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

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  1. Cuomo, Salvatore; Galletti, Ardelio; Giunta, Giulio; Marcellino, Livia: Reconstruction of implicit curves and surfaces via RBF interpolation (2017)
  2. Slattery, Stuart R.: Mesh-free data transfer algorithms for partitioned multiphysics problems: conservation, accuracy, and parallelism (2016)
  3. Farrell, Patricio; Pestana, Jennifer: Block preconditioners for linear systems arising from multiscale collocation with compactly supported RBFs. (2015)
  4. Xiao, Jianping; Wang, Lei; Boyd, John P.: RBF-vortex methods for the barotropic vorticity equation on a sphere (2015)
  5. Torres, C.E.; Parishani, H.; Ayala, O.; Rossi, L.F.; Wang, L.-P.: Analysis and parallel implementation of a forced $N$-body problem (2013)
  6. Ward, John Paul: $L^p$ error estimates for approximation by Sobolev splines and Wendland functions on $\Bbb R^d$ (2013)
  7. Yokota, Rio; Barba, L.A.: FMM-based vortex method for simulation of isotropic turbulence on GPUs, compared with a spectral method (2013)
  8. Deng, Quan; Driscoll, Tobin A.: A fast treecode for multiquadric interpolation with varying shape parameters (2012)
  9. Yokota, Rio; Barba, L.A.: Comparing the treecode with FMM on GPUs for vortex particle simulations of a leapfrogging vortex ring (2011)
  10. Yokota, Rio; Barba, L.A.; Knepley, Matthew G.: PetRBF - A parallel $O(N)$ algorithm for radial basis function interpolation with Gaussians (2010)
  11. Yokota, Rio; Barba, L.A.; Knepley, Matthew G.: Petrbf--A parallel $O(N)$ algorithm for radial basis function interpolation (2009) ioport