21.4.4.4. Radial Basis Functions

Radial Basis Functions (RBF) are a class of functions used for interpolation purposes. Their value depends on only on the distance that is the radius between the generic point and center of the particular function. The RBF method constructs the approximation function \(y\left( \mathbf{x} \right)\) to pass through all sampling points using radial basis function \({{B}_{i}}\left( \mathbf{x} \right)\) and polynomial basis function \({{X}_{j}}\left( \mathbf{x} \right)\).

\(y\left( \mathbf{x} \right)=\sum\limits_{i=1}^{{{n}_{s}}}{{{B}_{i}}\left( \mathbf{x} \right){{w}_{i}}}+\sum\limits_{j=1}^{m}{{{X}_{j}}\left( \mathbf{x} \right){{\beta }_{j}}}=\mathbf{B}{{\left( \mathbf{x} \right)}^{T}}\mathbf{w}+\mathbf{X}\left( \mathbf{x} \right)\mathbf{\beta }\)

where, \({{w}_{i}}\) is the weighting coefficient for \({{B}_{i}}\left( \mathbf{x} \right)\) and \({{\beta }_{j}}\) the coefficient for \({{X}_{j}}\left( \mathbf{x} \right)\).

A radial basis function has the following general form:

\({{B}_{i}}\left( \mathbf{x} \right)={{B}_{i}}\left( {{r}_{i}} \right)\)

where, \({{r}_{i}}\) is a distance between interpolating point \(\left( \mathbf{x} \right)\) and the ith sampled point \(\left( {{\mathbf{x}}_{i}} \right)\). In general, multiquadratics \(\left( \sqrt{{{r}^{2}}+{{\varepsilon }^{2}}} \right)\) and Gaussian spline \(\left( \exp \left( -{{r}^{2}} \right) \right)\) is widely used in the radial basis function.

In order to guarantee unique approximation, the following constraints are usually imposed to the polynomial term.

\(\sum\limits_{i=1}^{{{n}_{s}}}{{{X}_{j}}\left( {{\mathbf{x}}_{i}} \right){{w}_{i}}}=0,\text{ }j=1,2,...,m\)

It is expressed in matrix form as follows:

\(\left[ \begin{matrix} \mathbf{B} & \mathbf{X} \\ \mathbf{X} & \mathbf{0} \end{matrix} \right] \left\{ \begin{matrix} w \\ \beta \end{matrix} \right\} = \left\{ \begin{matrix} y \\ 0 \end{matrix} \right\}\)

As the distance is scalar value, the matrix \(\mathbf{B}\) is symmetric. Hence the unique solution is guaranteed if the inverse of matrix \(\mathbf{B}\) exists.

Reference

  1. Wang, J.G. and Liu, and G.R., “A point interpolation meshless method based on radial basis functions”, International Journal for Numerical Methods in Engineering, Vol. 54, pp. 1623-1648, 2002.

  2. Jin, R., Chen, W. and Simpson, T.W., “Comparative studies of metamodeling techniques under multiple modeling criteria”, Struct. Multidisc. Optim., Vol. 23, pp. 1-13, 2001.