On a parallel multilevel preconditioned Maxwell eigensolver

Abstract

We report on a parallel implementation of the Jacobi-Davidson algorithm to compute a few eigenvalues and corresponding eigenvectors of a large real symmetric generalized matrix eigenvalue problemThe eigenvalue problem stems from the design of cavities of particle accelerators. It is obtained by the finite element discretization of the time-harmonic Maxwell equation in weak form by a combination of Nédélec (edge) and Lagrange (node) elements.We found the Jacobi-Davidson (JD) method to be a very effective solver provided that a good preconditioner is available for the correction equations that have to be solved in each step of the JD iteration. The preconditioner of our choice is a combination of a hierarchical basis preconditioner and the ML smoothed aggregation AMG preconditioner. It is close to optimal regarding iteration count.The parallel code makes extensive use of the Trilinos software framework. In our examples from accelerator physics we observe satisfactory speedups and efficiencies.

Parallel Implementation

“…However, Davidson methods may present better performance computing interior eigenvalues and/or in generalized eigenproblems when exact solves with A − τ B are unaffordable, but some approximations are available, so-called preconditioners, [Davidson, 1975;van Lenthe and Pulay, 1990;Crouzeix et al, 1994;Heuveline et al, 1997;Arbenz et al, 2006;Genseberger, 2010].…”

Computing subdominant unstable modes of turbulent plasma with a parallel Jacobi–Davidson eigensolver

“…Despite their potential benefit, it is still difficult to find freely available parallel implementations of Davidson‐type eigensolvers, especially for the non‐symmetric case, although there are some publications dealing with parallel implementations of these methods employed for certain applications, see for instance 3–5. Parallel Davidson‐type methods can be found in PRIMME 6 and Anasazi 7.…”

“…Otherwise, depending on the performance of the matrix-vector (MV) product, the preconditioner and the orthogonalization, there exist Davidsontype variants that can be competitive with respect to Krylov-type eigensolvers. This paper illustrates an example of this.Despite their potential benefit, it is still difficult to find freely available parallel implementations of Davidson-type eigensolvers, especially for the non-symmetric case, although there are some publications dealing with parallel implementations of these methods employed for certain applications, see for instance [3][4][5]. Parallel Davidson-type methods can be found in PRIMME [6] and Anasazi [7].…”

A parallel subdomain by subdomain implementation of the implicitly restarted Arnoldi/Lanczos method

“…For large and sparse eigensystems this approach becomes prohibitive and iterative methods must be used. For the symmetric case, Arbenz et al [2,3] investigated the use of algebraic multigrid preconditioners with several eigensolvers, including the implicitly restarted Lanczos. With the use of an iterative method to solve the linear system of equations the influence of the matrix-vector (matvec) product in the overall performance becomes more relevant.…”