In this paper, we consider the tensor absolute value equations (TAVEs). When one tensor is row diagonal with odd order, we show that the TAVEs can be reduced to an algebraic equation; when it is row diagonal and nonsingular with even order, we prove that the TAVEs is equivalent to a polynomial complementary problem. When no tensor is row diagonal, we formulate the TAVEs equivalently as polynomial optimization problems in two different ways. Each of them can be solved by Lasserre’s hierarchy of semidefinite relaxations. The finite convergence properties are also discussed. Numerical experiments show the efficiency of the proposed methods.
{"title":"Semidefinite Relaxation Methods for Tensor Absolute Value Equations","authors":"Anwa Zhou, Kun Liu, Jinyan Fan","doi":"10.1137/22m1539137","DOIUrl":"https://doi.org/10.1137/22m1539137","url":null,"abstract":"In this paper, we consider the tensor absolute value equations (TAVEs). When one tensor is row diagonal with odd order, we show that the TAVEs can be reduced to an algebraic equation; when it is row diagonal and nonsingular with even order, we prove that the TAVEs is equivalent to a polynomial complementary problem. When no tensor is row diagonal, we formulate the TAVEs equivalently as polynomial optimization problems in two different ways. Each of them can be solved by Lasserre’s hierarchy of semidefinite relaxations. The finite convergence properties are also discussed. Numerical experiments show the efficiency of the proposed methods.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135475954","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}
Let $A$ be a real $(ntimes n)$-matrix. The piecewise linear equation system $z-Avert zvert =b$ is called an absolute value equation (AVE). It is well known to be uniquely solvable for all $binmathbb R^n$ if and only if a quantity called the sign-real spectral radius of $A$ is smaller than one. We construct a quantity similar to the sign-real spectral radius that we call the aligning spectral radius $rho^a$ of $A$. We prove that the AVE has mapping degree $1$ and thus an odd number of solutions for all $binmathbb R^n$ if the aligning spectral radius of $A$ is smaller than one. Under mild genericity assumptions on $A$ we also manage to prove a converse result. Structural properties of the aligning spectral radius are investigated. Due to the equivalence of the AVE to the linear complementarity problem, a side effect of our investigation are new sufficient and necessary conditions for $Q$-matrices.
设A是一个实数(n * n)矩阵。分段线性方程组$z- a vert zvert =b$称为绝对值方程(AVE)。众所周知，对于所有$binmathbb R^n$是唯一可解的，当且仅当一个称为$ a $的符号实谱半径的量小于1。我们构造一个类似于符号实谱半径的量，我们称之为对准谱半径$rho^a$ ($ a$)。我们证明了AVE具有映射度$1$，因此如果$A$的对准谱半径小于1，则所有$binmathbb R^n$都有奇数个解。在$A$的温和泛型假设下，我们还设法证明了一个相反的结果。研究了对准光谱半径的结构特性。由于AVE与线性互补问题的等价性，我们研究的一个副作用是$Q$-矩阵的新的充要条件。
{"title":"Generalized Perron Roots and Solvability of the Absolute Value Equation","authors":"Manuel Radons","doi":"10.1137/22m1517184","DOIUrl":"https://doi.org/10.1137/22m1517184","url":null,"abstract":"Let $A$ be a real $(ntimes n)$-matrix. The piecewise linear equation system $z-Avert zvert =b$ is called an absolute value equation (AVE). It is well known to be uniquely solvable for all $binmathbb R^n$ if and only if a quantity called the sign-real spectral radius of $A$ is smaller than one. We construct a quantity similar to the sign-real spectral radius that we call the aligning spectral radius $rho^a$ of $A$. We prove that the AVE has mapping degree $1$ and thus an odd number of solutions for all $binmathbb R^n$ if the aligning spectral radius of $A$ is smaller than one. Under mild genericity assumptions on $A$ we also manage to prove a converse result. Structural properties of the aligning spectral radius are investigated. Due to the equivalence of the AVE to the linear complementarity problem, a side effect of our investigation are new sufficient and necessary conditions for $Q$-matrices.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136018114","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}
Solving polynomial eigenvalue problems with eigenvector nonlinearities (PEPv) is an interesting computational challenge, outside the reach of the well-developed methods for nonlinear eigenvalue problems. We present a natural generalization of these methods which leads to a contour integration approach for computing all eigenvalues of a PEPv in a compact region of the complex plane. Our methods can be used to solve any suitably generic system of polynomial or rational function equations.
