Liya Gaynutdinova, Martin Ladecký, Ivana Pultarová, Miloslav Vlasák, Jan Zeman
This paper focuses on the design, analysis and implementation of a new preconditioning concept for linear second order partial differential equations, including the convection-diffusion-reaction problems discretized by Galerkin or discontinuous Galerkin methods. We expand on the approach introduced by Gergelits et al. and adapt it to the more general settings, assuming that both the original and preconditioning matrices are composed of sparse matrices of very low ranks, representing local contributions to the global matrices. When applied to a symmetric problem, the method provides bounds to all individual eigenvalues of the preconditioned matrix. We show that this preconditioning strategy works not only for Galerkin discretization, but also for the discontinuous Galerkin discretization, where local contributions are associated with individual edges of the triangulation. In the case of nonsymmetric problems, the method yields guaranteed bounds to real and imaginary parts of the resulting eigenvalues. We include some numerical experiments illustrating the method and its implementation, showcasing its effectiveness for the two variants of discretized (convection-)diffusion-reaction problems.
{"title":"Preconditioned discontinuous Galerkin method and convection-diffusion-reaction problems with guaranteed bounds to resulting spectra","authors":"Liya Gaynutdinova, Martin Ladecký, Ivana Pultarová, Miloslav Vlasák, Jan Zeman","doi":"10.1002/nla.2549","DOIUrl":"https://doi.org/10.1002/nla.2549","url":null,"abstract":"This paper focuses on the design, analysis and implementation of a new preconditioning concept for linear second order partial differential equations, including the convection-diffusion-reaction problems discretized by Galerkin or discontinuous Galerkin methods. We expand on the approach introduced by Gergelits et al. and adapt it to the more general settings, assuming that both the original and preconditioning matrices are composed of sparse matrices of very low ranks, representing local contributions to the global matrices. When applied to a symmetric problem, the method provides bounds to all individual eigenvalues of the preconditioned matrix. We show that this preconditioning strategy works not only for Galerkin discretization, but also for the discontinuous Galerkin discretization, where local contributions are associated with individual edges of the triangulation. In the case of nonsymmetric problems, the method yields guaranteed bounds to real and imaginary parts of the resulting eigenvalues. We include some numerical experiments illustrating the method and its implementation, showcasing its effectiveness for the two variants of discretized (convection-)diffusion-reaction problems.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"276 1","pages":""},"PeriodicalIF":4.3,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139768267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The problem of generalized tensor eigenvalue is the focus of this paper. To solve the problem, we suggest using the normalized Newton generalized eigenproblem approach (NNGEM). Since the rate of convergence of the spectral gradient projection method (SGP), the generalized eigenproblem adaptive power (GEAP), and other approaches is only linear, they are significantly improved by our proposed method, which is demonstrated to be locally and cubically convergent. Additionally, the modified normalized Newton method (MNNM), which converges to symmetric tensors Z-eigenpairs under the same -Newton stability requirement, is extended by the NNGEM technique. Using a Gröbner basis, a polynomial system solver (NSolve) generates all of the real eigenvalues for us. To illustrate the efficacy of our methodology, we present a few numerical findings.
