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Inexact inner–outer Golub–Kahan bidiagonalization method: A relaxation strategy 非精确内外Golub-Kahan双对角化方法:一种松弛策略
IF 4.3 3区 数学 Q1 MATHEMATICS Pub Date : 2022-12-12 DOI: 10.1002/nla.2484
Vincent Darrigrand, Andrei Dumitrasc, Carola Kruse, Ulrich Rüde
We study an inexact inner–outer generalized Golub–Kahan algorithm for the solution of saddle-point problems with a two-times-two block structure. In each outer iteration, an inner system has to be solved which in theory has to be done exactly. Whenever the system is getting large, an inner exact solver is, however, no longer efficient or even feasible and iterative methods must be used. We focus this article on a numerical study showing the influence of the accuracy of an inner iterative solution on the accuracy of the solution of the block system. Emphasis is further given on reducing the computational cost, which is defined as the total number of inner iterations. We develop relaxation techniques intended to dynamically change the inner tolerance for each outer iteration to further minimize the total number of inner iterations. We illustrate our findings on a Stokes problem and validate them on a mixed formulation of the Poisson problem.
研究了求解两乘二块结构鞍点问题的非精确内外广义Golub-Kahan算法。在每次外部迭代中,必须求解一个内部系统,这在理论上必须精确地完成。然而,当系统变得越来越大时,内部精确求解器就不再有效甚至可行,必须使用迭代方法。本文着重研究了内迭代解的精度对块系统解精度的影响。进一步强调降低计算成本,计算成本被定义为内部迭代的总次数。我们开发了旨在动态改变每个外部迭代的内部公差的松弛技术,以进一步减少内部迭代的总数。我们在Stokes问题上说明了我们的发现,并在泊松问题的混合公式上验证了它们。
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引用次数: 1
A fast and accurate algorithm for solving linear systems associated with a class of negative matrix 一类负矩阵线性方程组的快速精确求解算法
IF 4.3 3区 数学 Q1 MATHEMATICS Pub Date : 2022-12-08 DOI: 10.1002/nla.2483
Zhao Yang
A class of negative matrices including Vandermonde‐like matrices tends to be extremely ill‐conditioned, and linear systems associated with this class of matrices appear in the polynomial interpolation problems. In this article, we present a fast and accurate algorithm with O(n2)$$ Oleft({n}^2right) $$ complexity to solve the linear systems whose coefficient matrices belong to the class of negative matrix. We show that the inverse of any such matrix is generated in a subtraction‐free manner. Consequently, the solutions of linear systems associated with the class of negative matrix are accurately determined by parameterization matrices of coefficient matrices, and a pleasantly componentwise forward error is provided to illustrate that each component of the solution is computed to high accuracy. Numerical experiments are performed to confirm the claimed high accuracy.
包括类范德蒙德矩阵在内的一类负矩阵往往是极端病态的,与这类矩阵相关的线性系统出现在多项式插值问题中。在本文中,我们提出了一个具有O(n2)$$Oleft({n}^2 right)$$复杂度的快速精确算法来求解系数矩阵属于负矩阵类的线性系统。我们证明了任何这样的矩阵的逆都是以无减法的方式生成的。因此,与负矩阵类相关的线性系统的解由系数矩阵的参数化矩阵精确地确定,并且提供了令人愉快的分量前向误差,以说明解的每个分量都是高精度计算的。进行了数值实验以证实所声称的高精度。
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引用次数: 2
A class of improved conjugate gradient methods for nonconvex unconstrained optimization 非凸无约束优化的一类改进共轭梯度方法
IF 4.3 3区 数学 Q1 MATHEMATICS Pub Date : 2022-12-08 DOI: 10.1002/nla.2482
Qingjie Hu, Hongrun Zhang, Zhijuan Zhou, Yu Chen
In this paper, based on a new class of conjugate gradient methods which are proposed by Rivaie, Dai and Omer et al. we propose a class of improved conjugate gradient methods for nonconvex unconstrained optimization. Different from the above methods, our methods possess the following properties: (i) the search direction always satisfies the sufficient descent condition independent of any line search; (ii) these approaches are globally convergent with the standard Wolfe line search or standard Armijo line search without any convexity assumption. Moreover, our numerical results also demonstrated the efficiencies of the proposed methods.