{"title":"Contour Integration for Eigenvector Nonlinearities","authors":"Rob Claes, Karl Meerbergen, Simon Telen","doi":"10.1137/22m1497985","DOIUrl":"https://doi.org/10.1137/22m1497985","url":null,"abstract":"Solving polynomial eigenvalue problems with eigenvector nonlinearities (PEPv) is an interesting computational challenge, outside the reach of the well-developed methods for nonlinear eigenvalue problems. We present a natural generalization of these methods which leads to a contour integration approach for computing all eigenvalues of a PEPv in a compact region of the complex plane. Our methods can be used to solve any suitably generic system of polynomial or rational function equations.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136067415","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}
Michiel E. Hochstenbach, Christian Mehl, Bor Plestenjak
Generalized eigenvalue problems involving a singular pencil may be very challenging to solve, both with respect to accuracy and efficiency. While Part I presented a rank-completing addition to a singular pencil, we now develop two alternative methods. The first technique is based on a projection onto subspaces with dimension equal to the normal rank of the pencil while the second approach exploits an augmented matrix pencil. The projection approach seems to be the most attractive version for generic singular pencils because of its efficiency, while the augmented pencil approach may be suitable for applications where a linear system with the augmented pencil can be solved efficiently.
{"title":"Solving Singular Generalized Eigenvalue Problems. Part II: Projection and Augmentation","authors":"Michiel E. Hochstenbach, Christian Mehl, Bor Plestenjak","doi":"10.1137/22m1513174","DOIUrl":"https://doi.org/10.1137/22m1513174","url":null,"abstract":"Generalized eigenvalue problems involving a singular pencil may be very challenging to solve, both with respect to accuracy and efficiency. While Part I presented a rank-completing addition to a singular pencil, we now develop two alternative methods. The first technique is based on a projection onto subspaces with dimension equal to the normal rank of the pencil while the second approach exploits an augmented matrix pencil. The projection approach seems to be the most attractive version for generic singular pencils because of its efficiency, while the augmented pencil approach may be suitable for applications where a linear system with the augmented pencil can be solved efficiently.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135113202","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":"Robust Recovery of Low-Rank Matrices and Low-Tubal-Rank Tensors from Noisy Ketches","authors":"Anna Ma, Dominik Stöger, Yizhe Zhu","doi":"10.1137/22m150071x","DOIUrl":"https://doi.org/10.1137/22m150071x","url":null,"abstract":"","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135617657","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}
We consider the problem of iteratively solving large and sparse double saddle-point systems arising from the stationary Stokes–Darcy equations in two dimensions, discretized by the marker-and-cell finite difference method. We analyze the eigenvalue distribution of a few ideal block preconditioners. We then derive practical preconditioners that are based on approximations of Schur complements that arise in a block decomposition of the double saddle-point matrix. We show that including the interface conditions in the preconditioners is key in the pursuit of scalability. Numerical results show good convergence behavior of our preconditioned GMRES solver and demonstrate robustness of the proposed preconditioner with respect to the physical parameters of the problem.
{"title":"Block Preconditioners for the Marker-and-Cell Discretization of the Stokes–Darcy Equations","authors":"Chen Greif, Yunhui He","doi":"10.1137/22m1518384","DOIUrl":"https://doi.org/10.1137/22m1518384","url":null,"abstract":"We consider the problem of iteratively solving large and sparse double saddle-point systems arising from the stationary Stokes–Darcy equations in two dimensions, discretized by the marker-and-cell finite difference method. We analyze the eigenvalue distribution of a few ideal block preconditioners. We then derive practical preconditioners that are based on approximations of Schur complements that arise in a block decomposition of the double saddle-point matrix. We show that including the interface conditions in the preconditioners is key in the pursuit of scalability. Numerical results show good convergence behavior of our preconditioned GMRES solver and demonstrate robustness of the proposed preconditioner with respect to the physical parameters of the problem.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135884878","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}
H. Gernandt, F. Martínez Pería, F. Philipp, C. Trunk
The relationship between linear relations and matrix pencils is investigated. Given a linear relation, we introduce its Weyr characteristic. If the linear relation is the range (or the kernel) representation of a given matrix pencil, we show that there is a correspondence between this characteristic and the Kronecker canonical form of the pencil. This relationship is exploited to obtain estimations on the invariant characteristics of matrix pencils under rank-one perturbations.
