Past Event: Oden Institute Seminar

On the numerical solution of the Monge-Ampere Equation: An Operator-Splitting/Finite Element Approach

Roland Glowinski, Cullen Professor of Mathematics and Mechanical Engineering, University of Houston

Abstract

Monge-Ampère equations occur areas of Science and Engineering: Differential Geometry, Fluid Mechanics, Elasticity, Cosmology, Antenna and Windshield Design, Mesh Generation, Optimal Transport, Finance, Image Processing, etc. Our goal in this presentation is discuss the numerical solution of the canonical Monge-Ampère equation in dimensions 2 and 3. These last two-decades, the numerical solution of Monge-Ampere equation (MA-D) has motivated a relatively large number of publications, however most of the solution methods we know use either high order finite element approximations or wide-stencil finite difference ones. The method we going to present relies on: (i) piecewise affine approximations of the unknown function u and of its second order derivatives, (ii) a time discretization by operator-splitting of an initial value problem associated with an equivalent divergence formulation of problem (MA-D), (iii) a projection operator on the cone of the symmetric positive semi-definite matrices, (iv) a Tychonoff regularization method to approximate the second order derivatives. The resulting methodology is modular and can handle easily domains with curved boundaries. It is also robust in the sense that it can also handle efficiently non- smooth situations, the non-smoothness coming from the data (if, for example, forcing is a the positive multiple of Dirac measure), or from data incompatibility. The results of numerical experiments will be presented, including those associated with the solution of the following (nonlinear) eigenvalue problem for the Monge-Ampère operator.