Randolph E. Bank

Philip E. Gill

Michael Holst

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Adam Oberman

Simon Fraser University

Abstract:

Geometric Partial Differential Equations (PDEs) are at the forefront of current research in mathematics, as evidenced by Perelman's use of these equations in his proof of the Poincare Conjecture and Cedric Villani's Fields Medal in 2010 for his work on Optimal Transportation. They can be used to describe, manipulate and construct shapes based on intrinsic geometric properties such as curvatures, volumes, and geodesic lengths.

These equations are useful in modern applications (Image Registration, Computer Animation) which require geometric manipulation surfaces and volumes. Convergent numerical schemes are important in these applications in order to capture geometric features such as folds and corners, and avoid artificial singularities which arise from bad representations of the operators.

In general these equations are considered too difficult to solve, which is why linearized models or other approximations are commonly used. Progress has recently been made in building solvers for a class of Geometric PDEs. I'll discuss a few important geometric PDEs which can be solved using a numerical method called Wide Stencil finite difference schemes: Monge-Ampere, Convex Envelope, Infinity Laplace, Mean Curvature, and others.

Focusing in on the Monge-Ampere equation, which is the seminal geometric PDE, I'll show how naive schemes can work well for smooth solutions, but break down in the singular case. Several groups of researchers have proposed numerical schemes which fail to converge, or converge only in the case of smooth solutions. I'll present a convergent solver which which is fast: comparable to solving the Laplace equation a few times.

Tuesday, November 30, 2010

11:00AM AP&M 2402