Mesh adaptive multiple shooting for partial differential equations. Part I: linear quadratic optimal control problems

Abstract

We devise an indirect multiple shooting (MS) method for PDE constrained optimization with a posteriori error estimates and local mesh refinement. MS is applied to the KKT system as is exemplified for a linear quadratic optimal control problem. The setup of MS in function spaces and their discrete analogs is discussed. Error representation formulas based on Galerkin orthogonality are derived. They involve sensitivity analysis by an adjoint problem and employ standard error representation on subintervals combined with additional projection errors at the shooting nodes. A posteriori error estimates and mesh refinement indicators are derived from this error representation. We discuss several mesh structures originating from different restrictions to local refinement. Finally, we present numerical results for the heat equation and problems exhibiting simple dynamics. In a second article, we will address the embedding of the scheme into a nonlinear shooting method.