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PinT Preconditioner for Forward-Backward Evolutionary Equations 正反向进化方程的PinT预条件
IF 1.5 2区 数学
SIAM Journal on Matrix Analysis and Applications Pub Date : 2023-11-30 DOI: 10.1137/22m1516476
Shu-Lin Wu, Zhiyong Wang, Tao Zhou
{"title":"PinT Preconditioner for Forward-Backward Evolutionary Equations","authors":"Shu-Lin Wu, Zhiyong Wang, Tao Zhou","doi":"10.1137/22m1516476","DOIUrl":"https://doi.org/10.1137/22m1516476","url":null,"abstract":"SIAM Journal on Matrix Analysis and Applications, Volume 44, Issue 4, Page 1771-1798, December 2023. <br/> Abstract. Solving the linear system [math] is often the major computational burden when a forward-backward evolutionary equation must be solved in a problem, where [math] is the so-called all-at-once matrix of the forward subproblem after space-time discretization. An efficient solver requires a good preconditioner for [math]. Inspired by the structure of [math], we precondition [math] by [math] with [math] being a block [math]-circulant matrix constructed by replacing the Toeplitz matrices in [math] by the [math]-circulant matrices. By a block Fourier diagonalization of [math], the computation of the preconditioning step [math] is parallelizable for all the time steps. We give a spectral analysis for the preconditioned matrix [math] and prove that for any one-step stable time-integrator the eigenvalues of [math] spread in a mesh-independent interval [math] if the parameter [math] weakly scales in terms of the number of time steps [math] as [math], where [math] is a free constant. Two applications of the proposed preconditioner are illustrated: PDE-constrained optimal control problems and parabolic source identification problems. Numerical results for both problems indicate that spectral analysis predicts the convergence rate of the preconditioned conjugate gradient method very well.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"1 3","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138513874","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
Generalized Matrix Nearness Problems 广义矩阵逼近性问题
2区 数学
SIAM Journal on Matrix Analysis and Applications Pub Date : 2023-11-10 DOI: 10.1137/22m1526034
Zihao Li, Lek-Heng Lim
{"title":"Generalized Matrix Nearness Problems","authors":"Zihao Li, Lek-Heng Lim","doi":"10.1137/22m1526034","DOIUrl":"https://doi.org/10.1137/22m1526034","url":null,"abstract":"We show that the global minimum solution of can be found in closed form with singular value decompositions and generalized singular value decompositions for a variety of constraints on involving rank, norm, symmetry, two-sided product, and prescribed eigenvalue. This extends the solution of Friedland–Torokhti for the generalized rank-constrained approximation problem to other constraints and provides an alternative solution for rank constraint in terms of singular value decompositions. For more complicated constraints on involving structures such as Toeplitz, Hankel, circulant, nonnegativity, stochasticity, positive semidefiniteness, prescribed eigenvector, etc., we prove that a simple iterative method is linearly and globally convergent to the global minimum solution.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"120 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135137335","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
Introducing the Class of SemiDoubly Stochastic Matrices: A Novel Scaling Approach for Rectangular Matrices 引入一类半重随机矩阵:矩形矩阵的一种新的标度方法
2区 数学
SIAM Journal on Matrix Analysis and Applications Pub Date : 2023-11-10 DOI: 10.1137/22m1519791
Philip A. Knight, Luce le Gorrec, Sandrine Mouysset, Daniel Ruiz
{"title":"Introducing the Class of SemiDoubly Stochastic Matrices: A Novel Scaling Approach for Rectangular Matrices","authors":"Philip A. Knight, Luce le Gorrec, Sandrine Mouysset, Daniel Ruiz","doi":"10.1137/22m1519791","DOIUrl":"https://doi.org/10.1137/22m1519791","url":null,"abstract":"","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"120 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135137337","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
Stochastic Algebraic Riccati Equations Are Almost as Easy as Deterministic Ones Theoretically 随机代数Riccati方程在理论上几乎和确定性方程一样简单
2区 数学
SIAM Journal on Matrix Analysis and Applications Pub Date : 2023-11-10 DOI: 10.1137/22m1514647
Zhen-Chen Guo, Xin Liang
{"title":"Stochastic Algebraic Riccati Equations Are Almost as Easy as Deterministic Ones Theoretically","authors":"Zhen-Chen Guo, Xin Liang","doi":"10.1137/22m1514647","DOIUrl":"https://doi.org/10.1137/22m1514647","url":null,"abstract":"","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"120 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135137334","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
Kronecker Product Approximation of Operators in Spectral Norm via Alternating SDP 基于交替SDP的谱范数算子的Kronecker积逼近
2区 数学
SIAM Journal on Matrix Analysis and Applications Pub Date : 2023-11-09 DOI: 10.1137/22m1509953
Mareike Dressler, André Uschmajew, Venkat Chandrasekaran
{"title":"Kronecker Product Approximation of Operators in Spectral Norm via Alternating SDP","authors":"Mareike Dressler, André Uschmajew, Venkat Chandrasekaran","doi":"10.1137/22m1509953","DOIUrl":"https://doi.org/10.1137/22m1509953","url":null,"abstract":"The decomposition or approximation of a linear operator on a matrix space as a sum of Kronecker products plays an important role in matrix equations and low-rank modeling. The approximation problem in Frobenius norm admits a well-known solution via the singular value decomposition. However, the approximation problem in spectral norm, which is more natural for linear operators, is much more challenging. In particular, the Frobenius norm solution can be far from optimal in spectral norm. We describe an alternating optimization method based on semidefinite programming to obtain high-quality approximations in spectral norm, and we present computational experiments to illustrate the advantages of our approach.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":" 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135192437","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
Semidefinite Relaxation Methods for Tensor Absolute Value Equations 张量绝对值方程的半定松弛法
2区 数学
SIAM Journal on Matrix Analysis and Applications Pub Date : 2023-11-07 DOI: 10.1137/22m1539137
Anwa Zhou, Kun Liu, Jinyan Fan
{"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":"212 5","pages":"0"},"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}
引用次数: 0
Generalized Perron Roots and Solvability of the Absolute Value Equation 广义Perron根与绝对值方程的可解性
2区 数学
SIAM Journal on Matrix Analysis and Applications Pub Date : 2023-10-30 DOI: 10.1137/22m1517184
Manuel Radons
{"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":"327 ","pages":"0"},"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}
引用次数: 2
Contour Integration for Eigenvector Nonlinearities 特征向量非线性的轮廓积分
2区 数学
SIAM Journal on Matrix Analysis and Applications Pub Date : 2023-10-30 DOI: 10.1137/22m1497985
Rob Claes, Karl Meerbergen, Simon Telen
{"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":"46 2","pages":"0"},"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}
引用次数: 0
Solving Singular Generalized Eigenvalue Problems. Part II: Projection and Augmentation 求解奇异广义特征值问题。第二部分:投影和增强
2区 数学
SIAM Journal on Matrix Analysis and Applications Pub Date : 2023-10-25 DOI: 10.1137/22m1513174
Michiel E. Hochstenbach, Christian Mehl, Bor Plestenjak
{"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":"16 2","pages":"0"},"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}
引用次数: 3
Robust Recovery of Low-Rank Matrices and Low-Tubal-Rank Tensors from Noisy Ketches 低秩矩阵和低管秩张量在噪声Ketches中的鲁棒恢复
2区 数学
SIAM Journal on Matrix Analysis and Applications Pub Date : 2023-10-20 DOI: 10.1137/22m150071x
Anna Ma, Dominik Stöger, Yizhe Zhu
{"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":"159 1","pages":"0"},"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}
引用次数: 0
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