2015 Volume 8 Issue 2 Pages 171-180
This paper presents a technique to approximate heat diffusion on Riemannian manifolds. Specifically, the authors provide a framework to approximate the solution to the heat equation by using the idea of random walks of heat particles, governed by a continuous-time Markov chain, where the transition rates of the Markov chain are characterized by the metric of a given grid with non-uniformly placed nodes imposed on a manifold. The emphasis lies on the fact that nodes do not need to be distributed equidistant from each other, since such a regular grid is not effective on many manifolds. Some segments of the manifold may have a high curvature and require many nodes while other segments of the same manifold have zero curvature and require less nodes. The authors show how to characterize the Markov chain for a given grid in order to build a framework for numerical approximation of the solution to the heat equation on Riemannian manifolds. When viewing a grid as a graph, the infinitesimal generator of a random walk on a graph is also known as the graph Laplacian or discrete Laplace operator. Hence the authors effectively show how to choose the weights of the graph Laplacian such that the continuous Laplace-Beltrami operator which is used on Riemannian manifolds is approximated. Furthermore, we discuss advantages of this technique and provide examples and simulations of our results.