On the parallel iterative solution of linear systems arising in the FEAST algorithm for computing inner eigenvalues

Abstract

a b s t r a c tMethods for the solution of sparse eigenvalue problems that are based on spectral projectors and contour integration have recently attracted more and more attention. Such methods require the solution of many shifted sparse linear systems of full size. In most of the literature concerning these eigenvalue solvers, only few words are said on the solution of the linear systems, but they turn out to be very hard to solve by iterative linear solvers in practice. In this work we identify a row projection method for the solution of the inner linear systems encountered in the FEAST algorithm and introduce a novel hybrid parallel and fully iterative implementation of the eigenvalue solver. Our approach ultimately aims at achieving extreme parallelism by exploiting the algorithm's potential on several levels. We present numerical examples where graphene modeling is one of the target applications. In this application, several hundred or even thousands of eigenvalues from the interior of the spectrum are required, which is a big challenge for state-of-the-art numerical methods.

High-performance implementation of Chebyshev filter diagonalization for interior eigenvalue computations

“…However, evaluation of rational functions of sparse matrices is non-trivial. Standard iterative solvers require too many additional spMVMs to be competitive with polynomial filters, unless they can exploit additional structural or spectral properties of the matrix [43]. At present it is not clear whether non-polynomial filter functions can succeed in large-scale problems of the size reported here.…”

Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems

“…We can observe that as N c increases, the number of preconditioned GMRES iterations also increases. Indeed, iterative solvers are greatly affected by the location of the quadrature nodes ζ j , j =1,…, N c , with ζ j that lie closer to the real axis leading to slower convergence . By construction, higher values of N c will lead to some quadrature nodes being closer to the real axis.…”

“…Indeed, iterative solvers are greatly affected by the location of the quadrature nodes j , j = 1, … , N c , with j that lie closer to the real axis leading to slower convergence. 13 By construction, higher values of N c will lead to some quadrature nodes being closer to the real axis. Thus, when iterative solvers are exploited, setting N c to a low value, for example, N c = 1 or N c = 2, might in practice be a good choice.…”

“…This approach, however, is not always feasible due to the possible large amount of fill‐in in the triangular factors, for example, when factorizing matrices that originate from discretizations of 3D computational domains. One of the goals of this paper is to fill a part of the gap that exists between contour integration approaches and the use of hybrid iterative solvers within this context (see also the work of Galgon et al). In particular, we would like to accelerate iterative solutions of the linear systems encountered in the FEAST algorithm, by utilizing domain decomposition preconditioners to solve the resulting complex linear systems with multiple right‐hand sides.…”

GHOST: Building Blocks for High Performance Sparse Linear Algebra on Heterogeneous Systems

“…In the application areas we consider right now, methods such as incomplete factorization or multigrid can usually not be applied straightforwardly. The matrices that appear may not have an interpretation as physical quantities discretized on a mesh, they may be completely indefinite, and they may have relatively small diagonal entries and/or random elements [13].…”