广义张量特征值问题是本文的重点。为了解决这个问题，我们建议使用归一化牛顿广义特征问题方法（NNGEM）。由于频谱梯度投影法（SGP）、广义特征问题自适应功率法（GEAP）和其他方法的收敛速度只是线性的，而我们提出的方法能显著改善它们的收敛速度，并证明其具有局部收敛性和立方收敛性。此外，改进的归一化牛顿法（MNNM）在相同的 γ$$ gamma $$-Newton 稳定性要求下收敛于对称张量 Z 特征对，该方法由 NNGEM 技术扩展而来。多项式系统求解器（NSolve）使用格罗布纳基，为我们生成所有实特征值。为了说明我们的方法的有效性，我们给出了一些数值结果。
{"title":"Normalized Newton method to solve generalized tensor eigenvalue problems","authors":"Mehri Pakmanesh, Hamidreza Afshin, Masoud Hajarian","doi":"10.1002/nla.2547","DOIUrl":"https://doi.org/10.1002/nla.2547","url":null,"abstract":"The problem of generalized tensor eigenvalue is the focus of this paper. To solve the problem, we suggest using the normalized Newton generalized eigenproblem approach (NNGEM). Since the rate of convergence of the spectral gradient projection method (SGP), the generalized eigenproblem adaptive power (GEAP), and other approaches is only linear, they are significantly improved by our proposed method, which is demonstrated to be locally and cubically convergent. Additionally, the modified normalized Newton method (MNNM), which converges to symmetric tensors Z-eigenpairs under the same <math altimg=\"urn:x-wiley:nla:media:nla2547:nla2547-math-0003\" display=\"inline\" location=\"graphic/nla2547-math-0003.png\" overflow=\"scroll\">\u0000<semantics>\u0000<mrow>\u0000<mi>γ</mi>\u0000</mrow>\u0000$$ gamma $$</annotation>\u0000</semantics></math>-Newton stability requirement, is extended by the NNGEM technique. Using a Gröbner basis, a polynomial system solver (NSolve) generates all of the real eigenvalues for us. To illustrate the efficacy of our methodology, we present a few numerical findings.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"52 1","pages":""},"PeriodicalIF":4.3,"publicationDate":"2024-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139408676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Bogoya, Sven-Erik Ekström, S. Serra‐Capizzano, P. Vassalos
It is known that the generating function of a sequence of Toeplitz matrices may not describe the asymptotic distribution of the eigenvalues of the considered matrix sequence in the non‐Hermitian setting. In a recent work, under the assumption that the eigenvalues are real, admitting an asymptotic expansion whose first term is the distribution function, fast algorithms computing all the spectra were proposed in different settings. In the current work, we extend this idea to non‐Hermitian Toeplitz matrices with complex eigenvalues, in the case where the range of the generating function does not disconnect the complex field or the limiting set of the spectra, as the matrix‐size tends to infinity, has one nonclosed analytic arc. For a generating function having a power singularity, we prove the existence of an asymptotic expansion, that can be used as a theoretical base for the respective numerical algorithm. Different generating functions are explored, highlighting different numerical and theoretical aspects; for example, non‐Hermitian and complex symmetric matrix sequences, the reconstruction of the generating function, a consistent eigenvalue ordering, the requirements of high‐precision data types. Several numerical experiments are reported and critically discussed, and avenues of possible future research are presented.
{"title":"Matrix‐less methods for the spectral approximation of large non‐Hermitian Toeplitz matrices: A concise theoretical analysis and a numerical study","authors":"M. Bogoya, Sven-Erik Ekström, S. Serra‐Capizzano, P. Vassalos","doi":"10.1002/nla.2545","DOIUrl":"https://doi.org/10.1002/nla.2545","url":null,"abstract":"It is known that the generating function of a sequence of Toeplitz matrices may not describe the asymptotic distribution of the eigenvalues of the considered matrix sequence in the non‐Hermitian setting. In a recent work, under the assumption that the eigenvalues are real, admitting an asymptotic expansion whose first term is the distribution function, fast algorithms computing all the spectra were proposed in different settings. In the current work, we extend this idea to non‐Hermitian Toeplitz matrices with complex eigenvalues, in the case where the range of the generating function does not disconnect the complex field or the limiting set of the spectra, as the matrix‐size tends to infinity, has one nonclosed analytic arc. For a generating function having a power singularity, we prove the existence of an asymptotic expansion, that can be used as a theoretical base for the respective numerical algorithm. Different generating functions are explored, highlighting different numerical and theoretical aspects; for example, non‐Hermitian and complex symmetric matrix sequences, the reconstruction of the generating function, a consistent eigenvalue ordering, the requirements of high‐precision data types. Several numerical experiments are reported and critically discussed, and avenues of possible future research are presented.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"88 12","pages":""},"PeriodicalIF":4.3,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139386270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Based on sketching techniques, we propose two practical randomized algorithms for tensor ring (TR) decomposition. Specifically, on the basis of defining new tensor products and investigating their properties, the two algorithms are devised by applying the Kronecker sub-sampled randomized Fourier transform and TensorSketch to the alternating least squares subproblems derived from the minimization problem of TR decomposition. From the former, we find an algorithmic framework based on random projection for randomized TR decomposition. We compare our proposals with the existing methods using both synthetic and real data. Numerical results show that they have quite decent performance in accuracy and computing time.