本文在Rivaie、Dai和Omer等人提出的一类新的共轭梯度方法的基础上,提出了一类改进的非凸无约束优化共轭梯度方法。与上述方法不同,我们的方法具有以下性质:(i)搜索方向总是满足与任何直线搜索无关的充分下降条件;(ii)在没有任何凸性假设的情况下,这些方法与标准Wolfe线搜索或标准Armijo线搜索是全局收敛的。此外,我们的数值结果也证明了所提出的方法的有效性。
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引用次数: 0
Issue Information 问题信息
IF 4.3 3区 数学 Q1 MATHEMATICS Pub Date : 2022-12-01 DOI: 10.1002/nla.2449
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引用次数: 0
A parameterized extended shift‐splitting preconditioner for nonsymmetric saddle point problems 非对称鞍点问题的参数化扩展移位分裂预条件
IF 4.3 3区 数学 Q1 MATHEMATICS Pub Date : 2022-11-29 DOI: 10.1002/nla.2478
Seryas Vakili, G. Ebadi, C. Vuik
In this article, a parameterized extended shift‐splitting (PESS) method and its induced preconditioner are given for solving nonsingular and nonsymmetric saddle point problems with nonsymmetric positive definite (1,1) part. The convergence analysis of the PESS$$ PESS $$ iteration method is discussed. The distribution of eigenvalues of the preconditioned matrix is provided. A number of experiments are given to verify the efficiency of the PESS$$ PESS $$ method for solving nonsymmetric saddle‐point problems.
本文给出了求解具有非对称正定(1,1)部分的非奇异非对称鞍点问题的参数化扩展移位分裂(PESS)方法及其诱导预条件。讨论了PESS $$ PESS $$迭代法的收敛性分析。给出了预条件矩阵的特征值分布。通过实验验证了PESS $$ PESS $$方法求解非对称鞍点问题的有效性。
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引用次数: 1
On estimation of the optimal parameter of the modulus‐based matrix splitting algorithm for linear complementarity problems on second‐order cones 二阶锥上线性互补问题的基于模的矩阵分裂算法的最优参数估计
IF 4.3 3区 数学 Q1 MATHEMATICS Pub Date : 2022-11-29 DOI: 10.1002/nla.2480
Zhizhi Li, Huai Zhang
There are many studies on the well‐known modulus‐based matrix splitting (MMS) algorithm for solving complementarity problems, but very few studies on its optimal parameter, which is of theoretical and practical importance. Therefore and here, by introducing a novel mapping to explicitly cast the implicit fixed point equation and thus obtain the iteration matrix involved, we first present the estimation approach of the optimal parameter of each step of the MMS algorithm for solving linear complementarity problems on the direct product of second‐order cones (SOCLCPs). It also works on single second‐order cone and the non‐negative orthant. On this basis, we further propose an iteration‐independent optimal parameter selection strategy for practical usage. Finally, the practicability and effectiveness of the new proposal are verified by comparing with the experimental optimal parameter and the diagonal part of system matrix. In addition, with the optimal parameter, the effectiveness of the MMS algorithm can indeed be greatly improved, even better than the state‐of‐the‐art solvers SCS and SuperSCS that solve the equivalent SOC programming.
众所周知,求解互补问题的基于模量的矩阵分裂(MMS)算法有很多研究,但对其最优参数的研究却很少,这具有重要的理论和实际意义。因此,在这里,通过引入一种新的映射来显式地投射隐式不动点方程,从而得到所涉及的迭代矩阵,我们首先给出了求解二阶锥直积线性互补问题的MMS算法每一步最优参数的估计方法。它也适用于单二阶锥和非负正交。在此基础上,我们进一步提出了一种与迭代无关的最优参数选择策略。最后,通过与实验最优参数和系统矩阵对角部分的比较,验证了新方案的实用性和有效性。此外,通过优化参数,MMS算法的有效性确实可以大大提高,甚至优于解决等效SOC规划的最先进的解算器SCS和SuperSCS。
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引用次数: 1
Backward error analysis of specified eigenpairs for sparse matrix polynomials 稀疏矩阵多项式指定特征对的后向误差分析
IF 4.3 3区 数学 Q1 MATHEMATICS Pub Date : 2022-11-21 DOI: 10.1002/nla.2476
Sk. Safique Ahmad, Prince Kanhya
This article studies the unstructured and structured backward error analysis of specified eigenpairs for matrix polynomials. The structures we discuss include T$$ T $$ ‐symmetric, T$$ T $$ ‐skew‐symmetric, Hermitian, skew Hermitian, T$$ T $$ ‐even, T$$ T $$ ‐odd, H$$ H $$ ‐even, H$$ H $$ ‐odd, T$$ T $$ ‐palindromic, T$$ T $$ ‐anti‐palindromic, H$$ H $$ ‐palindromic, and H$$ H $$ ‐anti‐palindromic matrix polynomials. Minimally structured perturbations are constructed with respect to Frobenius norm such that specified eigenpairs become exact eigenpairs of an appropriately perturbed matrix polynomial that also preserves sparsity. Further, we have used our results to solve various quadratic inverse eigenvalue problems that arise from real‐life applications.
