Regularity and Conditioning of Variational Problems

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

While the concept of conditioning is well understood in numerical linear algebra, still little is known in that direction for problems beyond equations, and in particular for variational problems, finite- and infinite-dimensional, where the presence of constraints complicates the analysis considerably. In recent years it has become clear that the basic paradigm behind conditioning, the "distance to good behavior," can be extended to a vast variety of problems, linking it in one general picture with the development of error bounds measuring the effect of perturbations and approximations of a problem on its solutions as well as with the convergence rate of algorithms. The "good behavior" of a problem is usually understood as a regularity property of a mapping related to the problem at hand which describes desirable features of its solutions. In this project we will extend the conditioning paradigm to broad classes of variational problems, including problems in nonlinear programming, calculus of variations, and optimal control. This task requires background work in the general area of variational analysis as well as the development of highly technical tools for tackling specific problems. This project aims to establish theoretical foundations for conditioning of variational problems through rigorous analysis of regularity properties of mappings associated with such problems. The project will lead to a better understanding of the interplay between the theoretical features of a problem and the actual computation of solutions. Furthermore, it has the potential to have a direct impact on the scientific computing of variational problems, by providing tools for the development of preconditioning techniques and, consequently, new fast and efficient algorithms for solving large-scale problems that appear in science and technology.