PetRBF — A parallel O(N) algorithm for radial basis function interpolation with Gaussians

✍ Scribed by Rio Yokota; L.A. Barba; Matthew G. Knepley

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 772 KB
• Volume: 199
• Category: Article
• ISSN: 0045-7825
• DOI: 10.1016/j.cma.2010.02.008

✦ Synopsis

Order-N algorithms Particle methods Parallel computing

We have developed a parallel algorithm for radial basis function (RBF) interpolation that exhibits O(N) complexity, requires O(N) storage, and scales excellently up to a thousand processes. The algorithm uses a GMRES iterative solver with a restricted additive Schwarz method (RASM) as a preconditioner and a fast matrixvector algorithm. Previous fast RBF methodsachieving at most O(NlogN) complexitywere developed using multiquadric and polyharmonic basis functions. In contrast, the present method uses Gaussians with a small variance with respect to the domain, but with sufficient overlap. This is a common choice in particle methods for fluid simulation, our main target application. The fast decay of the Gaussian basis function allows rapid convergence of the iterative solver even when the subdomains in the RASM are very small. At the same time we show that the accuracy of the interpolation can achieve machine precision. The present method was implemented in parallel using the PETSc library (developer version). Numerical experiments demonstrate its capability in problems of RBF interpolation with more than 50 million data points, timing at 106 s (19 iterations for an error tolerance of 10 -15 ) on 1024 processors of a Blue Gene/L (700 MHz PowerPC processors). The parallel code is freely available in the open-source model.