本文研究了矩阵多项式指定特征对的非结构化和结构化后向误差分析。我们讨论的结构包括T $$ T $$对称、T $$ T $$偏对称、hermite、skew hermite、T $$ T $$偶、T $$ T $$奇、H $$ H $$偶、H $$ H $$奇、T $$ T $$回文、T $$ T $$反回文、H $$ H $$回文和H $$ H $$反回文矩阵多项式。基于Frobenius范数构造了最小结构摄动,使得指定的特征对成为适当摄动的矩阵多项式的精确特征对,并且保持了稀疏性。此外,我们已经使用我们的结果来解决实际应用中出现的各种二次型反特征值问题。
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引用次数: 0
Fast solution of three‐dimensional elliptic equations with randomly generated jumping coefficients by using tensor‐structured preconditioners 用张量结构预条件快速求解具有随机跳跃系数的三维椭圆方程
IF 4.3 3区 数学 Q1 MATHEMATICS Pub Date : 2022-11-17 DOI: 10.1002/nla.2477
B. Khoromskij, V. Khoromskaia
In this paper, we propose and analyze the numerical algorithms for fast solution of periodic elliptic problems in random media in ℝd$$ {mathbb{R}}^d $$ , d=2,3$$ d=2,3 $$ . Both the two‐dimensional (2D) and three‐dimensional (3D) elliptic problems are considered for the jumping equation coefficients built as a checkerboard type configuration of bumps randomly distributed on a large L×L$$ Ltimes L $$ , or L×L×L$$ Ltimes Ltimes L $$ lattice, respectively. The finite element method discretization procedure on a 3D n×n×n$$ ntimes ntimes n $$ uniform tensor grid is described in detail, and the Kronecker tensor product approach is proposed for fast generation of the stiffness matrix. We introduce tensor techniques for the construction of the low Kronecker rank spectrally equivalent preconditioner in a periodic setting to be used in the framework of the preconditioned conjugate gradient iteration. The discrete 3D periodic Laplacian pseudo‐inverse is first diagonalized in the Fourier basis, and then the diagonal matrix is reshaped into a fully populated third‐order tensor of size n×n×n$$ ntimes ntimes n $$ . The latter is approximated by a low‐rank canonical tensor by using the multigrid Tucker‐to‐canonical tensor transform. As an example, we apply the presented solver in numerical analysis of stochastic homogenization method where the 3D elliptic equation should be solved many hundred times, and where for every random sampling of the equation coefficient one has to construct the new stiffness matrix and the right‐hand side. The computational characteristics of the presented solver in terms of a lattice parameter L$$ L $$ and the grid‐size, nd$$ {n}^d $$ , in both 2D and 3D cases are illustrated in numerical tests. Our solver can be used in various applications where the elliptic problem should be solved for a number of different coefficients for example, in many‐particle dynamics, protein docking problems or stochastic modeling.
