Fast hierarchical algorithms for generating Gaussian random fields - Inria - Institut national de recherche en sciences et technologies du numérique Accéder directement au contenu
Rapport (Rapport De Recherche) Année : 2015

Fast hierarchical algorithms for generating Gaussian random fields

Résumé

Low-rank techniques for the approximation of matrices have become crucial tools in scientific computing in order to reduce the cost of storing matrices and compute usual matrix operations. Since standard techniques like the SVD or Cholesky do not scale well with the problem size N there has been a growing interest for alternative methods like randomized low-rank techniques. The main benefit of these methods lies in their simplicity, in fact only very basic operations like Matrix Vector Products (MVPs) or orthogonalizations are involved. As a result the cubic cost required to perform a matrix factor-ization with a standard technique is reduced to the quadratic cost required to apply a few MVPs, namely O(r × N 2) where r is the numerical rank of the matrix. We present a new efficient algorithm for performing MVPs in O(N) operations called the Uniform FMM (ufmm) based on polynomial interpolations combined with a hierarchical (data sparse) representation of a kernel matrix. The algorithm is close to the Black Box FMM by Fong and Darve [1], however it uses an interpolation scheme based on an equispaced grid, which allows the use of FFT and consequently reduce both running time and memory footprint but has implications on accuracy and stability. In this work the ufmm is used to speed-up the MVPs involved in the randomized SVD and consequently reduce its cost to O(r 2 ×N). Numerical experiments were performed on this accelerated randomized SVD in order to evaluate the computational cost of building a low-rank square root of a covariance matrix. Such method is of particular interest in a wide range of scientific fields where spatially correlated multivariate Gaussian random variables have to be generated with very large sets of points. As expected and confirmed by our experiments the hierarchical nature of the algorithm leads to very competitive performances when the distribution of point is heterogeneous.
Fichier principal
Vignette du fichier
RR-8811.pdf (898.29 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-01228519 , version 1 (13-11-2015)
hal-01228519 , version 2 (18-02-2016)

Identifiants

  • HAL Id : hal-01228519 , version 1

Citer

Pierre Blanchard, Olivier Coulaud, Eric Darve. Fast hierarchical algorithms for generating Gaussian random fields. [Research Report] Inria Bordeaux Sud-Ouest. 2015. ⟨hal-01228519v1⟩
557 Consultations
821 Téléchargements

Partager

Gmail Facebook X LinkedIn More