{"title":"Practical sketching-based randomized tensor ring decomposition","authors":"Yajie Yu, Hanyu Li","doi":"10.1002/nla.2548","DOIUrl":"https://doi.org/10.1002/nla.2548","url":null,"abstract":"Based on sketching techniques, we propose two practical randomized algorithms for tensor ring (TR) decomposition. Specifically, on the basis of defining new tensor products and investigating their properties, the two algorithms are devised by applying the Kronecker sub-sampled randomized Fourier transform and TensorSketch to the alternating least squares subproblems derived from the minimization problem of TR decomposition. From the former, we find an algorithmic framework based on random projection for randomized TR decomposition. We compare our proposals with the existing methods using both synthetic and real data. Numerical results show that they have quite decent performance in accuracy and computing time.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"27 1","pages":""},"PeriodicalIF":4.3,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139408905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper focuses on the analysis and the solution of the saddle-point problem arising from a three-field formulation of Biot's model of poroelasticity, discretized in time by the implicit Euler method. A block diagonal-preconditioner, based on the Schur complement, is analyzed on a functional level and compared with two other block-diagonal preconditioners having a similar structure. The problem is discretized in space using mixed finite elements and solved with appropriate iterative solvers, incorporating the investigated preconditioners. The solvers are tested on numerical examples inspired by geotechnical practice, with particular attention devoted to the solvers' robustness concerning strong heterogeneity in permeability.
{"title":"Robust block diagonal preconditioners for poroelastic problems with strongly heterogeneous material","authors":"Tomáš Luber, Stanislav Sysala","doi":"10.1002/nla.2546","DOIUrl":"https://doi.org/10.1002/nla.2546","url":null,"abstract":"This paper focuses on the analysis and the solution of the saddle-point problem arising from a three-field formulation of Biot's model of poroelasticity, discretized in time by the implicit Euler method. A block diagonal-preconditioner, based on the Schur complement, is analyzed on a functional level and compared with two other block-diagonal preconditioners having a similar structure. The problem is discretized in space using mixed finite elements and solved with appropriate iterative solvers, incorporating the investigated preconditioners. The solvers are tested on numerical examples inspired by geotechnical practice, with particular attention devoted to the solvers' robustness concerning strong heterogeneity in permeability.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"26 1","pages":""},"PeriodicalIF":4.3,"publicationDate":"2023-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139052970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tareq Zaman, Nicolas Nytko, Ali Taghibakhshi, Scott MacLachlan, Luke Olson, Matthew West
Algebraic multigrid (AMG) methods are often robust and effective solvers for solving the large and sparse linear systems that arise from discretized PDEs and other problems, relying on heuristic graph algorithms to achieve their performance. Reduction-based AMG (AMGr) algorithms attempt to formalize these heuristics by providing two-level convergence bounds that depend concretely on properties of the partitioning of the given matrix into its fine- and coarse-grid degrees of freedom. MacLachlan and Saad (SISC 2007) proved that the AMGr method yields provably robust two-level convergence for symmetric and positive-definite matrices that are diagonally dominant, with a convergence factor bounded as a function of a coarsening parameter. However, when applying AMGr algorithms to matrices that are not diagonally dominant, not only do the convergence factor bounds not hold, but measured performance is notably degraded. Here, we present modifications to the classical AMGr algorithm that improve its performance on matrices that are not diagonally dominant, making use of strength of connection, sparse approximate inverse (SPAI) techniques, and interpolation truncation and rescaling, to improve robustness while maintaining control of the algorithmic costs. We present numerical results demonstrating the robustness of this approach for both classical isotropic diffusion problems and for non-diagonally dominant systems coming from anisotropic diffusion.
