Computational Methods for Stochastic Eigenvalue Problems

University Of Maryland, College Park, College Park MD

Investigators

Abstract

This project concerns the development of numerical algorithms for efficiently computing eigenvalues of algebraic systems of equations derived from mathematical models of physical processes. The study and understanding of eigenvalues is of fundamental importance in numerous engineering applications. Examples include the design of buildings and bridges, for which eigenvalues correspond to resonant frequencies at which structures vibrate, and identification of qualities of flowing liquids or gases that establish whether or not the flows are turbulent. The focus of the project is on solving eigenvalue problems in cases where the underlying model is stochastic. This scenario arises when components of the model, such as elastic properties of the materials used in structures or permeabilities of the media in which flows take place, are not known with certainty but instead are treated as random variables. The resulting eigenvalue solutions are themselves also random variables. Having such solutions will enable scientists and engineers to incorporate new probabilistic methods into design, for example, by using new mathematical techniques to analyze the likelihood that a structure will buckle and devise methods to prevent it. The technical approaches the PI will use in the project include the study of solutions of stochastic eigenvalue problems to develop an enhanced understanding of stability of dynamical systems and the impact of uncertainty on mathematical models. For example, it is known that pseudospectral analysis reveals aspects of stability not shown by traditional linear stability analysis, but it is difficult to use pseudospectral methods to make quantitative statements. The PI expects to be able to assess the probability of a linearly stable process being unstable when pseudospectra suggest it is, and how such an assessment depends on the statistical properties of the random components of the model. He will also develop new and efficient computational algorithms for solving the eigenvalue problems, including algorithms designed for stochastic Galerkin finite element discretizations and stochastic collocation methods, and strategies for reduced-order models.