Radial basis functions for multivariable interpolation: a review
Algorithms for approximation
A spectral/B-spline method for the Navier-Stokes equations in unbounded domains
Journal of Computational Physics
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics (Scientific Computation)
The Runge phenomenon and spatially variable shape parameters in RBF interpolation
Computers & Mathematics with Applications
Radial basis function meshless method for the steady incompressible Navier-Stokes equations
International Journal of Computer Mathematics
Meshfree Approximation Methods with MATLAB
Meshfree Approximation Methods with MATLAB
Stable computation of multiquadric interpolants for all values of the shape parameter
Computers & Mathematics with Applications
A Rational Interpolation Scheme with Superpolynomial Rate of Convergence
SIAM Journal on Numerical Analysis
Towards inverse modeling of turbidity currents: The inverse lock-exchange problem
Computers & Geosciences
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We present a high order multivariate approximation scheme for scattered data sets. Each data point is represented as a Taylor series, and the high order derivatives in the Taylor series are treated as random variables. The approximation coefficients are then chosen to minimize an objective function at each point by solving an equality constrained least squares. The approximation is an interpolation when the data points are given as exact, or a nonlinear regression function when nonzero measurement errors are associated with the data points. Using this formulation, the gradient information on each data point can be used to significantly reduce the approximation error. All parameters of the approximation scheme can be computed automatically from the data points. An uncertainty bound of the approximation function is also produced by the scheme. Numerical experiments demonstrate that although this method is more computationally intensive than traditional methods, it produces more accurate approximation functions.