In this master's thesis an optimization problem in relation to a partial differential equation (PDE) called the discrete Laplace problem with Dirichlet boundary conditions is studied. The solution of the optimization problem will provide optimal Dirichlet boundary conditions that allow solution of the discrete Laplace problem giving a best possible approximation to a given finite subset.
Moreover, a number of methods that make use of various optimization tools to solve the aforementioned optimization problem are presented. Significant effort is also given to studies of various properties of the solution of the discrete Laplace problem.
Theory of partial differential equations and linear optimization are combined, and the reader is expected to have basic knowledge in these subjects. In addition, some knowledge of linear algebra will be beneficial.