Numerical Methods for Geometric Elliptic Partial Differential Equations
Simon Fraser University
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
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
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
Center for Computational Mathematics9500 Gilman Dr. #0112La Jolla, CA 92093-0112Tel: (858)534-9056