57 research outputs found

    Preconditioners for Krylov subspace methods: An overview

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    When simulating a mechanism from science or engineering, or an industrial process, one is frequently required to construct a mathematical model, and then resolve this model numerically. If accurate numerical solutions are necessary or desirable, this can involve solving large-scale systems of equations. One major class of solution methods is that of preconditioned iterative methods, involving preconditioners which are computationally cheap to apply while also capturing information contained in the linear system. In this article, we give a short survey of the field of preconditioning. We introduce a range of preconditioners for partial differential equations, followed by optimization problems, before discussing preconditioners constructed with less standard objectives in mind

    A preconditioned MINRES method for nonsymmetric Toeplitz matrices

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    Circulant preconditioning for symmetric Toeplitz linear systems is well established; theoretical guarantees of fast convergence for the conjugate gradient method are descriptive of the convergence seen in computations. This has led to robust and highly efficient solvers based on use of the fast Fourier transform exactly as originally envisaged in [G. Strang, Stud. Appl. Math., 74 (1986), pp. 171--176]. For nonsymmetric systems, the lack of generally descriptive convergence theory for most iterative methods of Krylov type has provided a barrier to such a comprehensive guarantee, though several methods have been proposed and some analysis of performance with the normal equations is available. In this paper, by the simple device of reordering, we rigorously establish a circulant preconditioned short recurrence Krylov subspace iterative method of minimum residual type for nonsymmetric (and possibly highly nonnormal) Toeplitz systems. Convergence estimates similar to those in the symmetric case are established

    Efficient preconditioning for sequences of parametric complex symmetric linear systems

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    Solution of sequences of complex symmetric linear systems of the form Ajxj = bj, j = 0,..., s, Aj = A + αjEj, A Hermitian, E0, ..., E a complex diagonal matrices and α0, ..., αa scalar complex parameters arise in a variety of challenging problems. This is the case of time dependent PDEs; lattice gauge computations in quantum chromodynamics; the Helmholtz equation; shift-and-invert and Jacobi-Davidson algorithms for large-scale eigenvalue calculations; problems in control theory and many others. If A is symmetric and has real entries then Aj is complex symmetric. The case A Hermitian positive semideflnite, Re(αj) ≥ 0 and such that the diagonal entries of E j, j = 0,..., s have nonnegative real part is considered here. Some strategies based on the update of incomplete factorizations of the matrix A and A-1 are introduced and analyzed. The numerical solution of sequences of algebraic linear systems from the discretization of the real and complex Helmholtz equation and of the diffusion equation in a rectangle illustrate the performance of the proposed approaches

    Symmetrization Techniques in Image Deblurring

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    This paper presents a couple of preconditioning techniques that can be used to enhance the performance of iterative regularization methods applied to image deblurring problems with a variety of point spread functions (PSFs) and boundary conditions. More precisely, we first consider the anti-identity preconditioner, which symmetrizes the coefficient matrix associated to problems with zero boundary conditions, allowing the use of MINRES as a regularization method. When considering more sophisticated boundary conditions and strongly nonsymmetric PSFs, the anti-identity preconditioner improves the performance of GMRES. We then consider both stationary and iteration-dependent regularizing circulant preconditioners that, applied in connection with the anti-identity matrix and both standard and flexible Krylov subspaces, speed up the iterations. A theoretical result about the clustering of the eigenvalues of the preconditioned matrices is proved in a special case. The results of many numerical experiments are reported to show the effectiveness of the new preconditioning techniques, including when considering the deblurring of sparse images

    Symbol Based Convergence Analysis in Block Multigrid Methods with applications for Stokes problems

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    The main focus of this paper is the study of efficient multigrid methods for large linear system with a particular saddle-point structure. In particular, we propose a symbol based convergence analysis for problems that have a hidden block Toeplitz structure. Then, they can be investigated focusing on the properties of the associated generating function f\mathbf{f}, which consequently is a matrix-valued function with dimension depending on the block of the problem. As numerical tests we focus on the matrix sequence stemming from the finite element approximation of the Stokes equation. We show the efficiency of the methods studying the hidden 9×99\times 9 block structure of the obtained matrix sequence proposing an efficient algebraic multigrid method with convergence rate independent of the matrix size. Moreover, we present several numerical tests comparing the results with different known strategies

    Preconditioners for symmetrized Toeplitz and multilevel Toeplitz matrices

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    When solving linear systems with nonsymmetric Toeplitz or multilevel Toeplitz matrices using Krylov subspace methods, the coefficient matrix may be symmetrized. The preconditioned MINRES method can then be applied to this symmetrized system, which allows rigorous upper bounds on the number of MINRES iterations to be obtained. However, effective preconditioners for symmetrized (multilevel) Toeplitz matrices are lacking. Here, we propose novel ideal preconditioners, and investigate the spectra of the preconditioned matrices. We show how these preconditioners can be approximated and demonstrate their effectiveness via numerical experiments

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

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    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 (−32,−12)⋃(12,32)\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 ±1\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
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