Spectral behavior of preconditioned non-Hermitian multilevel block Toeplitz matrices with matrix-valued symbol

This note is devoted to preconditioning strategies for non-Hermitian multilevel block Toeplitz linear systems associated with a multivariate Lebesgue integrable matrix-valued symbol. In particular, we consider special preconditioned matrices, where the preconditioner has a band multilevel block Toeplitz structure, and we complement known results on the localization of the spectrum with global distribution results for the eigenvalues of the preconditioned matrices. In this respect, our main result is as follows. Let $I_k:=(-\pi,\pi)^k$, let $\mathcal M_s$ be the linear space of complex $s\times s$ matrices, and let $f,g:I_k\to\mathcal M_s$ be functions whose components $f_{ij},\,g_{ij}:I_k\to\mathbb C,\ i,j=1,\ldots,s,$ belong to $L^\infty$. Consider the matrices $T_n^{-1}(g)T_n(f)$, where $n:=(n_1,\ldots,n_k)$ varies in $\mathbb N^k$ and $T_n(f),T_n(g)$ are the multilevel block Toeplitz matrices of size $n_1\cdots n_ks$ generated by $f,g$. Then $\{T_n^{-1}(g)T_n(f)\}_{n\in\mathbb N^k}\sim_\lambda g^{-1}f$, i.e. the family of matrices $\{T_n^{-1}(g)T_n(f)\}_{n\in\mathbb N^k}$ has a global (asymptotic) spectral distribution described by the function $g^{-1}f$, provided $g$ possesses certain properties (which ensure in particular the invertibility of $T_n^{-1}(g)$ for all $n$) and the following topological conditions are met: the essential range of $g^{-1}f$, defined as the union of the essential ranges of the eigenvalue functions $\lambda_j(g^{-1}f),\ j=1,\ldots,s$, does not disconnect the complex plane and has empty interior. This result generalizes the one obtained by Donatelli, Neytcheva, Serra-Capizzano in a previous work, concerning the non-preconditioned case $g=1$. The last part of this note is devoted to numerical experiments, which confirm the theoretical analysis and suggest the choice of optimal GMRES preconditioning techniques to be used for the considered linear systems.Comment: 18 pages, 26 figure

PRECONDITIONING STRATEGIES FOR 2D FINITE DIFFERENCE MATRIX SEQUENCES

In this paper we are concerned with the spectral analysis of the sequence of preconditioned matrices {P-1 n An(a, m1, m2, k)}n, where n = (n1, n2), N(n) = n1n2 and where An (a, m1, m2, k) 08 \u211dN(n) 7 N(n) is the symmetric two-level matrix coming from a high-order Finite Difference (FD) discretization of the problem (equation presented) with \u3bd denoting the unit outward normal direction and where m1 and m2 are parameters identifying the precision order of the used FD schemes. We assume that the coefficient a(x, y) is nonnegative and that the set of the possible zeros can be represented by a finite collection of curves. The proposed preconditioning matrix sequences correspond to two different choices: the Toeplitz sequence {An(1, m1, m2, k)}n and a Toeplitz based sequence that adds to the Toeplitz structure the informative content given by the suitable scaled diagonal part of An(a, m1, m2 k). The former case gives rise to optimal preconditioning sequences under the assumption of positivity and boundedness of a. With respect to the latter, the main result is the proof of the asymptotic clustering at unity of the eigenvalues of the preconditioned matrices, where the "strength" of the cluster depends on the order k, on the regularity features of a(x, y) and on the presence of zeros of a(x, y)

Block diagonal and schur complement preconditioners for block-toeplitz systems with small size blocks

In this paper we consider the solution of Hermitian positive definite block-Toeplitz systems with small size blocks. We propose and study block diagonal and Schur complement preconditioners for such block-Toeplitz matrices. We show that for some block-Toeplitz matrices, the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers where this fixed number depends only on the size of the block. Hence, conjugate gradient type methods, when applied to solving these preconditioned block-Toeplitz systems with small size blocks, converge very fast. Recursive computation of such block diagonal and Schur complement preconditioners is considered by using the nice matrix representation of the inverse of a block-Toeplitz matrix. Applications to block-Toeplitz systems arising from least squares filtering problems and queueing networks are presented. Numerical examples are given to demonstrate the effectiveness of the proposed method. © 2007 Society for Industrial and Applied Mathematics.published_or_final_versio

An Efficient Block Circulant Preconditioner For Simulating Fracture Using Large Fuse Networks