在本文中,我们提出并分析了随机介质中周期椭圆问题的快速求解的数值算法ℝd$${mathbb{R}}^d$$,d=2,3$$d=2,3$$。对于跳跃方程系数,分别考虑了二维(2D)和三维(3D)椭圆问题,该跳跃方程系数是随机分布在大L×L$$Ltimes L$$或L×L×L$LtimesL$$格上的凸块的棋盘型配置。详细描述了三维n×n×n$ntimestimesn$$均匀张量网格上的有限元离散化过程,并提出了Kronecker张量积方法来快速生成刚度矩阵。我们介绍了张量技术,用于在周期设置中构造低Kronecker秩谱等价预条件器,用于预条件共轭梯度迭代的框架中。离散的三维周期拉普拉斯伪逆首先在傅立叶基中对角化,然后将对角矩阵重塑为大小为n×n×n$$ntimestimesn$$的完全填充三阶张量。后者通过使用多重网格Tucker到正则张量变换由低阶正则张量近似。例如,我们将所提出的求解器应用于随机均匀化方法的数值分析中,其中三维椭圆方程应求解数百次,并且对于方程系数的每次随机采样,都必须构造新的刚度矩阵和右手边。在2D和3D情况下,所提出的求解器在晶格参数L$$L$$和网格大小nd$${n}^d$$方面的计算特性在数值测试中得到了说明。我们的求解器可用于各种应用,其中椭圆问题应针对许多不同的系数求解,例如,在多粒子动力学、蛋白质对接问题或随机建模中。
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引用次数: 1
QZ algorithm with two‐sided generalized Rayleigh quotient shifts 具有双侧广义瑞利商位移的QZ算法
IF 4.3 3区 数学 Q1 MATHEMATICS Pub Date : 2022-11-10 DOI: 10.1002/nla.2475
X. Chen, Hongguo Xu
We generalize the recently proposed two‐sided Rayleigh quotient single‐shift and the two‐sided Grassmann–Rayleigh quotient double‐shift used in the QR algorithm and apply the generalized versions to the QZ algorithm. With such shift strategies the QZ algorithm normally has a cubic local convergence rate. Our main focus is on the modified shift strategies and their corresponding truncated versions. Numerical examples are provided to demonstrate the convergence properties and the efficiency of the QZ algorithm equipped with the proposed shifts. For the truncated versions, local convergence analysis is not provided. Numerical examples show they outperform the modified shifts and the standard Rayleigh quotient single‐shift and Francis double‐shift.
我们推广了最近提出的QR算法中使用的双侧瑞利商单频移和双侧Grassmann–Rayleigh商双频移,并将广义版本应用于QZ算法。对于这样的移位策略,QZ算法通常具有三次局部收敛速度。我们的主要关注点是修改后的转换策略及其相应的截断版本。数值算例证明了带有所提出的移位的QZ算法的收敛性和效率。对于截断的版本,不提供局部收敛性分析。数值算例表明,它们优于修正位移和标准瑞利商单位移和弗朗西斯双位移。
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引用次数: 0
Enhanced algebraic substructuring for symmetric generalized eigenvalue problems 对称广义特征值问题的增强代数子结构
IF 4.3 3区 数学 Q1 MATHEMATICS Pub Date : 2022-10-17 DOI: 10.1002/nla.2473
V. Kalantzis, L. Horesh
This article proposes a new substructuring algorithm to approximate the algebraically smallest eigenvalues and corresponding eigenvectors of a symmetric positive‐definite matrix pencil (A,M)$$ left(A,Mright) $$ . The proposed approach partitions the graph associated with (A,M)$$ left(A,Mright) $$ into a number of algebraic substructures and builds a Rayleigh–Ritz projection subspace by combining spectral information associated with the interior and interface variables of the algebraic domain. The subspace associated with interior variables is built by computing substructural eigenvectors and truncated Neumann series expansions of resolvent matrices. The subspace associated with interface variables is built by computing eigenvectors and associated leading derivatives of linearized spectral Schur complements. The proposed algorithm can take advantage of multilevel partitionings when the size of the pencil. Experiments performed on problems stemming from discretizations of model problems showcase the efficiency of the proposed algorithm and verify that adding eigenvector derivatives can enhance the overall accuracy of the approximate eigenpairs, especially those associated with eigenvalues located near the origin.
本文提出了一种新的子结构算法来近似对称正定矩阵pencil (a,M) $$ left(A,Mright) $$的代数最小特征值和相应的特征向量。该方法将与(A,M) $$ left(A,Mright) $$相关的图划分为多个代数子结构,并结合与代数域的内部变量和界面变量相关的谱信息构建Rayleigh-Ritz投影子空间。通过计算子结构特征向量和求解矩阵的截断诺伊曼级数展开式,建立了与内部变量相关的子空间。通过计算线性化谱舒尔补的特征向量和相关导导数,建立了与界面变量相关的子空间。该算法可以在铅笔大小不同的情况下利用多层分区的优势。在模型问题离散化过程中进行的实验证明了该算法的有效性,并验证了添加特征向量导数可以提高近似特征对的整体精度,特别是那些与原点附近的特征值相关的特征对。
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引用次数: 0
期刊
Numerical Linear Algebra with Applications
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