{"title":"Generalizing reduction-based algebraic multigrid","authors":"Tareq Zaman, Nicolas Nytko, Ali Taghibakhshi, Scott MacLachlan, Luke Olson, Matthew West","doi":"10.1002/nla.2543","DOIUrl":"https://doi.org/10.1002/nla.2543","url":null,"abstract":"Algebraic multigrid (AMG) methods are often robust and effective solvers for solving the large and sparse linear systems that arise from discretized PDEs and other problems, relying on heuristic graph algorithms to achieve their performance. Reduction-based AMG (AMGr) algorithms attempt to formalize these heuristics by providing two-level convergence bounds that depend concretely on properties of the partitioning of the given matrix into its fine- and coarse-grid degrees of freedom. MacLachlan and Saad (SISC 2007) proved that the AMGr method yields provably robust two-level convergence for symmetric and positive-definite matrices that are diagonally dominant, with a convergence factor bounded as a function of a coarsening parameter. However, when applying AMGr algorithms to matrices that are not diagonally dominant, not only do the convergence factor bounds not hold, but measured performance is notably degraded. Here, we present modifications to the classical AMGr algorithm that improve its performance on matrices that are not diagonally dominant, making use of strength of connection, sparse approximate inverse (SPAI) techniques, and interpolation truncation and rescaling, to improve robustness while maintaining control of the algorithmic costs. We present numerical results demonstrating the robustness of this approach for both classical isotropic diffusion problems and for non-diagonally dominant systems coming from anisotropic diffusion.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"13 1","pages":""},"PeriodicalIF":4.3,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138824407","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Robust low-rank tensor completion plays an important role in multidimensional data analysis against different degradations, such as sparse noise, and missing entries, and has a variety of applications in image processing and computer vision. In this paper, an optimization model for low-rank tensor completion problems is proposed and a block coordinate descent algorithm is developed to solve this model. It is shown that for one of the subproblems, the closed-form solution exists and for the other, a Riemannian conjugate gradient algorithm is used. In particular, when all elements are known, that is, no missing values, the block coordinate descent is simplified in the sense that both subproblems have closed-form solutions. The convergence analysis is established without requiring the latter subproblem to be solved exactly. Numerical experiments illustrate that the proposed model with the algorithm is feasible and effective.
{"title":"An iterative algorithm for low-rank tensor completion problem with sparse noise and missing values","authors":"Jianheng Chen, Wen Huang","doi":"10.1002/nla.2544","DOIUrl":"https://doi.org/10.1002/nla.2544","url":null,"abstract":"Robust low-rank tensor completion plays an important role in multidimensional data analysis against different degradations, such as sparse noise, and missing entries, and has a variety of applications in image processing and computer vision. In this paper, an optimization model for low-rank tensor completion problems is proposed and a block coordinate descent algorithm is developed to solve this model. It is shown that for one of the subproblems, the closed-form solution exists and for the other, a Riemannian conjugate gradient algorithm is used. In particular, when all elements are known, that is, no missing values, the block coordinate descent is simplified in the sense that both subproblems have closed-form solutions. The convergence analysis is established without requiring the latter subproblem to be solved exactly. Numerical experiments illustrate that the proposed model with the algorithm is feasible and effective.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"38 1","pages":""},"PeriodicalIF":4.3,"publicationDate":"2023-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138716923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tensor ring (TR) decomposition has been widely applied as an effective approach in a variety of applications to discover the hidden low-rank patterns in multidimensional and higher-order data. A well-known method for TR decomposition is the alternating least squares (ALS). However, solving the ALS subproblems often suffers from high cost issue, especially for large-scale tensors. In this paper, we provide two strategies to tackle this issue and design three ALS-based algorithms. Specifically, the first strategy is used to simplify the calculation of the coefficient matrices of the normal equations for the ALS subproblems, which takes full advantage of the structure of the coefficient matrices of the subproblems and hence makes the corresponding algorithm perform much better than the regular ALS method in terms of computing time. The second strategy is to stabilize the ALS subproblems by QR factorizations on TR-cores, and hence the corresponding algorithms are more numerically stable compared with our first algorithm. Extensive numerical experiments on synthetic and real data are given to illustrate and confirm the above results. In addition, we also present the complexity analyses of the proposed algorithms.
