Standard finite difference discretizations of the elliptic Monge-Ampere equation

Start: 11/13/2013 - 1:15pm
End  : 11/13/2013 - 2:15pm

Applied Math Seminar

Gerard Awanou (University of Illinois, Chicago)


Abstract: Given an orthogonal lattice with mesh length $ h $ on a bounded convex domain $ \Omega $, we show that the Aleksandrov solution of the Monge-Amp\`ere equation is the uniform limit on compact subsets of $ \Omega $ of mesh functions $ u_h $ which solve a discrete Monge-Amp\`ere equation with the Hessian discretized using the standard finite difference method. The result provides the mathematical foundation of methods based on the standard finite difference method and designed to numerically converge to non-smooth solutions. The numerical resolution of the Monge-Amp\`ere equation has become an active research field because of several applications of increasing importance e.g. optimal transport, reflector design etc.

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