Smooth surface reconstruction from noisy range data
Proceedings of the 1st international conference on Computer graphics and interactive techniques in Australasia and South East Asia
Rapid evaluation of radial basis functions
Journal of Computational and Applied Mathematics
Fast multipole method for the biharmonic equation in three dimensions
Journal of Computational Physics
Short note: A kernel independent fast multipole algorithm for radial basis functions
Journal of Computational Physics
Computers & Mathematics with Applications
A univariate quasi-multiquadric interpolationwith better smoothness
Computers & Mathematics with Applications
Application of the RBF meshless method to the solution of the radiative transport equation
Journal of Computational Physics
Rapid evaluation of radial basis functions
Journal of Computational and Applied Mathematics
Defining, contouring, and visualizing scalar functions on point-sampled surfaces
Computer-Aided Design
Fast Evaluation of Multiquadric RBF Sums by a Cartesian Treecode
SIAM Journal on Scientific Computing
CAD and mesh repair with Radial Basis Functions
Journal of Computational Physics
A massively parallel adaptive fast multipole method on heterogeneous architectures
Communications of the ACM
An adaptive hybrid surrogate model
Structural and Multidisciplinary Optimization
Least-squares hermite radial basis functions implicits with adaptive sampling
Proceedings of Graphics Interface 2013
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A generalized multiquadric radial basis function is a function of the form $ s(x) = \sum_{i=1}^N d_i \phi(|x -t_i|), $ where $ \phi(r)= ( r^2+\tau^2 )^{k/2}$, $x \in \mathbb{R}^n$, and $k\in\mathbb{Z}$ is odd. The direct evaluation of an N center generalized multiquadric radial basis function at m points requires $\mathcal{O}(m N)$ flops, which is prohibitive when m and N are large. Similar considerations apparently rule out fitting an interpolating N center generalized multiquadric to N data points by either direct or iterative solution of the associated system of linear equations in realistic problems.In this paper we will develop far field expansions, recurrence relations for efficient formation of the expansions, error estimates, and translation formulas for generalized multiquadric radial basis functions in n-variables. These pieces are combined in a hierarchical fast evaluator requiring only $\mathcal{O}((m+N)\log N|\log\epsilon|^{n+1})$ flops for evaluation of an N center generalized multiquadric at m points. This flop count is significantly less than that of the direct method. Moreover, used to compute matrix-vector products, the fast evaluator provides a basis for fast iterative fitting strategies.