Krylov Subspace Acceleration of Waveform Relaxation

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 conjugate gradient algorithm and two convolution variants of the GMRES algorithm and analyze their convergence behavior. We prove that the convolution Krylov-subspace algorithms for initial-value problems have the same rate of convergence as their linear algebra counterparts. Analytical examples are given to illustrate the operation of convolution Krylov subspace methods. Experimental results compare the convergence of waveform relaxation, wavform GMRES, convolution SOR, and convolution CG applied to solving a linear IVP, as well as that of CG for solving the associated linear algebraic equation.