### Abstract

The fast Gauss transform allows for the calculation of the sum of N Gaussians at M points in O(N + M) time. Here, we extend the algorithm to a wider class of kernels, motivated by quadrature issues that arise in using integral equation methods for solving the heat equation on moving domains. In particular, robust high-order product integration methods require convolution with O(q) distinct Gaussian-type kernels in order to obtain qth-order accuracy in time. The generalized Gauss transform permits the computation of each of these kernels and, thus, the construction of fast solvers with optimal computational complexity. We also develop plane-wave representations of these Gaussian-type fields, permitting the "diagonal translation" version of the Gauss transform to be applied. When the sources and targets lie on a uniform grid, or a hierarchy of uniform grids, we show that the curse of dimensionality (the exponential growth of complexity constants with dimension) can be mitigated. Under these conditions, the algorithm has a lower operation count than the fast Fourier transform even for modest values of N and M.

Original language | English (US) |
---|---|

Pages (from-to) | 3092-3107 |

Number of pages | 16 |

Journal | SIAM Journal on Scientific Computing |

Volume | 32 |

Issue number | 5 |

DOIs | |

State | Published - 2010 |

### Fingerprint

### Keywords

- Fast algorithms
- Gauss transform
- Heat potentials
- High-order accuracy
- Tensorproduct grids

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics

### Cite this

*SIAM Journal on Scientific Computing*,

*32*(5), 3092-3107. https://doi.org/10.1137/100790744

**The fast generalized Gauss transform.** / Spivak, Marina; Veerapaneni, Shravan K.; Greengard, Leslie.

Research output: Contribution to journal › Article

*SIAM Journal on Scientific Computing*, vol. 32, no. 5, pp. 3092-3107. https://doi.org/10.1137/100790744

}

TY - JOUR

T1 - The fast generalized Gauss transform

AU - Spivak, Marina

AU - Veerapaneni, Shravan K.

AU - Greengard, Leslie

PY - 2010

Y1 - 2010

N2 - The fast Gauss transform allows for the calculation of the sum of N Gaussians at M points in O(N + M) time. Here, we extend the algorithm to a wider class of kernels, motivated by quadrature issues that arise in using integral equation methods for solving the heat equation on moving domains. In particular, robust high-order product integration methods require convolution with O(q) distinct Gaussian-type kernels in order to obtain qth-order accuracy in time. The generalized Gauss transform permits the computation of each of these kernels and, thus, the construction of fast solvers with optimal computational complexity. We also develop plane-wave representations of these Gaussian-type fields, permitting the "diagonal translation" version of the Gauss transform to be applied. When the sources and targets lie on a uniform grid, or a hierarchy of uniform grids, we show that the curse of dimensionality (the exponential growth of complexity constants with dimension) can be mitigated. Under these conditions, the algorithm has a lower operation count than the fast Fourier transform even for modest values of N and M.

AB - The fast Gauss transform allows for the calculation of the sum of N Gaussians at M points in O(N + M) time. Here, we extend the algorithm to a wider class of kernels, motivated by quadrature issues that arise in using integral equation methods for solving the heat equation on moving domains. In particular, robust high-order product integration methods require convolution with O(q) distinct Gaussian-type kernels in order to obtain qth-order accuracy in time. The generalized Gauss transform permits the computation of each of these kernels and, thus, the construction of fast solvers with optimal computational complexity. We also develop plane-wave representations of these Gaussian-type fields, permitting the "diagonal translation" version of the Gauss transform to be applied. When the sources and targets lie on a uniform grid, or a hierarchy of uniform grids, we show that the curse of dimensionality (the exponential growth of complexity constants with dimension) can be mitigated. Under these conditions, the algorithm has a lower operation count than the fast Fourier transform even for modest values of N and M.

KW - Fast algorithms

KW - Gauss transform

KW - Heat potentials

KW - High-order accuracy

KW - Tensorproduct grids

UR - http://www.scopus.com/inward/record.url?scp=78149337561&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78149337561&partnerID=8YFLogxK

U2 - 10.1137/100790744

DO - 10.1137/100790744

M3 - Article

AN - SCOPUS:78149337561

VL - 32

SP - 3092

EP - 3107

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

SN - 1064-8275

IS - 5

ER -