Tang
SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1821-A1849, June 2024. Abstract. We present a novel solver technique for the anisotropic heat flux equation, aimed at the high level of anisotropy seen in magnetic confinement fusion plasmas. Such problems pose two major challenges: (i) discretization accuracy and (ii) efficient implicit linear solvers. We simultaneously address each of these challenges by constructing a new finite element discretization with excellent accuracy properties, tailored to a novel solver approach based on algebraic multigrid (AMG) methods designed for advective operators. We pose the problem in a mixed formulation, introducing the directional temperature gradient as an auxiliary variable. The temperature and auxiliary fields are discretized in a scalar discontinuous Galerkin space with upwinding principles used for discretizations of advection. We demonstrate the proposed discretization’s superior accuracy over other discretizations of anisotropic heat flux, achieving error [math] smaller for anisotropy ratio of [math], for closed field lines. The block matrix system is reordered and solved in an approach where the two advection operators are inverted using AMG solvers based on approximate ideal restriction, which is particularly efficient for upwind discontinuous Galerkin discretizations of advection. To ensure that the advection operators are nonsingular, in this paper we restrict ourselves to considering open (acyclic) magnetic field lines for the linear solvers. We demonstrate fast convergence of the proposed iterative solver in highly anisotropic regimes where other diffusion-based AMG methods fail.
{"title":"A Fast Algebraic Multigrid Solver and Accurate Discretization for Highly Anisotropic Heat Flux I: Open Field Lines","authors":"Golo A. Wimmer, Ben S. Southworth, Thomas J. Gregory, Xian-Zhu Tang","doi":"10.1137/23m155918x","DOIUrl":"https://doi.org/10.1137/23m155918x","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1821-A1849, June 2024. <br/> Abstract. We present a novel solver technique for the anisotropic heat flux equation, aimed at the high level of anisotropy seen in magnetic confinement fusion plasmas. Such problems pose two major challenges: (i) discretization accuracy and (ii) efficient implicit linear solvers. We simultaneously address each of these challenges by constructing a new finite element discretization with excellent accuracy properties, tailored to a novel solver approach based on algebraic multigrid (AMG) methods designed for advective operators. We pose the problem in a mixed formulation, introducing the directional temperature gradient as an auxiliary variable. The temperature and auxiliary fields are discretized in a scalar discontinuous Galerkin space with upwinding principles used for discretizations of advection. We demonstrate the proposed discretization’s superior accuracy over other discretizations of anisotropic heat flux, achieving error [math] smaller for anisotropy ratio of [math], for closed field lines. The block matrix system is reordered and solved in an approach where the two advection operators are inverted using AMG solvers based on approximate ideal restriction, which is particularly efficient for upwind discontinuous Galerkin discretizations of advection. To ensure that the advection operators are nonsingular, in this paper we restrict ourselves to considering open (acyclic) magnetic field lines for the linear solvers. We demonstrate fast convergence of the proposed iterative solver in highly anisotropic regimes where other diffusion-based AMG methods fail.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141150418","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}
{"title":"A New Preconditioned Nonlinear Conjugate Gradient Method in Real Arithmetic for Computing the Ground States of Rotational Bose–Einstein Condensate","authors":"Tianqi Zhang, Fei Xue","doi":"10.1137/23m1590317","DOIUrl":"https://doi.org/10.1137/23m1590317","url":null,"abstract":"","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141103497","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 Scientific Computing, Volume 46, Issue 3, Page A1747-A1763, June 2024. Abstract. A randomized algorithm for computing a compressed representation of a given rank-structured matrix [math] is presented. The algorithm interacts with [math] only through its action on vectors. Specifically, it draws two tall thin matrices [math] from a suitable distribution, and then reconstructs [math] from the information contained in the set [math]. For the specific case of a “Hierarchically Block Separable (HBS)” matrix (a.k.a. Hierarchically Semi-Separable matrix) of block rank [math], the number of samples [math] required satisfies [math], with [math] being representative. While a number of randomized algorithms for compressing rank-structured matrices have previously been published, the current algorithm appears to be the first that is both of truly linear complexity (no [math] factors in the complexity bound) and fully “black box” in the sense that no matrix entry evaluation is required. Further, all samples can be extracted in parallel, enabling the algorithm to work in a “streaming” or “single view” mode.