{"title":"Practical alternating least squares for tensor ring decomposition","authors":"Yajie Yu, Hanyu Li","doi":"10.1002/nla.2542","DOIUrl":"https://doi.org/10.1002/nla.2542","url":null,"abstract":"Tensor ring (TR) decomposition has been widely applied as an effective approach in a variety of applications to discover the hidden low-rank patterns in multidimensional and higher-order data. A well-known method for TR decomposition is the alternating least squares (ALS). However, solving the ALS subproblems often suffers from high cost issue, especially for large-scale tensors. In this paper, we provide two strategies to tackle this issue and design three ALS-based algorithms. Specifically, the first strategy is used to simplify the calculation of the coefficient matrices of the normal equations for the ALS subproblems, which takes full advantage of the structure of the coefficient matrices of the subproblems and hence makes the corresponding algorithm perform much better than the regular ALS method in terms of computing time. The second strategy is to stabilize the ALS subproblems by QR factorizations on TR-cores, and hence the corresponding algorithms are more numerically stable compared with our first algorithm. Extensive numerical experiments on synthetic and real data are given to illustrate and confirm the above results. In addition, we also present the complexity analyses of the proposed algorithms.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"4 1","pages":""},"PeriodicalIF":4.3,"publicationDate":"2023-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138632252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Zhao‐Li Shen, Bruno Carpentieri, Chun Wen, Jian‐Jun Wang, Stefano Serra‐Capizzano, Shi‐Ping Du
Abstract The PageRank model, which was first proposed by Google for its web search engine application, has since become a popular computational tool in a wide range of scientific fields, including chemistry, bioinformatics, neuroscience, bibliometrics, social networks, and others. PageRank calculations necessitate the use of fast computational techniques with low algorithmic and memory complexity. In recent years, much attention has been paid to Krylov subspace algorithms for solving difficult PageRank linear systems, such as those with large damping parameters close to one. In this article, we examine the full orthogonalization method (FOM). We present a convergence study of the method that extends and clarifies part of the conclusions reached in Zhang et al. (J Comput Appl Math. 2016; 296:397–409.). Furthermore, we demonstrate that FOM is breakdown free when solving singular PageRank linear systems with index one and we investigate the effect of using weighted inner‐products instead of conventional inner‐products in the orthonormalization procedure on FOM convergence. Finally, we develop a shifted polynomial preconditioner that takes advantage of the special structure of the PageRank linear system and has a good ability to cluster most of the eigenvalues, making it a good choice for an iterative method like FOM or GMRES. Numerical experiments are presented to support the theoretical findings and to evaluate the performance of the new weighted preconditioned FOM PageRank solver in comparison to other established solvers for this class of problem, including conventional stationary methods, hybrid combinations of stationary and Krylov subspace methods, and multi‐step splitting strategies.
