CAREER: Extrapolation Methods for Matrix and Tensor Eigenvalue Problems

University Of Florida, Gainesville FL

Investigators

Abstract

Eigenvalue problems arise naturally throughout many areas of mathematics and data science, and their numerical solution is crucial for understanding the behavior of many complex systems. Applications include structural mechanics, epidemiology, image processing, medical imaging, search-engine technology, modeling of population dynamics, and stability of numerical algorithms. Eigenvalue problems are often challenging to solve, as solutions can generally only be found by generating sequences of successive approximations. This work will develop efficient, robust and theoretically sound technologies that will accelerate convergence to solutions of matrix and tensor eigenvalue problems. In response to the increased prevalence of remote-learning, the integrated educational plan will develop stand-alone apps to aid in the delivery of standard and advanced topics in numerical analysis and linear algebra. The apps developed will include exposition of ideas and methods closely related to the research program. The technical aim of this work is the development and analysis of both novel and long-standing extrapolation techniques for eigenvalue problems. Extrapolation techniques are low-cost methods that combine a history of iterates and update steps to form the next approximation in a sequence. In this work they will be used to accelerate convergence of power-type iterations for challenging matrix and tensor eigenvalue problems. The main components of the research for matrix problems are development of novel methods that (1) damp multiple modes simultaneously; (2) resolve multiple modes simultaneously; and (3) extend the target problem class to indefinite matrices. For tensor problems, the main outcomes will be (1) development and convergence analysis of methods that accelerate robust but linearly converging power-type iterations; (2) extensions to accelerated versions of adaptively-shifted power-type iterations and generalized problems; and (3) studies on stability and clustering, and the development of fast techniques to capture complete sets of eigenvalues. This investigation of tensor methods is expected to advance the state of the art by introducing fast but low-complexity methods that are well suited to high-order problems. 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.