### 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 language | English (US) |
---|---|

Pages (from-to) | 529-551 |

Number of pages | 23 |

Journal | Advances in Computational Mathematics |

Volume | 41 |

Issue number | 3 |

DOIs | |

State | Published - Jul 4 2014 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics

### Cite this

*Advances in Computational Mathematics*,

*41*(3), 529-551. https://doi.org/10.1007/s10444-014-9372-1

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

Research output: Contribution to journal › Article

*Advances in Computational Mathematics*, vol. 41, no. 3, pp. 529-551. https://doi.org/10.1007/s10444-014-9372-1

}

TY - JOUR

T1 - Efficient sum-of-exponentials approximations for the heat kernel and their applications

AU - Jiang, Shidong

AU - Greengard, Leslie

AU - Wang, Shaobo

PY - 2014/7/4

Y1 - 2014/7/4

N2 - 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 NS points in the spatial discretization and NT 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.

AB - 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 NS points in the spatial discretization and NT 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.

KW - Heat kernels

KW - Heat potentials

KW - Inverse laplace transform

KW - Sum-of-exponentials approximation

UR - http://www.scopus.com/inward/record.url?scp=84937191308&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84937191308&partnerID=8YFLogxK

U2 - 10.1007/s10444-014-9372-1

DO - 10.1007/s10444-014-9372-1

M3 - Article

AN - SCOPUS:84937191308

VL - 41

SP - 529

EP - 551

JO - Advances in Computational Mathematics

JF - Advances in Computational Mathematics

SN - 1019-7168

IS - 3

ER -