Convergent semi-Lagrangian methods for the Monge-Ampère equation on unstructured grids

Dr. Max Jensen, University of Sussex

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Feb 15, 2018
from 02:00 PM to 03:00 PM


MAB 119

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In this presentation I will discuss a semi-Lagrangian discretisation of the Monge-Ampère operator on P1 finite element spaces. The wide stencil of the scheme is designed to ensure uniform stability of numerical solutions. Monge-Ampère equations arise for example in the Inverse Reflector Problem where the geometry of a reflecting surface is reconstructed from the illumination pattern on a target screen and the characteristics of the light source. 

Monge-Ampère type equations, along with Hamilton-Jacobi-Bellman type equations are two major classes of fully nonlinear second order partial differential equations (PDEs). From the PDE point of view, Monge-Ampère type equations are well understood. On the other hand, from the numerical point of view, the situation is far from ideal. Very few numerical methods, which can reliably and efficiently approximate viscosity solutions of Monge-Ampère type PDEs on general convex domains. There are two main difficulties which contribute to the situation: 

* Firstly, it is well known that the fully nonlinear structure and nonvariational concept of viscosity solutions of the PDEs prevent a direct formulation of any Galerkin-type numerical methods. 

* Secondly, the Monge-Ampère operator is not an elliptic operator in generality, instead, it is only elliptic in the set of convex functions and the uniqueness of viscosity solutions only holds in that space. This convexity constraint, imposed on the admissible space, causes a daunting challenge for constructing convergent numerical methods; it indeed screens out any trivial finite difference and finite element analysis because the set of convex finite element functions is not dense in the set of convex functions 

The goal of our work is to develop a new approach for constructing convergent numerical methods for the Monge-Ampère Dirichlet problem, in particular, by focusing on overcoming the second difficulty caused by the convexity constraint. The crux of the approach is to first establish an equivalent (in the viscosity sense) Bellman formulation of the Monge-Ampère equation and then to design monotone numerical methods for the resulting Bellman equation on general triangular grids. An aim in the design of the numerical schemes was to make Howard's algorithm available, which is a globally superlinearly converging semi-smooth Newton solver as this allows us to robustly compute numerical approximations on very fine meshes of non-smooth viscosity solutions. An advantage of the rigorous convergence analysis of the numerical solutions is the comparison principle for the Bellman operator, which extends to non-convex functions. We deviate from the established Barles-Souganidis framework in the treatment of the boundary conditions to address challenges arising from consistency and comparison. The proposed approach also bridges the gap between advances on numerical methods for these two classes of second order fully nonlinear PDEs. 

The contents of the presentation is based on joint work with X. Feng from the University of Tennessee. 

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