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

Abstract

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 $O(\log(\frac{T}δ) (\log(\frac{1}ε)+\log\log(\frac{T}δ)))$ for any $x\in\bbR$ and $δ\leq t \leq T$, where $ε$ is the desired precision. In all higher dimensions, the corresponding heat kernel admits an approximation involving only $O(\log^2(\frac{T}δ))$ terms for fixed accuracy $ε$. 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 $O(N_S N_T \log^2 \frac{T}δ)$ in terms of both memory and CPU time for fixed accuracy $ε$. The algorithms can be parallelized in a straightforward manner. Several numerical examples are presented to illustrate the accuracy and stability of these approximations.