Efficient Solution of Parabolic Equations by Krylov Approximation Methods

Abstract

In this paper we take a new look at numerical techniques for solving parabolic equations by the method of lines. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus, the resulting approximation consists of applying an evolution operator of very small dimension to a known vector which is, in turn, computed accurately by exploiting high-order rational Chebyshev and Padé approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications, and as a result the algorithm can easily be parallelized and vectorized. Further parallelism is introduced by expanding the rational approximations into partial fractions. Some relevant approximation and stability issues are discussed. We present some numerical experiments with the method and compare its performance with a few explicit and implicit algorithms.

Three‐dimensional Model of the Structure and Evolution of Coronal Mass Ejections

“…A detailed description of the mathematical theory behind these methods is beyond the scope of this paper. We instead brieÑy describe the main ideas behind the construction of an exponential propagation scheme and refer the interested reader to references (Hochbruck & Lubich 1997 ;Hochbruck et al 1998 ;Gallopoulos & Saad 1992) for a more exhaustive treatment of the subject.…”

A new investigation of the extended Krylov subspace method for matrix function evaluations

“…The development of numerical procedures for approximating the action of f (A) to a vector has received considerable attention in the last two decades. This is possibly related to the significant increase of methods for the numerical solution of partial differential equations that either directly approximate the exact solution (see, e.g., [20], [47], [24], [23]), or employ matrix functional integrators; see, e.g., [30], [28], [29]. In addition, large-scale advanced scientific applications often require function evaluations of matrices; see, e.g., [5], [43], [55], [17].…”

Krylov subspace methods for computing hydrodynamic interactions in Brownian dynamics simulations

“…Such polynomial approximations of the square root of a matrix times a vector (and in general, any function of a matrix times a vector) appeared soon afterward in the numerical analysis literature, [15][16][17] but these studies were unaware of Fixman's contribution. An essential feature of these approximations is that p(D) is not needed and is never computed explicitly; rather, only p(D)z is required, which can be computed much more efficiently.…”

“…11,13 However, when multiple beads are assembled into compact structures, the TEA method does not show the correct scaling of translational diffusivity with N. 14 In this paper, we study Krylov subspace methods for computing correlated Brownian displacements for use in BD. The methods are not new in the numerical analysis community, [15][16][17] but to the best of our knowledge, they appear to be unknown in the BD literature. These methods have two major advantages over Chebyshev polynomial approximations.…”