This is a set of Matlab functions to interpolate scattered data with Radial Basis Functions (RBF).

## Getting Started

```
y = RBFinterp(xs, ys, x, RBFtype, R)
```

interpolates to find y, the values of the function y=f(x) at the points x.

Xs must be a matrix of size [N,Dx], with N the number of data points and Dx the dimension of the points in xs and x.

Ys must be a matrix of size [N,Dy], with N the number of known values at points in Xs, and Dy the dimension of the y values.

X must be a matrix of size [M,Dx], with M the number of query points.

RBFtype specifies the radial basis functions (RBF) to be used.

- The available global support RBFs are:
- ‘R1’ - linear spline
- ‘R3’ - cubic spline
- ‘TPS2’ - thin plate spline
- ‘Q’ - quadric
- ‘MQ’ - multiquadric
- ‘IMQ’ - inverse multiquadric
- ‘IQ’ - inverse quadric
- ‘GS’ - Gauss

RBF name | Abbreviation | |
---|---|---|

Linear spline | R1 | |

Cubic splie | R3 | |

Thin plate spline | TPS2 | |

Quadric | Q | |

Multiquadric | MQ | |

Inverse multiquadric | IMQ | |

Inverse quadric | IQ | |

Gauss | GS |

- The available compact support RBFs are (see Wendland H., Konstruktion und Untersuchung radialer Basisfunktionen mit kompaktem Träger. PhD thesis, Göttingen, Georg-August-Universität zu Göttingen, Diss, 1996):
- ‘CP_C0’
- ‘CP_C2’
- ‘CP_C4’
- ‘CP_C6’
- ‘CTPS_C0’
- ‘CTPS_C1’
- ‘CTPS_C2a’
- ‘CTPS_C2b’

Compact support functions have the form

RBF name | |
---|---|

R is either the support radius for the compact support RBFs or a parameter to make the distance values dimensionless for the global support RBFs.

```
[fPar, M] = RBFparam(xs, ys, RBFtype, R)
```

returns the weights in the RBF summation and the polynomial coefficients in a column vector fPar by solving a linear system

```
[y] = RBFeval(xs, x, fPar, RBFtype, R)
```

returns the values of the interpolation weighted function at points x by performing the matrix-vector product

## Running the example

An example case can be run just by typing in the Matlab command line

```
test
```

## References

- Beckert, Armin and Wendland, Holger. Multivariate interpolation for fluid-structure-interaction problems using radial basis functions.
*Aerospace Science and Technology*, 5 (2), p. 125-134, 2001. - Wendland, Holger.
*Konstruktion und Untersuchung radialer Basisfunktionen mit kompaktem Träger}*. PhD thesis, Göttingen, Georg-August-Universität zu Göttingen, Diss, 1996. - De Boer, A and Van der Schoot, MS and Bijl, Hester. Mesh deformation based on radial basis function interpolation.
*Computers & structures*, 85 (11-14), p. 784-795, 2007. - Biancolini, Marco Evangelos.
*Fast Radial Basis Functions for Engineering Applications*. Springer, 2018.