Hybrid Gaussian-cubic radial basis functions for scattered data interpolation

• Published in: Computational geosciences Vol. 22; no. 5; pp. 1203 - 1218
• Main Authors: Mishra, Pankaj K., Nath, Sankar K., Sen, Mrinal K., Fasshauer, Gregory E.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.10.2018; Springer Nature B.V
• Subjects: Basis functions; Coefficients; Conditioning; Data; Data processing; Datasets; Differential equations; Earth and Environmental Science; Earth Sciences; Finite element method; Geostatistics; Geotechnical Engineering & Applied Earth Sciences; Hybridization; Hydrogeology; Interpolation; Kriging interpolation; Mathematical Modeling and Industrial Mathematics; Mathematical models; Meshless methods; Neural networks; Numerical methods; Original Paper; Parameters; Partial differential equations; Particle swarm optimization; Radial basis function; Shape; Shape functions; Soil Science & Conservation; Stability; Statistical methods; Radial basis function; Multivariate interpolation; 68; Particle swarm optimization; Spatial data analysis; 65
• ISSN: 1420-0597; 1573-1499
• DOI: 10.1007/s10596-018-9747-3

Scattered data interpolation schemes using kriging and radial basis functions (RBFs) have the advantage of being meshless and dimensional independent; however, for the datasets having insufficient observations, RBFs have the advantage over geostatistical methods as the latter requires variogram study and statistical expertise. Moreover, RBFs can be used for scattered data interpolation with very good convergence, which makes them desirable for shape function interpolation in meshless methods for numerical solution of partial differential equations. For interpolation of large datasets, however, RBFs in their usual form, lead to solving an ill-conditioned system of equations, for which, a small error in the data can cause a significantly large error in the interpolated solution. In order to reduce this limitation, we propose a hybrid kernel by using the conventional Gaussian and a shape parameter independent cubic kernel. Global particle swarm optimization method has been used to analyze the optimal values of the shape parameter as well as the weight coefficients controlling the Gaussian and the cubic part in the hybridization. Through a series of numerical tests, we demonstrate that such hybridization stabilizes the interpolation scheme by yielding a far superior implementation compared to those obtained by using only the Gaussian or cubic kernels. The proposed kernel maintains the accuracy and stability at small shape parameter as well as relatively large degrees of freedom, which exhibit its potential for scattered data interpolation and intrigues its application in global as well as local meshless methods for numerical solution of PDEs.