Feast Eigensolver for Non-Hermitian Problems

Abstract

Abstract. A detailed new upgrade of the FEAST eigensolver targeting non-Hermitian eigenvalue problems is presented and thoroughly discussed. It aims at broadening the class of eigenproblems that can be addressed within the framework of the FEAST algorithm. The algorithm is ideally suited for computing selected interior eigenvalues and their associated right/left bi-orthogonal eigenvectors, located within a subset of the complex plane. It combines subspace iteration with efficient contour integration techniques that approximate the left and right spectral projectors. We discuss the various algorithmic choices that have been made to improve the stability and usability of the new nonHermitian eigensolver. The latter retains the convergence property and multi-level parallelism of Hermitian FEAST, making it a valuable new software tool for the scientific community.Key words. non-Hermitian eigenproblem, FEAST, spectral projectors, contour integration, right/left eigenvectors, bi-orthogonal vectors AMS subject classifications. 65F15, 15A18 34L16 65Y05 35P991. Introduction. The generalized eigenvalue problem AX = BXΛ with A and B square matrices and Λ diagonal, is a central topic in numerical linear algebra and arises from a broad and diverse set of disciplines in mathematics, science and engineering (the problem is said "standard" if B ≡ I or "generalized" otherwise). Solving the interior eigenvalue problem consists of determining nontrivial solutions {λ i , x i } (i.e. eigenpairs with x i = Xe i and λ i = Λ i,i ) located anywhere inside the spectrum. Most common numerical applications lead to symmetric eigenvalue problems where A is real symmetric or complex Hermitian, B is symmetric or Hermitian positive definite (hpd), and all the obtained eigenvalues λ i are real. Non-symmetric and non-Hermitian eigenvalue problem (including the case where A is complex symmetric) can also be encountered in a variety of situations resulting in complex values for λ i . In this case x i is called the right eigenvector associated with λ i , while one can also define a left eigenvector. Although many software packages are available for symmetric (or Hermitian) matrices (see e.g. [17,23,33,18,25,5,36,20]), relatively few algorithms and software can handle the non-Hermitian problem [22,4,23,3,13]. The FEAST eigensolver [29, 10], proven to be a robust and efficient tool for computing the partial eigenspectrum of Hermitian system matrices [38], can also be generalized and applied to arbitrary non-Hermitian systems [21,41,37].FEAST is a subspace iteration method that uses the Rayleigh-Ritz projection and an approximate spectral projector as a filter [38]. Given a Hermitian generalized eigenvalue problem AX = BXΛ of size n, the algorithm in Figure 1 outlines the main steps of a generic Rayleigh-Ritz subspace iteration procedure for computing m eigenpairs. At convergence, the algorithm yields the B-orthonormal eigensubspace Y m ≡ X m = {x 1 , x 2 , . . . , x m } n×m and associated eigenvalues Λ Q m ≡ Λ m . Taking ρ(B −1 A) = B −1 A, yields the...

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

“…The accuracy of the approximate eigenpairs computed by a Rayleigh–Ritz projection on the subspace created by can be improved by repeating the procedure in , using the most recent approximate eigenvectors as the new matrix V to multiply . If direct solvers are used to solve the complex linear systems in , this approach essentially amounts to subspace iteration with the matrix A replaced by ρ ( A ), that is, the FEAST package ; see also other works . It is also possible to consider contour integrals of other rational functions, for example, the scalar function u ∗ ( ζ I − A ) −1 v , with , as proposed in other works .…”

Krylov eigenvalue strategy using the FEAST algorithm with inexact system solves

“…In general, however, FEAST, and all of the results in this paper, can be extended to non-Hermitian matrices as well. 5,6 From now on, we will restrict our attention to the standard eigenvalue problem case B = I, that is,…”

“…A recent parallel FEAST (PFEAST) implementation was proposed for solving larger system sizes of this kind, taking advantage of distributed-memory sparse linear system solvers and domain decomposition techniques. 5,14 In very large-scale applications, however, the structure of the matrix A causes the factorization step to be extremely slow and expensive to perform, and the storage of the factorization may even be impossible due to memory constraints. In some other cases, matrix A is too large and dense to be stored at all and is instead being represented implicitly by a rule for performing fast matrix-vector products (see the work of Giannozzi et al 15 for an example of an application where this approach is used).…”

“…For the sake of simplicity, we consider only regions that are intervals on the real number line, thereby restricting our attention to Hermitian matrices. In general, however, FEAST, and all of the results in this paper, can be extended to non‐Hermitian matrices as well …”

“…A direct solver requires that one be able to form and store a factorization of the matrices ( z k I − A ). A recent parallel FEAST (PFEAST) implementation was proposed for solving larger system sizes of this kind, taking advantage of distributed‐memory sparse linear system solvers and domain decomposition techniques . In very large‐scale applications, however, the structure of the matrix A causes the factorization step to be extremely slow and expensive to perform, and the storage of the factorization may even be impossible due to memory constraints.…”