Approximation of matrix operators applied to multiple vectors

Abstract

In this paper we propose a numerical method for approximating the product of a matrix function with multiple vectors by Krylov subspace methods combined with a QR decomposition of these vectors. This problem arises in the implementation of exponential integrators for semilinear parabolic problems. We will derive reliable stopping criteria and we suggest variants using up-and downdating techniques. Moreover, we show how Ritz vectors can be included in order to speed up the computation even further. By a number of numerical examples, we will illustrate that the proposed method will reduce the total number of Krylov steps significantly compared to a standard implementation if the vectors correspond to the evaluation of a smooth function at certain quadrature points.

A block Krylov subspace time‐exact solution method for linear ordinary differential equation systems

“…where R k , V kC1 , H kC1,k , E k are from (18) and (20) and u.t / is the solution of the projected IVP (14). Furthermore, the error e k .t / Á y.t/ y k .t /, with y.t/ being the exact solution of (7), is given by…”

“…An attractive feature of exponential time integrators is a combination of excellent stability and accuracy properties, with the latter being usually better than in the standard implicit time integrators [3][4][5][6]. The interest in exponential time integration is due to the new, challenging applications [7-9] as well as to the recent progress in Krylov subspace techniques to compute actions of matrix functions for large matrices (e.g., [10][11][12][13][14][15][16][17][18][19][20]). Other methods to compute actions of large matrix functions have also been developed, such as ones based on Chebyshev and Taylor polynomials, for example, [12,20,21].In some applications in which explicit time integration is by far inefficient, a gain with implicit or exponential time integration is not guaranteed [22,23].…”

Exponential Rosenbrock-Type Methods

“…The latter three make use of Krylov subspace approximations. To improve the efficiency of the Krogstad method, we reused information from previously computed Krylov subspaces, an approach proposed in [13]. Since an adaptive step-size control based on embedding is not possible for Krogstad's method, we ran this method with constant step size.…”

“…Note that, it is impossible to give a reformulation of Krogstad's method in such a way that only one expensive Krylov subspace is required in each step. The gain achieved by reusing previously computed Krylov subspaces [13] does not compensate this disadvantage. Moreover, Krogstad's method has four stages and uses even more matrix functions than exprb43.…”

Towards parallel-in-time simulations of turbulent Rayleigh-Bénard convection

“…Consider the thin QR factorization of G, G = QR, where Q ∈ R n×s has orthonormal columns and R ∈ R s×s is an upper triangular matrix with nonnegative diagonal entries. As Theorem 1 in [76] states, the entries r jk of R satisfy…”