Reduced-Order and Low-Rank Methods for Parameter-Dependent Partial Differential Equations

University Of Maryland, College Park, College Park MD

Investigators

Abstract

The goal of this study is to develop new computer algorithms to be used to simulate engineering models and physical phenomena, with the aim of predicting behavior of systems in the physical world. Examples of the use of these techniques include modeling of the flow of pollutants in groundwater, assessing the stability of structures such as airplane wings in the face of stresses such as high temperatures or pressures, and simulating the effects of magnetic fields on performance of semiconductors or reactions in nuclear fusion. Such algorithms help engineers and scientists to make good decisions about construction and use of new technology, but they are only practically useful if demonstrated to be efficient (that is, use modest amounts of computer time and memory) and accurate. The aim of this work is to advance the development of efficient algorithms of this type and to demonstrate their utility for models of transient phenomena such as fluid flows and stability of physical structures. The technical goals of the project are to study solution algorithms for parameter-dependent partial differential equations by constructing approximate solutions of low-rank structure. Parameterized problems of this type arise when underlying terms figuring in the differential operators of the system depend on a set of unknown or random parameters. Examples include unknown permeabilities in models of diffusion or velocity fields that are affected by temperatures. In this scenario, the solutions sought also depend on parameters, but they can often be approximated well in a space of low dimension, i.e., solutions for all parameter values can be represented as a linear combination of a small finite set of functions. Such a reduced representation offers the prospect of significant reduction of costs required to compute solutions. The project will explore the utility this approach for two types of problems: (i) dynamical systems, for which the dependence on time requires new methods to develop reduced-order models that are accurate over long time periods, and (ii) eigenvalue problems, with emphasis on techniques for stability analysis of dynamical systems. 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.