Abstract
Rational filter functions can be used to improve convergence of contour-based eigensolvers, a popular family of algorithms for the solution of the interior eigenvalue problem. We present a framework for the optimization of rational filters based on a non-convex weighted Least-Squares scheme. When used in combination with a contour based eigensolvers library, our filters out-perform existing ones on a large and representative set of benchmark problems. This work provides a detailed description of: (1) a set up of the optimization process that exploits symmetries of the filter function for Hermitian eigenproblems, (2) a formulation of the gradient descent and Levenberg-Marquardt algorithms that exploits the symmetries, (3) a method to select the starting position for the optimization algorithms that reliably produces effective filters, (4) a constrained optimization scheme that produces filter functions with specific properties that may be beneficial to the performance of the eigensolver that employs them.
Highlights
The last fifteen years have witnessed a proliferation of papers on contour based methods for the solution of the Hermitian interior eigenvalue problem [1,2,3,4]
For a problem with m eigenpairs inside the search interval we select a size of the subspace iteration of p = 1.5m, the value that FEAST recommends for the Gauss filter
We propose η-Symmetric non-Linear Optimized Least-Squares (SLiSe) as a replacement for the Elliptic filter currently used in contour-based subspace iteration methods
Summary
The last fifteen years have witnessed a proliferation of papers on contour based methods for the solution of the Hermitian interior eigenvalue problem [1,2,3,4]. A key insight to the use of numerical quadrature is the reinterpretation of the contour integration of the matrix resolvent as a matrix-valued rational filter that maps eigenvalues inside and outside [a, b] to one and zero respectively From this point of view, the mathematical formalism of contour based methods can be considered closely related to the richer mathematical field of rational function approximation. In this paper we conduct a detailed investigation of how a carefully crafted rational function can improve the efficacy of contour based eigensolvers To this end, we look at the the numerical quadrature of a complex-valued resolvent (t − z)−1 as an approximation of an ideal filter represented by the standard indicator function. We illustrate the mathematical background, the construction of the optimization framework, and provide a software implementation of it in Julia [5] that generates readyto-use rational filters
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