Parallelization of the Rational Arnoldi Algorithm

Abstract

Abstract. Rational Krylov methods are applicable to a wide range of scientific computing problems, and the rational Arnoldi algorithm is a commonly used procedure for computing an orthonormal basis of a rational Krylov space. Typically, the computationally most expensive component of this algorithm is the solution of a large linear system of equations at each iteration. We explore the option of solving several linear systems simultaneously, thus constructing the rational Krylov basis in parallel. If this is not done carefully, the basis being orthogonalized may become badly conditioned, leading to numerical instabilities in the orthogonalization process. We introduce the new concept of continuation pairs, which gives rise to a near-optimal parallelization strategy that allows us to control the growth of the condition number of this nonorthogonal basis. As a consequence we obtain a significantly more accurate and reliable parallel rational Arnoldi algorithm. The computational benefits are illustrated using several numerical examples from different application areas.Key words. rational Krylov, orthogonalization, parallelization AMS subject classifications. 68W10, 65F25, 65G50, 65F15 DOI. 10.1137/16M10791781. Introduction. Rational Krylov methods have become an indispensable tool of scientific computing. Invented by Ruhe for the solution of large sparse eigenvalue problems (see, e.g., [26,27]) and frequently advocated for this purpose (see, e.g., [23,32] and [2, sect. 8.5]), these methods have seen an increasing number of other applications over the last two decades or so. Examples of rational Krylov applications can be found in model order reduction [11,16,17,22], matrix function approximation [10,13,19], matrix equations [3,9,24], nonlinear eigenvalue problems [20,21,31], and nonlinear rational least squares fitting [5,6].At the core of most rational Krylov applications is the rational Arnoldi algorithm, which is a Gram-Schmidt procedure for generating an orthonormal basis of a rational Krylov space. Given a matrix A ∈ C N,N , a vector b ∈ C N , and a polynomial q m of degree at most m and such that q m (A) is nonsingular, the rational Krylov space of order m is defined as

The nonlinear eigenvalue problem

“…(2014), which also supports exploitation of low-rank structure in the NEP: The CORK method described in Van Beeumen et al. (2015 a ), which implements the compact representation of Krylov basis vectors (6.22), exploitation of low-rank terms, and implicit restarting: The function in the Rational Krylov Toolbox, which we demonstrated in Figure 6.3, as well as the function which implements the (parallel) rational Arnoldi algorithm described in Berljafa and Güttel (2017): …”

“…• The util nleigs function in the Rational Krylov Toolbox, which we demonstrated in Figure 6.3, as well as the rat krylov function which implements the (parallel) rational Arnoldi algorithm described in Berljafa and Güttel (2017):…”

Improved ParaDiag via low-rank updates and interpolation

“…In the following, we discuss the application of low-rank solvers to the correction equation (8). In view of the (assumed) invertibility of M and B 2 we replace (8) by the following Sylvester matrix equation:…”

“…In this situation, the cost reduces to Op C sys `pn`n t q 2 `nn t logpn t qq for each of the n t cores. A further reduction can be obtained by making use of parallel implementations of RKSM [8] and FFT.…”