Abstract
In this paper we describe and analyze Krylov subspace techniques for accelerating the convergence of waveform relaxation for solving time-dependent problems. A new class of accelerated waveform methods, convolution Krylov subspace methods, is presented. In particular, we give convolution variants of the CG algorithm and the GMRES algorithm and analyze their convergence behavior. We prove that the convolution Krylov subspace algorithms for initial value problems have the same convergence bounds as their linear algebra counterparts. Analytical examples are given to illustrate the operation of convolution Krylov subspace methods. Experimental results are presented which show the convergence behavior of traditional and convolution waveform methods applied to solving a linear initial value problem as well as the convergence behavior of static Krylov subspace methods applied to solving the associated linear algebraic equation.
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