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

TL;DR

This paper develops efficient sum-of-exponentials approximations for the heat kernel in any dimension, enabling faster boundary value problem solutions with near-optimal algorithms and demonstrated numerical stability.

Contribution

The paper introduces a novel method for constructing sum-of-exponentials approximations of the heat kernel applicable in any dimension, significantly improving computational efficiency.

Findings

Approximation involves O(log(T/δ) (log(1/ε)+loglog(T/δ))) terms in 1D.

Higher dimensions require only O(log^2(T/δ)) terms for fixed accuracy.

Algorithms achieve O(N_S N_T log^2(T/δ)) complexity, are nearly optimal and parallelizable.

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