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

Shidong Jiang, Leslie Greengard, Shaobo Wang

Research output: Contribution to journalArticle

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 (Formula Presented.) for any x∈ℝR and δ≤t≤T, where (Formula Presented.) is the desired precision. In all higher dimensions, the corresponding heat kernel admits an approximation involving only (Formula Presented.) terms for fixed accuracy (Formula Presented.). 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<inf>S</inf> points in the spatial discretization and N<inf>T</inf> time steps, the cost is (Formula Presented.) in terms of both memory and CPU time for fixed accuracy (Formula Presented.). The algorithms can be parallelized in a straightforward manner. Several numerical examples are presented to illustrate the accuracy and stability of these approximations.

Original languageEnglish (US)
Pages (from-to)529-551
Number of pages23
JournalAdvances in Computational Mathematics
Volume41
Issue number3
DOIs
StatePublished - Jul 4 2014

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Heat Kernel
Approximation
Boundary value problems
Integral equations
Program processors
Complex Geometry
Term
CPU Time
Data storage equipment
Heat Equation
Higher Dimensions
Accelerate
Geometry
Hot Temperature
Integral Equations
Discretization
Boundary Value Problem
Costs
Numerical Examples

Keywords

  • Heat kernels
  • Heat potentials
  • Inverse laplace transform
  • Sum-of-exponentials approximation

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics

Cite this

Efficient sum-of-exponentials approximations for the heat kernel and their applications. / Jiang, Shidong; Greengard, Leslie; Wang, Shaobo.

In: Advances in Computational Mathematics, Vol. 41, No. 3, 04.07.2014, p. 529-551.

Research output: Contribution to journalArticle

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