Least squares simulation of distributed systems

This paper addresses the application of linear optimal control theory to the least squares functional approximation of linear initial-boundary value problems. The method described produces the optimal approximate solution by the realization of a linear quadratic servo configuration imposed on the Galerkin simulation for the problem. It is shown that appropriate formulation of the servo problem guarantees a stable numerical solution, even when the Galerkin simulation itself is unstable, a situation not uncommon with certain hyperbolic partial differential equations. Theoretical least squares and Galerkin properties are comparatively discussed, and numerical examples demonstrating least squares convergence in the face of Galerkin divergence are presented.