A framework of the harmonic Arnoldi method for evaluating ph functions with applications to exponential integrators
In recent years, a great deal of attention has been focused on exponential integrators. The important ingredient to the implementation of exponential integrators is the efficient and accurate evaluation of the so called ph functions on a given vector. The Krylov subspace method is an important technique for this problem. For this type of method, however, restarts become essential for the sake of storage requirements or due to computational complexities of evaluating matrix function on a reduced matrix of growing size. Another problem in computing ph functions is the lack of a clear residual notion. The contribution of this work is threefold. First, we introduce a framework of the harmonic Arnoldi method for ph functions, which is based on the residual and the oblique projection technique. Second, we establish the relationship between the harmonic Arnoldi approximation and the Arnoldi approximation, and compare the harmonic Arnoldi method and the Arnoldi method from a theoretical point of view. Third, we apply the thick restarting strategy to the harmonic Arnoldi method, and propose a thick restarted harmonic Arnoldi algorithm for evaluating ph functions. An advantage of the new algorithm is that we can compute several ph functions simultaneously in the same search subspace after restarting. The relationship between the error and the residual of the harmonic Arnoldi approximation is also investigated. Numerical experiments show the superiority of our new algorithm over many state of the art algorithms for computing ph functions.
Adv. Comput. Math.