Steepest descent using smoothed gradients

Abstract

This paper reviews solving differential equations using the least-squares steepest descent method with Sobolev gradients. The method's superiority over standard steepest descent with a Euclidean gradient is explained in terms of stability and the classical Courant–Freiderichs–Lewy condition for the path of steepest descent. The spectra for several of the operators arising from a cononical example are also computed. Kantorovich's inequality then gives explicit estimates on the rate of convergence for the two processes. In this way, use of the Sobolev gradient is viewed as a very effective preconditioning strategy for the linear part of the differential equation.