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

• Published in: Computer methods in applied mechanics and engineering Vol. 199; no. 25; pp. 1793 - 1804
• Main Authors: Yokota, Rio, Barba, L.A., Knepley, Matthew G.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 01.05.2010; Elsevier
• Subjects: Algorithms; Applied sciences; Basis functions; Computational techniques; Computer science; control theory; systems; Computer simulation; Computer systems and distributed systems. User interface; Domain decomposition methods; Exact sciences and technology; Gaussian; gmres; Interpolation; Iterative methods; Mathematical methods in physics; Mathematical models; Mathematics; Methods of scientific computing (including symbolic computation, algebraic computation); Numerical analysis. Scientific computation; Order- N algorithms; Parallel computing; Particle methods; Physics; Processors; Radial basis function interpolation; Sciences and techniques of general use; Software; Domain decomposition methods; Radial basis function interpolation; Parallel computing; gmres; Order- N algorithms; Particle methods; Quadric; Radial function; Krylov subspace method; Parallel algorithms; Distributed computing; Open source software; Variance; Complexity; Schwarz method; GMRES; Parallel processing; Libraries; Fast algorithm; Iterative methods; Least square fit; Overlay; Capability index; Order-N algorithms; Experimental study; Timed system; Harmonic function; N order; Radial basis function; Gaussian processes; Timing; Domain decomposition; Preconditioning
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2010.02.008

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( Nlog N) complexity — were 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.