{\it Critical slowing down} associated with the iterative solvers close to the critical point often hinders large-scale numerical simulation of fracture using discrete lattice networks. This paper presents a block circlant preconditioner for iterative solvers for the simulation of progressive fracture in disordered, quasi-brittle materials using large discrete lattice networks. The average computational cost of the present alorithm per iteration is $O(rs log s) + delops$, where the stiffness matrix ${\bf A}$ is partioned into $r$-by-$r$ blocks such that each block is an $s$-by-$s$ matrix, and $delops$ represents the operational count associated with solving a block-diagonal matrix with $r$-by-$r$ dense matrix blocks. This algorithm using the block circulant preconditioner is faster than the Fourier accelerated preconditioned conjugate gradient (PCG) algorithm, and alleviates the {\it critical slowing down} that is especially severe close to the critical point. Numerical results using random resistor networks substantiate the efficiency of the present algorithm.Comment: 16 pages including 2 figure

A preconditioned MINRES method for optimal control of wave equations and its asymptotic spectral distribution theory

In this work, we propose a novel preconditioned Krylov subspace method for solving an optimal control problem of wave equations, after explicitly identifying the asymptotic spectral distribution of the involved sequence of linear coefficient matrices from the optimal control problem. Namely, we first show that the all-at-once system stemming from the wave control problem is associated to a structured coefficient matrix-sequence possessing an eigenvalue distribution. Then, based on such a spectral distribution of which the symbol is explicitly identified, we develop an ideal preconditioner and two parallel-in-time preconditioners for the saddle point system composed of two block Toeplitz matrices. For the ideal preconditioner, we show that the eigenvalues of the preconditioned matrix-sequence all belong to the set $\left(-\frac{3}{2},-\frac{1}{2}\right)\bigcup \left(\frac{1}{2},\frac{3}{2}\right)$ well separated from zero, leading to mesh-independent convergence when the minimal residual method is employed. The proposed {parallel-in-time} preconditioners can be implemented efficiently using fast Fourier transforms or discrete sine transforms, and their effectiveness is theoretically shown in the sense that the eigenvalues of the preconditioned matrix-sequences are clustered around $\pm 1$, which leads to rapid convergence. When these parallel-in-time preconditioners are not fast diagonalizable, we further propose modified versions which can be efficiently inverted. Several numerical examples are reported to verify our derived localization and spectral distribution result and to support the effectiveness of our proposed preconditioners and the related advantages with respect to the relevant literature

Approximation and spectral analysis for large structured linear systems.

In this work we are interested in standard and less standard structured linear systems coming from applications in various _elds of computational mathematics and often modeled by integral and/or di_erential equations. Starting from classical Toeplitz and Circulant structures, we consider some extensions as g-Toeplitz and g-Circulants matrices appearing in several contexts in numerical analysis and applications. Then we consider special matrices arising from collocation methods for di_erential equations: also in this case, under suitable assumptions we observe a Toeplitz structure. More in detail we _rst propose a detailed study of singular values and eigenvalues of g-circulant matrices and then we provide an analysis of distribution of g-Toeplitz sequences. Furthermore, when possible, we consider Krylov space methods with special attention to the minimization of the computational work. When the involved dimensions are large, the Preconditioned Conjugate Gradient (PCG) method is recommended because of the much stronger robustness with respect to the propagation of errors. In that case, crucial issues are the convergence speed of this iterative solver, the use of special techniques (preconditioning, multilevel techniques) for accelerating the convergence, and a careful study of the spectral properties of such matrices. Finally, the use of radial basis functions allow of determining and studying the asymptotic behavior of the spectral radii of collocation matrices approximating elliptic boundary value problems

Approximation and spectral analysis for large structured linear systems.

In this work we are interested in standard and less standard structured linear systems coming from applications in various _elds of computational mathematics and often modeled by integral and/or di_erential equations. Starting from classical Toeplitz and Circulant structures, we consider some extensions as g-Toeplitz and g-Circulants matrices appearing in several contexts in numerical analysis and applications. Then we consider special matrices arising from collocation methods for di_erential equations: also in this case, under suitable assumptions we observe a Toeplitz structure. More in detail we _rst propose a detailed study of singular values and eigenvalues of g-circulant matrices and then we provide an analysis of distribution of g-Toeplitz sequences. Furthermore, when possible, we consider Krylov space methods with special attention to the minimization of the computational work. When the involved dimensions are large, the Preconditioned Conjugate Gradient (PCG) method is recommended because of the much stronger robustness with respect to the propagation of errors. In that case, crucial issues are the convergence speed of this iterative solver, the use of special techniques (preconditioning, multilevel techniques) for accelerating the convergence, and a careful study of the spectral properties of such matrices. Finally, the use of radial basis functions allow of determining and studying the asymptotic behavior of the spectral radii of collocation matrices approximating elliptic boundary value problems