Applying GMRES to the Helmholtz equation with strong trapping: how does the number of iterations depend on the frequency?

Abstract

We consider GMRES applied to discretisations of the high-frequency Helmholtz equation with strong trapping; recall that in this situation the problem is exponentially ill-conditioned through an increasing sequence of frequencies. Our main focus is on boundary-integral-equation formulations of the exterior Dirichlet and Neumann obstacle problems in 2- and 3-d. Under certain assumptions about the distribution of the eigenvalues of the integral operators, we prove upper bounds on how the number of GMRES iterations grows with the frequency; we then investigate numerically the sharpness (in terms of dependence on frequency) of both our bounds and various quantities entering our bounds. This paper is therefore the first comprehensive study of the frequency-dependence of the number of GMRES iterations for Helmholtz boundary-integral equations under trapping.

Eigenvalues of the Truncated Helmholtz Solution Operator under Strong Trapping

“…Indeed, the functions u \ell in the quasimode construction in [4] are based on the family of eigenfunctions of the ellipse localizing around the periodic orbit \{ (0, x 2 ) : | x 2 | \leq a 2 \} ; when the eigenfunctions are sufficiently localized, the eigenfunctions multiplied by a suitable cut-off function form a quasimode, with frequencies k \ell equal to the square roots of eigenvalues of the ellipse. By separation of variables, k \ell can be expressed as the solution of a multiparametric spectral problem involving Mathieu functions; see see [4, Appendix A] and [38,Appendix E].…”

“…When giving specific values of k \ell below, we use the notation from [4, Appendix A] and [38,Appendix E] that k e m,n and k o m,n are the frequencies associated with the eigenfunctions of the ellipse that are even/odd, respectively, in the angular variable, with m zeros in the radial direction (other than at the center or the boundary) and n zeros in the angular variable in the interval [0, \pi ).…”

“…Implications of the main results for numerical analysis of the Helmholtz exterior Dirichlet problem. Theorems 1.5 and 1.8 are the first step towards rigorously understanding how iterative solvers such as the generalized minimum residual method (GMRES) behave when applied to discretizations of high-frequency Helmholtz problems under strong trapping (the subject of the companion paper [38]). We now explain this in more detail.…”

“…The paper [38] analyzes GMRES applied to discretizations of Helmholtz problems with strong trapping, using the ``cluster plus outliers"" GMRES convergence theory from [6] (with this idea arising in the context of the conjugate gradient method [32] and used subsequently in, e.g., [18]). The paper [38] obtains bounds on how the number of GMRES iterations depends on the frequency under various assumptions about the eigenvalues. In particular, Theorem 1.5 rigorously justifies [38,Observation O2(b)] for the standard variational formulation of the truncated exterior Dirichlet problem.…”

“…The paper [38] obtains bounds on how the number of GMRES iterations depends on the frequency under various assumptions about the eigenvalues. In particular, Theorem 1.5 rigorously justifies [38,Observation O2(b)] for the standard variational formulation of the truncated exterior Dirichlet problem. We highlight that, although the results in [38] are about unpreconditioned systems, they give insight into the design of preconditioners.…”