{"title":"On Characteristic Invariants of Matrix Pencils and Linear Relations","authors":"H. Gernandt, F. Martínez Pería, F. Philipp, C. Trunk","doi":"10.1137/22m1535449","DOIUrl":"https://doi.org/10.1137/22m1535449","url":null,"abstract":"The relationship between linear relations and matrix pencils is investigated. Given a linear relation, we introduce its Weyr characteristic. If the linear relation is the range (or the kernel) representation of a given matrix pencil, we show that there is a correspondence between this characteristic and the Kronecker canonical form of the pencil. This relationship is exploited to obtain estimations on the invariant characteristics of matrix pencils under rank-one perturbations.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135995980","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 Preconditioned MINRES Method for Optimal Control of Wave Equations and its Asymptotic Spectral Distribution Theory","authors":"Sean Hon, Jiamei Dong, Stefano Serra-Capizzano","doi":"10.1137/23m1547251","DOIUrl":"https://doi.org/10.1137/23m1547251","url":null,"abstract":"","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136112542","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}
Let be invertible, unknown, and given. We are interested in approximate solutions: vectors such that is small. We prove that for all , there is a composition of orthogonal projections onto the hyperplanes generated by the rows of , where , which maps the origin to a vector satisfying . We note that this upper bound on is independent of the matrix . This procedure is stable in the sense that . The existence proof is based on a probabilistically refined analysis of the randomized Kaczmarz method, which seems to achieve this rate when solving for with high likelihood. We also prove a general version for matrices with and full rank.
{"title":"Approximate Solutions of Linear Systems at a Universal Rate","authors":"Stefan Steinerberger","doi":"10.1137/22m1517196","DOIUrl":"https://doi.org/10.1137/22m1517196","url":null,"abstract":"Let be invertible, unknown, and given. We are interested in approximate solutions: vectors such that is small. We prove that for all , there is a composition of orthogonal projections onto the hyperplanes generated by the rows of , where , which maps the origin to a vector satisfying . We note that this upper bound on is independent of the matrix . This procedure is stable in the sense that . The existence proof is based on a probabilistically refined analysis of the randomized Kaczmarz method, which seems to achieve this rate when solving for with high likelihood. We also prove a general version for matrices with and full rank.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135863460","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}
The set of covariance matrices equipped with the Bures–Wasserstein distance is the orbit space of the smooth, proper, and isometric action of the orthogonal group on the Euclidean space of square matrices. This construction induces a natural orbit stratification on covariance matrices, which is exactly the stratification by the rank. Thus, the strata are the manifolds of symmetric positive semidefinite matrices of fixed rank endowed with the Bures–Wasserstein Riemannian metric. In this work, we study the geodesics of the Bures–Wasserstein distance. First, we complete the literature on geodesics in each stratum by clarifying the set of preimages of the exponential map and by specifying the injectivity domain. We also give explicit formulae of the horizontal lift, the exponential map, and the Riemannian logarithms that were kept implicit in previous works. Second, we give the expression of all the minimizing geodesic segments joining two covariance matrices of any rank. More precisely, we show that the set of all minimizing geodesics between two covariance matrices and is parametrized by the closed unit ball of for the spectral norm, where are the respective ranks of . In particular, the minimizing geodesic is unique if and only if . Otherwise, there are infinitely many. As a secondary contribution, we provide a review of the definitions related to geodesics in metric spaces, affine connection manifolds, and Riemannian manifolds, which is helpful for the study of other spaces.
{"title":"Bures–Wasserstein Minimizing Geodesics between Covariance Matrices of Different Ranks","authors":"Yann Thanwerdas, Xavier Pennec","doi":"10.1137/22m149168x","DOIUrl":"https://doi.org/10.1137/22m149168x","url":null,"abstract":"The set of covariance matrices equipped with the Bures–Wasserstein distance is the orbit space of the smooth, proper, and isometric action of the orthogonal group on the Euclidean space of square matrices. This construction induces a natural orbit stratification on covariance matrices, which is exactly the stratification by the rank. Thus, the strata are the manifolds of symmetric positive semidefinite matrices of fixed rank endowed with the Bures–Wasserstein Riemannian metric. In this work, we study the geodesics of the Bures–Wasserstein distance. First, we complete the literature on geodesics in each stratum by clarifying the set of preimages of the exponential map and by specifying the injectivity domain. We also give explicit formulae of the horizontal lift, the exponential map, and the Riemannian logarithms that were kept implicit in previous works. Second, we give the expression of all the minimizing geodesic segments joining two covariance matrices of any rank. More precisely, we show that the set of all minimizing geodesics between two covariance matrices and is parametrized by the closed unit ball of for the spectral norm, where are the respective ranks of . In particular, the minimizing geodesic is unique if and only if . Otherwise, there are infinitely many. As a secondary contribution, we provide a review of the definitions related to geodesics in metric spaces, affine connection manifolds, and Riemannian manifolds, which is helpful for the study of other spaces.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135768754","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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