A shifted Laplace rational filter for large-scale eigenvalue problems

TL;DR

This work tackles the problem of computing all eigenvalues in $(0,\gamma]$ of large-scale symmetric definite generalized eigenvalue problems $\mathbf{A}\mathbf{x}=\lambda\mathbf{B}\mathbf{x}$. It introduces the Shifted Laplace Rational Filter (SLRF), where poles lie on two straight lines $\mathcal{L}_{\pm}: y = x(1 \pm i\alpha)$ and weights are optimized to approximate a step function on the positive real axis, balancing eigenvalue separation with efficient linear solves. The poles/weights are obtained via an optimization problem and the strategy is combined with a shifted-Laplace preconditioning idea to ease the linear systems $(\mathbf{A}-\sigma\mathbf{B})\mathbf{x}=\mathbf{f}$, enhancing overall convergence within a subspace projection framework. Numerical experiments on finite element vibration models show that SLRF often reduces the average cost of linear solves compared to quadrature-based filters, while delivering strong convergence for all eigenpairs in the interval; a notable limitation appears when $\mathbf{B}$ is singular. Overall, the method provides a scalable, parallel-friendly alternative to contour-based filtering for large-scale eigenvalue problems with real positive spectra.

Abstract

We present a rational filter for computing all eigenvalues of a symmetric definite eigenvalue problem lying in an interval on the real axis. The linear systems arising from the filter embedded in the subspace iteration framework, are solved via a preconditioned Krylov method. The choice of the poles of the filter is based on two criteria. On the one hand, the filter should enhance the eigenvalues in the interval of interest, which suggests that the poles should be chosen close to or in the interval. On the other hand, the choice of poles has an important impact on the convergence speed of the iterative method. For the solution of problems arising from vibrations, the two criteria contradict each other, since fast convergence of the eigensolver requires poles to be in or close to the interval, whereas the iterative linear system solver becomes cheaper when the poles lie further away from the eigenvalues. In the paper, we propose a selection of poles inspired by the shifted Laplace preconditioner for the Helmholtz equation. We show numerical experiments from finite element models of vibrations. We compare the shifted Laplace rational filter with rational filters based on quadrature rules for contour integration.

Figures (8)

• Figure 1: The rational filters $\Phi(x)$ are plotted over the interval $[-1, 2]$ for $\gamma=1$ and $\alpha = 1$, where $\beta = 1$ (left) and $\beta = 0.01$ (right) for different number of poles $N = 1, 2, 4, 8$.
• Figure 2: Comparison of separation factors$|\Phi'(1)|$ of the rational filter using $N = 1, 2, 4, 8$ poles and different slope $\alpha$.
• Figure 3: The convergence curves of the maximum relative residual norm for the 2D beam model are presented for different filter parameters. The results are shown for $N = 2$ (top), $N = 4$ (middle), $N = 8$ (bottom), with variations in the slope: $\alpha = 1$ (left) and $\alpha = 0.5$ (right), and filter parameters $\gamma=1$, $\beta = 1$ and $\beta = 0.01$ in each subfigure. The maximum residual norm is reported for approximate eigenvalues located within the interval (0, 337.4505], which contains the first 100 smallest eigenvalues, with a search space dimension of $L=120$. The associated linear systems are solved by a complete LU factorization
• Figure 4: Left: the rational filter $\Phi(x)$ for $\gamma=1$ over the interval $x\in[-1, 2]$ for $N = 3$; right: a zoomed-in view at $x=1$.
• Figure 5: The corresponding poles position of four rational filters in the upper half plane.

Theorems & Definitions (4)

• Lemma 2.1
• Proposition 2.1
• Proposition 2.2
• Definition 3.1