Non-linear Least-Squares Optimization of Rational Filters for the Solution of Interior Hermitian Eigenvalue Problems

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 the FEAST 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.

Krylov eigenvalue strategy using the FEAST algorithm with inexact system solves

“…Examples of quadrature rules that can be used for discretizing the integration are the traditional trapezoidal rule or Gauss quadrature, or more advanced methods such as Zolotarev integration 10 or least squares optimization. 11,12 The rate of convergence of the FEAST algorithm is related to both the dimension of the search subspace and to the accuracy with which the original integral in Equation (5) is approximated; the more linear systems that we solve for the quadrature rule (6), the better the integral is approximated and the fewer FEAST subspace iterations are required to converge to the desired level of accuracy. One of the benefits of this is that, because the linear systems can be solved independently of each other, the use of additional parallel processing power can be translated directly into a faster convergence rate simply by solving more linear systems in parallel.…”

“…The weights and locations of the shifts are determined by using complex contour integration in conjunction with a quadrature rule. Examples of quadrature rules that can be used for discretizing the integration are the traditional trapezoidal rule or Gauss quadrature, or more advanced methods such as Zolotarev integration or least squares optimization . The rate of convergence of the FEAST algorithm is related to both the dimension of the search subspace and to the accuracy with which the original integral in Equation is approximated; the more linear systems that we solve for the quadrature rule , the better the integral is approximated and the fewer FEAST subspace iterations are required to converge to the desired level of accuracy.…”

Fast randomized non-Hermitian eigensolver based on rational filtering and matrix partitioning

“…Compared to shift-and-invert, contour integral eigensolvers are generally oblivious to the location of the sought eigenvalues inside the disk D, and enjoy enhanced scalability when implemented in distributed memory computing environments [1,18,21,24,50]. Other rational filters, though not necessarily based on contour integration, can be found in [4,5,15,25,28,31,41,47,48,49].…”

A FEAST variant incorporated with a power iteration

“…Theoretical analysis exists in [37] and a comparison with some existing standard Krylov subspace eigensolvers was presented in [14]. The numerical computation and analysis of (1.4) have been discussed in depth in [15,41], and rational filters other than that induced by (1.4) were proposed in [39,40] through least-squares processes. For real and symmetric matrices, it is possible to limit all the computations to real arithmetic [1].…”