A radial basis function method for solving PDE-constrained optimization problems

In this article, we apply the theory of meshfree methods to the problem of PDE-constrained optimization. We derive new collocation-type methods to solve the distributed control problem with Dirichlet boundary conditions and also discuss the Neumann boundary control problem, both involving Poisson’s equation. We prove results concerning invertibility of the matrix systems we generate, and discuss a modification to guarantee invertibility. We implement these methods using Matlab, and produce numerical results to demonstrate the methods’ capability. We also comment on the methods’ effectiveness in comparison to the widely-used finite element formulation of the problem, and make some recommendations as to how this work may be extended.