Solving Boundary Value Problems Numerically Using Steepest Descent In Sobolev Spaces

Abstract

Elementary boundary value problems are used as a vehicle to introduce upper level undergraduate students or first year graduate students to descent algorithms. The paper is expository in nature and includes references for Euclidean and Sobolev steepest descent while serving as an introduction to optimization techniques such as variable metric and conjugate gradient methods. Numerical algorithms for solving boundary value problems with both Euclidean and Sobolev descent are developed. Results for constrained, unconstrained, and singular problems are displayed and properties of descent algorithms are outlined.