Applications of Adaptive Finite Element Methods to Problems in Estimation and Control for Partial Differential Equations.

摘要

The purpose of this grant has been to undertake research in the general area of adaptive-grid finite element methods in estimation and control problems. The overall goal has been to investigate adaptive gridding numerical algorithms and to develop a solid theory to handle the mathematical questions of convergence and stability which arise in the use of such algorithms. The P.I. has met this goal through the study of a number of adaptive-grid finite element methods for parameter estimation problems governed by distributed parameter systems. In addition, stability problems encountered in the implementation of such algorithm have led the P. I. to make extensive studies of the ill-posed nature of these problems. In particular, theoretical and numerical studies were undertaken concerning regularization schemes for a number of ill-posed estimation problems. Specific applications considered in this latter study include the inverse heat conduction problem and regularizing aspects of descent methods as used to minimize fit-to-data functions associated with linear estimation problems.