Efficient sum-of-exponentials approximations for the heat kernel and their applications

摘要

In this paper, we show that efficient separated sum-of-exponentials approximations can be constructed for the heat kernel in any dimension. In one space dimension, the heat kernel admits an approximation involving a number of terms that is of the order for any and delta a parts per thousand currency signta parts per thousand currency signT, where oee- is the desired precision. In all higher dimensions, the corresponding heat kernel admits an approximation involving only terms for fixed accuracy oee-. These approximations can be used to accelerate integral equation-based methods for boundary value problems governed by the heat equation in complex geometry. The resulting algorithms are nearly optimal. For N (S) points in the spatial discretization and N (T) time steps, the cost is in terms of both memory and CPU time for fixed accuracy oee-. The algorithms can be parallelized in a straightforward manner. Several numerical examples are presented to illustrate the accuracy and stability of these approximations.