{"title":"Linear-Complexity Black-Box Randomized Compression of Rank-Structured Matrices","authors":"James Levitt, Per-Gunnar Martinsson","doi":"10.1137/22m1528574","DOIUrl":"https://doi.org/10.1137/22m1528574","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1747-A1763, June 2024. <br/> Abstract. A randomized algorithm for computing a compressed representation of a given rank-structured matrix [math] is presented. The algorithm interacts with [math] only through its action on vectors. Specifically, it draws two tall thin matrices [math] from a suitable distribution, and then reconstructs [math] from the information contained in the set [math]. For the specific case of a “Hierarchically Block Separable (HBS)” matrix (a.k.a. Hierarchically Semi-Separable matrix) of block rank [math], the number of samples [math] required satisfies [math], with [math] being representative. While a number of randomized algorithms for compressing rank-structured matrices have previously been published, the current algorithm appears to be the first that is both of truly linear complexity (no [math] factors in the complexity bound) and fully “black box” in the sense that no matrix entry evaluation is required. Further, all samples can be extracted in parallel, enabling the algorithm to work in a “streaming” or “single view” mode.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141063181","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 Scientific Computing, Volume 46, Issue 3, Page A1714-A1746, June 2024. Abstract. Tensor wheel (TW) decomposition is an elegant compromise of the popular tensor ring decomposition and fully connected tensor network decomposition, and it has many applications. In this work, we investigate the computation of this decomposition. Three randomized algorithms based on random sampling or random projection are proposed. Specifically, by defining a new tensor product called the subwheel product, the structures of the coefficient matrices of the alternating least squares subproblems from the minimization problem of TW decomposition are first figured out. Then, using the structures and the properties of the subwheel product, a random sampling algorithm based on leverage sampling and two random projection algorithms respectively based on Kronecker subsampled randomized Fourier transform and TensorSketch are derived. These algorithms can implement the sampling and projection on TW factors and hence can avoid forming the full coefficient matrices of subproblems. We present the complexity analysis and numerical performance on synthetic data, real data, and image reconstruction for our algorithms. Experimental results show that, compared with the deterministic algorithm in the literature, they need much less computing time while achieving similar accuracy and reconstruction effect. We also apply the proposed algorithms to tensor completion and find that the sampling-based algorithm always has excellent performance and the projection-based algorithms behave well when the sampling rate is higher than 50%.
{"title":"Randomized Tensor Wheel Decomposition","authors":"Mengyu Wang, Yajie Yu, Hanyu Li","doi":"10.1137/23m1583934","DOIUrl":"https://doi.org/10.1137/23m1583934","url":null,"abstract":"SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1714-A1746, June 2024. <br/> Abstract. Tensor wheel (TW) decomposition is an elegant compromise of the popular tensor ring decomposition and fully connected tensor network decomposition, and it has many applications. In this work, we investigate the computation of this decomposition. Three randomized algorithms based on random sampling or random projection are proposed. Specifically, by defining a new tensor product called the subwheel product, the structures of the coefficient matrices of the alternating least squares subproblems from the minimization problem of TW decomposition are first figured out. Then, using the structures and the properties of the subwheel product, a random sampling algorithm based on leverage sampling and two random projection algorithms respectively based on Kronecker subsampled randomized Fourier transform and TensorSketch are derived. These algorithms can implement the sampling and projection on TW factors and hence can avoid forming the full coefficient matrices of subproblems. We present the complexity analysis and numerical performance on synthetic data, real data, and image reconstruction for our algorithms. Experimental results show that, compared with the deterministic algorithm in the literature, they need much less computing time while achieving similar accuracy and reconstruction effect. We also apply the proposed algorithms to tensor completion and find that the sampling-based algorithm always has excellent performance and the projection-based algorithms behave well when the sampling rate is higher than 50%.","PeriodicalId":49526,"journal":{"name":"SIAM Journal on Scientific Computing","volume":null,"pages":null},"PeriodicalIF":3.1,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141257897","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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