Solving linear partial differential equations via semidefinite optimization

Abstract

Using recent progress on moment problems, and their connections with semidefinite optimization, we present in this thesis a new methodology based on semidefinite optimization, to obtain a hierarchy of upper and lower bounds on both linear and nonlinear functionals defined on solutions of linear partial differential equations. We apply the proposed methods to examples of PDEs in one and two dimensions with very encouraging results. We also provide computational evidence that the semidefinite constraints are critically important in improving the quality of the bounds, that is without them the bounds are weak.