1,721,011 research outputs found
Reliable preconditioned iterative linear solvers for some numerical integrators
No full textImplicit time-step numerical integrators for ordinary and evolutionary partial differential equations need, at each step, the solution of linear algebraic equations that are unsymmetric and often large and sparse. Recently, a block preconditioner based on circulant approximations for the linear systems arising in the boundary value methods (BVMs) was introduced by the author. Here, some circulant approximations are compared and a further new type is considered. Numerical experiments are presented to check the effectiveness of the various approximations that can be used in the underlying block preconditioner. Copyright © 2001 John Wiley & Sons, Ltd
The spectrum of circulant-like preconditioners for some general linear multistep formulas for linear boundary value problems
No full textThe spectrum of the eigenvalues, the conditioning, and other related properties of circulant-like matrices used to build up block preconditioners for the nonsymmetric algebraic linear equations of time-step integrators for linear boundary value problems are analyzed. Moreover, results concerning the entries of a class of Toeplitz matrices related to the latter are proposed. Generalizations of implicit linear multistep formulas in boundary value form are considered in more detail. It is proven that there exists a new class of approximations which are well conditioned and whose eigenvalues have positive and bounded real and bounded imaginary part. Moreover, it is observed that preconditioners based on other circulant-like approximations, which are well suited for Hermitian linear systems, can be severely ill conditioned even if the matrices of the nonpreconditioned system are well conditioned
Symposium: Nonlinear computational techniques with applications to image restoration and cochlear modeling
No full textComputing Function of Large Matrices by a Preconditioned Rational Krylov Method
No full textRational Krylov methods are a powerful alternative for computing the product of a function of a large matrix times a given vector. However, the creation of the underlying rational subspaces requires solving sequences of large linear systems, a delicate task that can require intensive computational resources and should be monitored to avoid the creation of subspace different to those required whenever, e.g., the underlying matrices are ill-conditioned. We propose the use of robust preconditioned iterative techniques to speedup the underlying process. We also discuss briefly how the inexact solution of these linear systems can affect the computed subspace. A preliminary test approximating a fractional power of the Laplacian matrix is included
Computing functions of very large matrices with small TT/QTT ranks by quadrature formulas
No full textThe computation of matrix functions using quadrature formulas and rational approximations of very large structured matrices using tensor trains (TT), and quantized tensor trains (QTT) is considered here. The focus is on matrices with a small TT/QTT rank. Some analysis of the error produced by the use of the TT/QTT representation and the underlying approximation formula used is also provided. Promising experiments on exponential, power, Mittag-Leffler and logarithm function of multilevel Toeplitz matrices, that are among those which generate a low TT/QTT rank representation, are also provided, confirming that the proposed approach is feasible
Efficient preconditioning for sequences of parametric complex symmetric linear systems
Full text linkSolution 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
Band-Toeplitz Preconditioned GMRES Iterations for Time-dependent PDEs
No full textNonsymmetric linear systems of algebraic equations which are small rank perturbations of block band-Toeplitz matrices from discretization of time-dependent PDEs are considered. With a combination of analytical and experimental results, we examine the convergence characteristics of the GMRES method with circulant-like block preconditioning for solving these systems
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