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

TL;DR

This paper introduces a parallel algorithm for radial basis function interpolation with linear complexity, leveraging Gaussian basis functions and iterative solvers, enabling efficient processing of extremely large datasets on high-performance computing systems.

Contribution

The paper presents a novel parallel O(N) RBF interpolation algorithm using Gaussian basis functions, a GMRES solver, and a fast matrix-vector method, scalable to thousands of processors.

Findings

Successfully interpolated over 50 million data points.

Achieved a runtime of 106 seconds on 1024 processors.

Demonstrated excellent scalability and convergence properties.

Abstract

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.