Preconditioning Lanczos Approximations to the Matrix Exponential

Abstract

Abstract. The Lanczos method is an iterative procedure to compute an orthogonal basis for the Krylov subspace generated by a symmetric matrix A and a starting vector v. An interesting application of this method is the computation of the matrix exponential exp(−τ A)v. This vector plays an important role in the solution of parabolic equations where A results from some form of discretization of an elliptic operator. In the present paper we will argue that for these applications the convergence behavior of this method can be unsatisfactory. We will propose a modified method that resolves this by a simple preconditioned transformation at the cost of an innerouter iteration. A priori error bounds are presented that are independent of the norm of A. This shows that the worst case convergence speed is independent of the mesh width in the spatial discretization of the elliptic operator. We discuss, furthermore, a posteriori error estimation and the tuning of the coupling between the inner and outer iteration. We conclude with several numerical experiments with the proposed method.

Approximation Algorithms for Semidefinite Packing Problems with Applications to Maxcut and Graph Coloring

“…Thus, this solves systems of linear equations in O((n + m) log(1/ )) iterations where > 0 is the relative error and m is the number of nonzeros in the matrix A. Overall, Theorem 3.3 of [30] indicates that O(log 2 (1/ )) iterations are required to approximate each column of the exponential. This results an overall complexity of O(n(n + m) log 3 (1/ )).…”

“…The main improvement in [30] results from using Krylov subspaces generated by (I + γA) −1 , where γ > 0 (see also [15]). Therefore, at each iteration, we need to compute a solution to the linear system y k+1 = (I + γA)y k .…”

“…We use a method which we refer to as SI-Lanczos, or the shift-and-invert Lanczos, developed by van den Eshof and Hochbruck [30]. The main improvement in [30] results from using Krylov subspaces generated by (I + γA) −1 , where γ > 0 (see also [15]).…”

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

“…Another possibility, first suggested for the exponential function in [31], is given by the following estimate…”

“…Given a real parameter γ > 0, the Shift-Invert Lanczos method constructs the Krylov subspace K m ((I + γA) −1 , v), computes the projection and restriction T m of the shifted and inverted matrix (I + γA) −1 , and then computes an approximation as Q m f (γ −1 (T −1 m − I))e 1 , where the columns of Q m form an orthonormal basis of K m ((I + γA) −1 , v). The method was analyzed in [40] for a class of functions, and in [31] for the exponential function. In [39] a study of the parameter γ was performed, and in the symmetric non-singular case the value γ = 1/ √ λ min λ max was obtained as a quasi-optimal estimate.…”