Vectorial interpolation using radial-basis-like functions
The authors deal with vector field interpolants \(\sigma: \mathbb{R}^3\to \mathbb{R}^3\) on scattered data. By solving the variational spline problem \[ |f|^2_{m,\rho}= \rho|\text{div } f|^2_{m- 1}+ |\text{rot } f|^2_{m- 1}, \] with a parameter \(\rho\) which governs the balance of the divergence part and the rotational part of the interpolant and with some seminorm \(|\cdot|_{m-1}\), where \(m\) determines the regularity of the interpolant, possible connections between the components of the interpolant were taken into account. The authors prove that there exists exactly one interpolant \(\sigma_{m,\rho}\) which minimizes \(|\cdot|_{m,\rho}\). This interpolant has the form \[ \sigma_{m,\rho}(x)= \sum^n_{i=1} \Phi(x- x^i)a^i+ p_{m-1}(x), \] where \(\Phi\) is some \(3\times 3\) matrix functions, \(a^i\) are \(\mathbb{R}^3\)-vectors and \(p_{m-1}\) is a vectorial polynomial of degree \(m-1\). For \(\rho= 1\), one finds the usual \(m\)-harmonic spline in each component of \(\sigma_{m,\rho}\). Numerical examples demonstrate the power of the method.
scientific article; zbMATH DE number 1781510