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":"55 1","pages":"0"},"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":"30 1","pages":"0"},"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":"53 1","pages":"0"},"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":"32 1","pages":"0"},"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":"22 1","pages":"0"},"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}
{"title":"An Apocalypse-Free First-Order Low-Rank Optimization Algorithm with at Most One Rank Reduction Attempt per Iteration","authors":"Guillaume Olikier, P.-A. Absil","doi":"10.1137/22m1518256","DOIUrl":"https://doi.org/10.1137/22m1518256","url":null,"abstract":"","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136060490","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}
Nonnegative matrix factorization (NMF) is a popular model in the field of pattern recognition. It aims to find a low rank approximation for nonnegative data M by a product of two nonnegative matrices W and H. In general, NMF is NP-hard to solve while it can be solved efficiently under separability assumption, which requires the columns of factor matrix are equal to columns of the input matrix. In this paper, we generalize separability assumption based on 3-factor NMF M=P_1SP_2, and require that S is a sub-matrix of the input matrix. We refer to this NMF as a Co-Separable NMF (CoS-NMF). We discuss some mathematics properties of CoS-NMF, and present the relationships with other related matrix factorizations such as CUR decomposition, generalized separable NMF(GS-NMF), and bi-orthogonal tri-factorization (BiOR-NM3F). An optimization model for CoS-NMF is proposed and alternated fast gradient method is employed to solve the model. Numerical experiments on synthetic datasets, document datasets and facial databases are conducted to verify the effectiveness of our CoS-NMF model. Compared to state-of-the-art methods, CoS-NMF model performs very well in co-clustering task, and preserves a good approximation to the input data matrix as well.
{"title":"Coseparable Nonnegative Matrix Factorization","authors":"Junjun Pan, Michael K. Ng","doi":"10.1137/22m1510509","DOIUrl":"https://doi.org/10.1137/22m1510509","url":null,"abstract":"Nonnegative matrix factorization (NMF) is a popular model in the field of pattern recognition. It aims to find a low rank approximation for nonnegative data M by a product of two nonnegative matrices W and H. In general, NMF is NP-hard to solve while it can be solved efficiently under separability assumption, which requires the columns of factor matrix are equal to columns of the input matrix. In this paper, we generalize separability assumption based on 3-factor NMF M=P_1SP_2, and require that S is a sub-matrix of the input matrix. We refer to this NMF as a Co-Separable NMF (CoS-NMF). We discuss some mathematics properties of CoS-NMF, and present the relationships with other related matrix factorizations such as CUR decomposition, generalized separable NMF(GS-NMF), and bi-orthogonal tri-factorization (BiOR-NM3F). An optimization model for CoS-NMF is proposed and alternated fast gradient method is employed to solve the model. Numerical experiments on synthetic datasets, document datasets and facial databases are conducted to verify the effectiveness of our CoS-NMF model. Compared to state-of-the-art methods, CoS-NMF model performs very well in co-clustering task, and preserves a good approximation to the input data matrix as well.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"237 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135353866","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}
Vivak Patel, Mohammad Jahangoshahi, Daniel Adrian Maldonado
. Randomized linear solvers leverage randomization to structure-blindly compress and solve a linear system to produce an inexpensive solution. While such a property is highly desirable, randomized linear solvers often suffer when it comes to performance as either (1) problem structure is not being exploited, and (2) hardware is inefficiently used. Thus, randomized adaptive solvers are starting to appear that use the benefits of randomness while attempting to still exploit problem structure and reduce hardware inefficiencies. Unfortunately, such randomized adaptive solvers are likely to be without a theoretical foundation to show that they will work (i.e., find a solution). Accordingly, here, we distill three general criteria for randomized block adaptive solvers, which, as we show, will guarantee convergence of the randomized adaptive solver and supply a worst-case rate of convergence. We will demonstrate that these results apply to existing randomized block adaptive solvers, and to several that we devise for demonstrative purposes.
