A Convergent Scheme for the infinity Laplacian
Building a convergent finite difference scheme for the Infinity Laplacian is challenging because:
- the equation lack regularity. Even in two dimensions there are known solutions which are not twice differentiable
- viscosity solutions much be captured, which requires a monotone scheme be built
- the equation is highly degenerate: in fact, it is degenerate along every direction except the gradient
- the obvious thing won't work. Unlike the equation for motion by mean curvature, if you simply try the obvious scheme, the solutions obtained are often completely wrong. The equation for motion by mean curvature is more forgiving, because it is time dependent, whereas this equation is not.
The publication is:
- A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions, Mathematics of Computation, 74 (2005) Number 251, 1217-1230.
Impact
When last checked, this paper was cited 32 times on google scholar
Cited by
- Tug-of-war and the infinity Laplacian, Peres, Shramm, Sheffield, and Wilson, Journal of the AMS http://www.ams.org.proxy.lib.sfu.ca/jams/2009-22-01/S0894-0347-08-00606-1/home.html
- On geometric variational models for inpainting surface holes, http://portal.acm.org/citation.cfm?id=1410574
- Brain and surface warping via minimizing Lipschitz Extensions, Memoli, Sapiro, and Thompson http://www.ima.umn.edu/preprints/jan2006/2092.pdf
An interesting connection between Infinity Laplace and Games has been studied. This is the topic of a conference
- New Connections Between Differential and Random Turn Games, PDE's and Image Processing Vancouver, BC, July 27-31, 2009
Related work
For background on building convergence finite difference schemes for viscosity solutions, refer to