A Comparison of Factorization-Free Eigensolvers with Application to Cavity Resonators

Abstract

Abstract. We investigate eigensolvers for the generalized eigenvalue problem Ax = λM x with symmetric A and symmetric positive definite M that do not require matrix factorizations. We compare various variants of Rayleigh quotient minimization and the Jacobi-Davidson algorithm by means large-scale finite element problems originating from the design of resonant cavities of particle accelerators.

Preconditioning bandgap eigenvalue problems in three-dimensional photonic crystals simulations

“…To study how we may solve the eigenvalue problems arising in three-dimensional photonic crystals, we first propose using shift-and-invert Krylov-Schur method and Jacobi-Davidson method to solve the generalized eigenvalue problem (1). We then suggest several preconditioning schemes for the associated linear systems.…”

“…These methods use shift-and-invert technique to compute interior eigenpairs and the computational cost for solving the corresponding linear systems can be excessive. On the other hand, we can also use Jacobi-Davidson method [1][2][3][4]15,24,41,42] to find the interior target eigenvalues without using the shift-and-invert technique. However, Jacobi-Davidson method still needs to solve linear systems approximately in each of the iterations.…”

“…Furthermore, due to the presence of a large null space associated with (1) [8,12], these target eigenvalues are actually located in the interior of the spectrum of (1). The eigenvalue problem (1) can thus be solved by inverse power method [2,11,12,21], various Lanczos [2,40] or Arnoldi method [4]. These methods use shift-and-invert technique to compute interior eigenpairs and the computational cost for solving the corresponding linear systems can be excessive.…”