PetRBF--A parallel O(N) algorithm for radial basis function interpolation

• Published in: arXiv.org
• Main Authors: Yokota, Rio, Barba, L A, Knepley, Matthew G
• Format: Paper Journal Article
• Language: English; Japanese
• Published: Ithaca Cornell University Library, arXiv.org 29.09.2009
• Subjects: Algorithms; Basis functions; Complexity; Computer Science - Distributed, Parallel, and Cluster Computing; Computer Science - Mathematical Software; Computer Science - Numerical Analysis; Computer simulation; Data points; Decay rate; Harmonic functions; Interpolation; Iterative methods; Mathematical analysis; Mathematical models; Matrix algebra; Matrix methods; Neural networks; Particle decay; Particle methods (mathematics); Processors; Radial basis function
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.0909.5413

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 matrix-vector algorithm. Previous fast RBF methods, --,achieving at most O(NlogN) complexity,--, were developed using multiquadric and polyharmonic basis functions. In contrast, the present method uses Gaussians with a small variance (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. 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 seconds (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.