AF: Small: Collaborative Research: Effective Numerical Algorithms and Software for Nonlinear Eigenvalue Problems

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The eigenvalue problem is a central topic in science and engineering arising from a wide range of applications and posing major numerical challenges. For decades, it has been the focus of numerous theoretical research activities for developing various efficient numerical algorithms. These efforts have led to the development of new software that is essential to assist the everyday work of many engineers and scientists. In spite of progress made on solving the eigenvalue problem, methods available for handling these problems remain limited in their scope and they have not resulted in effective general-purpose software so far. The primary goal of this project is to fill this gap by advancing the state of the art in solution methods for nonlinear eigenvalue problems which are both mathematically and practically far more challenging than the traditional linear eigenvalue problems. The combined expertise of the investigating team is well suited for exploring new algorithms in this arena, analyzing them, and developing new effective software that can universally impact a wide range of disciplines (engineering, physics, chemistry, and biology). The outcome of the project are expected to open new and efficient ways to solve nonlinear eigenvalue problems. A new suite of state of the art numerical routines will be developed, fully tested, and publicly released.The goal of this project is to advance the state-of-the-art in solution methods for nonlinear eigenvalue problems. The new approaches that are envisioned are expected to be particularly effective for solving large-scale problems using parallelism. The main thrust of the project is the development of novel eigenvalue algorithms based on generalizations of Cauchy integral type methods for the nonlinear case, combined with projection methods such as Krylov and subspace iteration. A starting point in this investigation is the FEAST approach which will be adapted to the nonlinear context. Because the problems under consideration are expected to be large and sparse, the team will investigate methods that rely on domain decomposition where the original physical domain is partitioned into a number of subdomains in order to exploit parallelism. Among other goals, the team will carefully study the extension of the tools that are exploited in the linear case, such as spectrum slicing (computing eigenvalues by parts), block methods, and iterative linear solves, to the nonlinear case.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.