PageRank模型最早是由Google为其网络搜索引擎应用而提出的，现已成为广泛应用于化学、生物信息学、神经科学、文献计量学、社会网络等科学领域的一种流行计算工具。PageRank计算需要使用低算法和内存复杂度的快速计算技术。近年来，Krylov子空间算法得到了广泛的关注，用于求解复杂的PageRank线性系统，如具有接近1的大阻尼参数的系统。在本文中，我们研究了完全正交化方法(FOM)。我们对该方法进行了收敛性研究，扩展并澄清了Zhang等人得出的部分结论(J computer apple Math. 2016;296:397 - 409)。进一步，我们证明了FOM在求解索引为1的奇异PageRank线性系统时是无击穿的，并研究了在标准正交化过程中使用加权内积代替常规内积对FOM收敛性的影响。最后，我们开发了一个移位多项式预调节器，利用PageRank线性系统的特殊结构，具有很好的聚类大部分特征值的能力，使其成为FOM或GMRES等迭代方法的良好选择。通过数值实验来支持理论发现，并与其他已建立的此类问题的求解器(包括传统的平稳方法、平稳和Krylov子空间方法的混合组合以及多步分裂策略)进行了比较，评估了新的加权预条件FOM PageRank求解器的性能。
{"title":"Preconditioned weighted full orothogonalization method for solving singular linear systems from PageRank problems","authors":"Zhao‐Li Shen, Bruno Carpentieri, Chun Wen, Jian‐Jun Wang, Stefano Serra‐Capizzano, Shi‐Ping Du","doi":"10.1002/nla.2541","DOIUrl":"https://doi.org/10.1002/nla.2541","url":null,"abstract":"Abstract The PageRank model, which was first proposed by Google for its web search engine application, has since become a popular computational tool in a wide range of scientific fields, including chemistry, bioinformatics, neuroscience, bibliometrics, social networks, and others. PageRank calculations necessitate the use of fast computational techniques with low algorithmic and memory complexity. In recent years, much attention has been paid to Krylov subspace algorithms for solving difficult PageRank linear systems, such as those with large damping parameters close to one. In this article, we examine the full orthogonalization method (FOM). We present a convergence study of the method that extends and clarifies part of the conclusions reached in Zhang et al. (J Comput Appl Math. 2016; 296:397–409.). Furthermore, we demonstrate that FOM is breakdown free when solving singular PageRank linear systems with index one and we investigate the effect of using weighted inner‐products instead of conventional inner‐products in the orthonormalization procedure on FOM convergence. Finally, we develop a shifted polynomial preconditioner that takes advantage of the special structure of the PageRank linear system and has a good ability to cluster most of the eigenvalues, making it a good choice for an iterative method like FOM or GMRES. Numerical experiments are presented to support the theoretical findings and to evaluate the performance of the new weighted preconditioned FOM PageRank solver in comparison to other established solvers for this class of problem, including conventional stationary methods, hybrid combinations of stationary and Krylov subspace methods, and multi‐step splitting strategies.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"114 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":"135138404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Michiel E. Hochstenbach, Tomaž Košir, Bor Plestenjak
Abstract Standard multiparameter eigenvalue problems (MEPs) are systems of linear ‐parameter square matrix pencils. Recently, a new form of multiparameter eigenvalue problems has emerged: a rectangular MEP (RMEP) with only one multivariate rectangular matrix pencil, where we are looking for combinations of the parameters for which the rank of the pencil is not full. Applications include finding the optimal least squares autoregressive moving average (ARMA) model and the optimal least squares realization of autonomous linear time‐invariant (LTI) dynamical system. For linear and polynomial RMEPs, we give the number of solutions and show how these problems can be solved numerically by a transformation into a standard MEP. For the transformation we provide new linearizations for quadratic multivariate matrix polynomials with a specific structure of monomials and consider mixed systems of rectangular and square multivariate matrix polynomials. This numerical approach seems computationally considerably more attractive than the block Macaulay method, the only other currently available numerical method for polynomial RMEPs.
{"title":"Numerical methods for rectangular multiparameter eigenvalue problems, with applications to finding optimal ARMA and LTI models","authors":"Michiel E. Hochstenbach, Tomaž Košir, Bor Plestenjak","doi":"10.1002/nla.2540","DOIUrl":"https://doi.org/10.1002/nla.2540","url":null,"abstract":"Abstract Standard multiparameter eigenvalue problems (MEPs) are systems of linear ‐parameter square matrix pencils. Recently, a new form of multiparameter eigenvalue problems has emerged: a rectangular MEP (RMEP) with only one multivariate rectangular matrix pencil, where we are looking for combinations of the parameters for which the rank of the pencil is not full. Applications include finding the optimal least squares autoregressive moving average (ARMA) model and the optimal least squares realization of autonomous linear time‐invariant (LTI) dynamical system. For linear and polynomial RMEPs, we give the number of solutions and show how these problems can be solved numerically by a transformation into a standard MEP. For the transformation we provide new linearizations for quadratic multivariate matrix polynomials with a specific structure of monomials and consider mixed systems of rectangular and square multivariate matrix polynomials. This numerical approach seems computationally considerably more attractive than the block Macaulay method, the only other currently available numerical method for polynomial RMEPs.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":" 17","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135242136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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