1. Examples

2. Motivation

3. References

Radial basis function (RBF) interpolation is a method for constructing high-accuracy interpolants of unstructured data, possibly in high dimensions. The interpolant is a weighted sum of radial basis functions. RBF interpolation is a mesh-free method, meaning the nodes (points in the domain) need not lie on a structured grid or mesh, and does not require the formation of a mesh. It is often spectrally accurate{{cn|date=April 2019}} and stable for large numbers of nodes even in high dimensions.

Many interpolation methods can be used as the theoretical foundation of algorithms for approximating linear operators, and RBF interpolation is no exception. RBF interpolation has been used to approximate differential operators, integral operators, and surface differential operators. These algorithms have been used to find highly accurate solutions of many differential equations including Navier–Stokes^[1], Cahn–Hilliard, and the shallow water equations^[2][3].

Examples

Let and let be 15 equally spaced points on the interval . We will form where is a radial basis function, and choose such that ( interpolates at the chosen points). In matrix notation this can be written as

Choosing , the Gaussian, with a shape parameter of , we can then solve the matrix equation for the weights and plot the interpolant. Plotting the interpolating function below, we see that it is visually the same everywhere except near the left boundary, where it is still a very close approximation. More precisely the maximum error is roughly .

Motivation

The Mairhuber–Curtis theorem says that for any vector space with dimension higher than 2, and linearly independent functions on , there exists a set of points in the domain such that the interpolation matrix

is not singular.^[4]

This means that if one wishes to have a general interpolation algorithm, one must choose the basis functions to depend on the interpolation points. In 1971, Rolland Hardy developed a method of interpolating scattered data using interpolants of the form . This is interpolation using a basis of shifted multiquadric functions, now more commonly written as , and is the first instance of radial basis function interpolation. ^[5] It has been shown that the resulting interpolation matrix will always be non-singular. This does not violate the Mairhuber–Curtis theorem since the basis functions depend on the points of interpolation. Choosing a radial kernel such that the interpolation matrix is non-singular is exactly the definition of a radial basis function. It has been shown that any function that is completely monotone will have this property, including the Gaussian, inverse quadratic, and inverse multiquadric functions.^[6]

References

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• Dikran Chökürian