Finite Difference Methods for Nonlinear Elliptic Equations with Application to Optimal Transport

Abstract: We describe the use of finite difference methods for solving nonlinear elliptic partial differential equations (PDEs). We show that simple techniques, which work for linear equations, may fail for nonlinear equations. We describe a framework for developing convergent finite difference methods for nonlinear degenerate elliptic equations. Focusing specifically on optimal transport, a challenging problem that is important to both theoretical and applied mathematics, we construct robust numerical methods for the Monge-Ampere equation with transport boundary conditions. A range of computational examples demonstrate the effectiveness and efficiency of the method.