{"title":"Randomized Block Adaptive Linear System Solvers","authors":"Vivak Patel, Mohammad Jahangoshahi, Daniel Adrian Maldonado","doi":"10.1137/22m1488715","DOIUrl":"https://doi.org/10.1137/22m1488715","url":null,"abstract":". Randomized linear solvers leverage randomization to structure-blindly compress and solve a linear system to produce an inexpensive solution. While such a property is highly desirable, randomized linear solvers often suffer when it comes to performance as either (1) problem structure is not being exploited, and (2) hardware is inefficiently used. Thus, randomized adaptive solvers are starting to appear that use the benefits of randomness while attempting to still exploit problem structure and reduce hardware inefficiencies. Unfortunately, such randomized adaptive solvers are likely to be without a theoretical foundation to show that they will work (i.e., find a solution). Accordingly, here, we distill three general criteria for randomized block adaptive solvers, which, as we show, will guarantee convergence of the randomized adaptive solver and supply a worst-case rate of convergence. We will demonstrate that these results apply to existing randomized block adaptive solvers, and to several that we devise for demonstrative purposes.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"44 1","pages":"1349-1369"},"PeriodicalIF":1.5,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64315847","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}
When analyzing complex networks, an important task is the identification of those nodes which play a leading role for the overall communicability of the network. In the context of modifying networks (or making them robust against targeted attacks or outages), it is also relevant to know how sensitive the network’s communicability reacts to changes in certain nodes or edges. Recently, the concept of total network sensitivity was introduced in [O. De la Cruz Cabrera, J. Jin, S. Noschese, and L. Reichel, Appl. Numer. Math., 172 (2022) pp. 186–205], which allows one to measure how sensitive the total communicability of a network is to the addition or removal of certain edges. One shortcoming of this concept is that sensitivities are extremely costly to compute when using a straightforward approach (orders of magnitude more expensive than the corresponding communicability measures). In this work, we present computational procedures for estimating network sensitivity with a cost that is essentially linear in the number of nodes for many real-world complex networks. Additionally, we extend the sensitivity concept such that it also covers sensitivity of subgraph centrality and the Estrada index, and we discuss the case of node removal. We propose a priori bounds for these sensitivities which capture well the qualitative behavior and give insight into the general behavior of matrix function based network indices under perturbations. These bounds are based on decay results for Fréchet derivatives of matrix functions with structured, low-rank direction terms which might be of independent interest also for applications other than network analysis.
在分析复杂网络时，一个重要的任务是识别对网络整体通信起主导作用的节点。在修改网络(或使其对目标攻击或中断具有健壮性)的上下文中，了解网络的可通信性对某些节点或边缘的变化的反应敏感程度也是相关的。最近，在[0]中引入了全网络灵敏度的概念。De la Cruz Cabrera, J. Jin, S. Noschese和L. Reichel, apple。号码。数学。， 172 (2022) pp. 186-205]，它允许人们测量网络的总通信能力对某些边的添加或移除有多敏感。这个概念的一个缺点是，当使用直接的方法时，灵敏度的计算成本非常高(比相应的通信度量要高几个数量级)。在这项工作中，我们提出了用于估计网络灵敏度的计算程序，其成本在许多现实世界的复杂网络的节点数量中基本上是线性的。此外，我们扩展了灵敏度概念，使其涵盖了子图中心性和Estrada指数的灵敏度，并讨论了节点移除的情况。我们提出了这些敏感性的先验界，它很好地捕捉了定性行为，并深入了解了基于矩阵函数的网络指标在扰动下的一般行为。这些边界是基于矩阵函数的fr导数的衰减结果，这些函数具有结构化的、低秩的方向项，这对于除网络分析以外的应用也可能是独立的。
{"title":"Sensitivity of Matrix Function Based Network Communicability Measures: Computational Methods and A Priori Bounds","authors":"Marcel Schweitzer","doi":"10.1137/23m1556708","DOIUrl":"https://doi.org/10.1137/23m1556708","url":null,"abstract":"When analyzing complex networks, an important task is the identification of those nodes which play a leading role for the overall communicability of the network. In the context of modifying networks (or making them robust against targeted attacks or outages), it is also relevant to know how sensitive the network’s communicability reacts to changes in certain nodes or edges. Recently, the concept of total network sensitivity was introduced in [O. De la Cruz Cabrera, J. Jin, S. Noschese, and L. Reichel, Appl. Numer. Math., 172 (2022) pp. 186–205], which allows one to measure how sensitive the total communicability of a network is to the addition or removal of certain edges. One shortcoming of this concept is that sensitivities are extremely costly to compute when using a straightforward approach (orders of magnitude more expensive than the corresponding communicability measures). In this work, we present computational procedures for estimating network sensitivity with a cost that is essentially linear in the number of nodes for many real-world complex networks. Additionally, we extend the sensitivity concept such that it also covers sensitivity of subgraph centrality and the Estrada index, and we discuss the case of node removal. We propose a priori bounds for these sensitivities which capture well the qualitative behavior and give insight into the general behavior of matrix function based network indices under perturbations. These bounds are based on decay results for Fréchet derivatives of matrix functions with structured, low-rank direction terms which might be of independent interest also for applications other than network analysis.","PeriodicalId":49538,"journal":{"name":"SIAM Journal on Matrix Analysis and Applications","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135890679","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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