2:00 pm Tuesday, March 4, 2014
Numerical Analysis Seminar: Numerical Methods for a class of nonlinear elliptic PDEs with emphasis on the Monge-Ampère equation by Adam Oberman (McGill University) in POB 6.304
Nonlinear elliptic and parabolic Partial Differential Equations (PDEs) have applications to image processing, first arrival times in wave propagation, homogenization, mathematical finance, stochastic control and games theory. In order to capture geometric features such as folds and corners, and avoid artificial singularities which arise from bad representations of the operators, it is important to use convergent numerical schemes. In the first part of the talk, I will introduce the class of equations, and present some of the examples mentioned above. I'll also explain how to build simple nonlinear finite difference methods for the obstacle problem. In the second part of the talk, I'll focus on a specific equation and explain the method. The (elliptic) Monge-Ampère Partial Differential Equation is a classical nonlinear PDE arising in geometry. It has been studied recently due to its connections with Optimal Transportation theory. Starting with the Dirichlet problem, I will present a finite difference scheme which is the only scheme proven to converge to weak (viscosity) solutions. Building on the original discretization, I'll describe modifications which improve the accuracy and solution speed. Finally, I will show how to solve the problem with Optimal Transportation boundary conditions. This is joint work with Jean-David Benamou and Brittany Froese. Submitted by
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