SIAM Journal on Matrix Analysis and Applications最新文献

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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 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications

Pub Date : 2023-11-30 DOI: 10.1137/22m1516476

Shu-Lin Wu, Zhiyong Wang, Tao Zhou

SIAM Journal on Matrix Analysis and Applications, Volume 44, Issue 4, Page 1771-1798, December 2023.
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.

矩阵分析与应用，第44卷，第4期，第1771-1798页，2023年12月。摘要。求解线性系统[数学]往往是主要的计算负担，当必须在一个问题中求解一个向前向后的进化方程时，其中[数学]是所谓的时空离散化后的前向子问题的一次性矩阵。一个有效的解算器需要一个好的[数学]前提条件。受[math]结构的启发，我们将[math]作为[math]的先决条件，其中[math]是一个块[math]-循环矩阵，通过将[math]中的Toeplitz矩阵替换为[math]-循环矩阵来构建。通过[math]的块傅立叶对角化，预处理步骤[math]的计算可以对所有时间步骤并行化。我们给出了预条件矩阵[math]的谱分析，并证明了对于任何一步稳定时间积分器[math]，如果参数[math]在[math]的时间步数[math]方面弱缩放[math]，则[math]的特征值在网格无关区间[math]中传播，其中[math]是一个自由常数。给出了该预调节器的两种应用:pde约束最优控制问题和抛物型源识别问题。数值结果表明，谱分析能很好地预测预条件共轭梯度法的收敛速度。

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引用次数: 0

Generalized Matrix Nearness Problems 广义矩阵逼近性问题

2区数学 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications

Pub Date : 2023-11-10 DOI: 10.1137/22m1526034

Zihao Li, Lek-Heng Lim

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.

对于涉及秩、范数、对称、双边积和规定特征值的各种约束，用奇异值分解和广义奇异值分解可以找到全局最小解的封闭形式。这将Friedland-Torokhti对广义秩约束近似问题的解推广到其他约束，并提供了秩约束在奇异值分解方面的另一种解。对于Toeplitz、Hankel、循环、非负性、随机性、正半正定性、规定特征向量等更复杂的涉及结构约束，证明了一种简单的迭代方法是线性且全局收敛于全局最小解的。

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 Q2 MATHEMATICS, APPLIED

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

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 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications

Pub Date : 2023-11-10 DOI: 10.1137/22m1514647

Zhen-Chen Guo, Xin Liang

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 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications

Pub Date : 2023-11-09 DOI: 10.1137/22m1509953

Mareike Dressler, André Uschmajew, Venkat Chandrasekaran

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.

线性算子在矩阵空间上的分解或近似为Kronecker积的和在矩阵方程和低秩建模中起着重要的作用。Frobenius范数中的近似问题有一个众所周知的解，即奇异值分解。然而，谱范数的逼近问题更具有挑战性，因为谱范数对线性算子来说更自然。特别是，在谱范数上，Frobenius范数解可能远非最优。我们描述了一种基于半定规划的交替优化方法，以获得高质量的谱范数近似，并给出了计算实验来说明我们方法的优点。

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 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications

Pub Date : 2023-11-07 DOI: 10.1137/22m1539137

Anwa Zhou, Kun Liu, Jinyan Fan

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.

本文考虑张量绝对值方程(TAVEs)。当一个张量是奇数阶的行对角线时，我们证明了TAVEs可以简化为一个代数方程;当它是行对角且是非奇异的偶阶问题时，我们证明了TAVEs等价于一个多项式互补问题。当没有张量是行对角线时，我们以两种不同的方式将TAVEs等效地表述为多项式优化问题。它们中的每一个都可以用Lasserre的半定松弛层次来求解。讨论了有限收敛性质。数值实验证明了所提方法的有效性。

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 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications

Pub Date : 2023-10-30 DOI: 10.1137/22m1517184

Manuel Radons

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。

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 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications

Pub Date : 2023-10-30 DOI: 10.1137/22m1497985

Rob Claes, Karl Meerbergen, Simon Telen

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.

求解具有特征向量非线性(PEPv)的多项式特征值问题是一个有趣的计算挑战，超出了非线性特征值问题的成熟方法的范围。我们提出了这些方法的自然推广，这导致了计算复平面紧致区域中PEPv的所有特征值的轮廓积分方法。我们的方法可用于求解任何适当的多项式或有理函数方程的一般系统。

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 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications

Pub Date : 2023-10-25 DOI: 10.1137/22m1513174

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.

广义特征值问题涉及一个奇异铅笔可能是非常具有挑战性的解决，无论是在准确性和效率方面。在第一部分中，我们给出了对单个铅笔进行排序补全的加法，现在我们开发了两种替代方法。第一种技术是基于维度等于铅笔的法秩的子空间上的投影，而第二种方法是利用增广矩阵铅笔。投影法因其效率而成为一般奇异铅笔最具吸引力的版本，而增广铅笔法可能适用于具有增广铅笔的线性系统可以有效求解的应用。

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 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications

Pub Date : 2023-10-20 DOI: 10.1137/22m150071x

Anna Ma, Dominik Stöger, Yizhe Zhu

引用次数: 0

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