Hybrid Gaussian-cubic radial basis functions for scattered data interpolation

Abstract

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.

Appendices

Appendix 1: LOOCV

Following the notations used in problem 2.1, let us write the datasites without the k^th data as,

$$\boldsymbol{x}^{[k]} = [\boldsymbol{x}_{1},\ldots,\boldsymbol{x}_{k-1}, \boldsymbol{x}_{k + 1},\ldots,\boldsymbol{x}_{N} ]^{T}. $$

The removed point has been indicated by the superscript [k]. This superscript will differentiate the quantities computed with “full” dataset and partial dataset without the k^th point. Hence, the partial RBF interpolant $\mathcal {F}(\boldsymbol {x})$ of the given data f(x) can be written as,

$$\mathcal{F}(\boldsymbol{x}) = \sum\limits_{j = 1}^{N-1} {c}_{j}^{[k]} \phi (\parallel \boldsymbol{x}-{\boldsymbol{x}}_{j}^{[k]}\parallel). $$

The error estimator can, therefore, be written as,

$$e_{k} = \boldsymbol{f}(\boldsymbol{x}_{k}) - \mathcal{F}^{[k]}(\boldsymbol{x}_{k}). $$

The norm of the error vector e = [e₁,…,e_N]^T, obtained by removing each one point and comparing the interpolant to the known value at the excluded point determines the quality of the interpolation. This norm serves as the “cost function” which is the function of the kernel parameters ε, α, and β. We consider l₂ norm of the error vectors for our purpose. The algorithm for constructing the “cost function” for RBF interpolation via LOOCV has been summarized in Algorithm 1. We recommend [7, 9] for some more insights of the application of LOOCV in radial basis interpolation problems. Here c_k is the k^th coefficient for the interpolant on “full data” set and ${\mathbf {A}}_{kk}^{-1}$ is the k^th diagonal element in the inverse of the interpolation matrix for “full data”.

Appendix 2: Particle swarm optimization

The term optimization refers to the process of finding a set of parameters corresponding to a given criterion among many possible sets of parameters. One such optimization algorithm is particle swarm optimization (PSO), proposed by James Kennedy and Russell Eberhart in 1995 [4, 5]. PSO is known as an algorithm which is inspired by the exercise of living organisms like bird flocking and fish schooling. In PSO, the system is initiated with many possible random solutions and it finds optima in the given search space by updating the solutions over the specified number of generations. The possible solutions corresponding to a user defined criterion are termed as particles. At each generation, the algorithm decides optimum particle towards which all the particles fly in the problem space. The rate of change in the position of a particle in the problem space is termed as particle velocity. In each generation, all the particles are given two variables which are known as pbest and gbest. The first variable (pbest) stores the best solution by a particle after a typical number of iteration. The second variable (gbest) stores the global best solution, obtained so far by any particle in the search space [34, 35]. Once the algorithm finds these two parameters, it updates the velocity and the position of all the particles according to the following pseudo-codes,

$$\begin{array}{@{}rcl@{}} v[.] &=& v[.]+c_{1}*rand(.)*(pbest[.] - present[.])\\ && + c_{2} * rand(.) * (gbest[.] - present[.]), \end{array} $$

$$\begin{array}{@{}rcl@{}} present[.] = present[.] + v[.] \end{array} $$

Where, v[.] is the particle velocity, present[.] is the particle at current generation, and c₁ and c₂ are learning factors. According to the studies of Perez and Behdinan [27], the particle swarm algorithm is stable only if the following conditions are fulfilled;

$$0< c_{1}+c_{2} < 4 $$

$$\left(\frac{c_{1}+c_{2}}{2}\right)-1 < w<1 $$

The optimization of the parameters of hybrid Gaussian-cubic kernel using particle swarm optimization is summarized in the flowchart given in the Fig. 10.

Fig. 10

Flowchart of particle swarm optimization in the context of numerical tests

Appendix 3: Variogram models of ‘normal fault’ data

Fig. 11

Variogram models for the data used in Section 6.5. We have used the following MATLAB package for the same https://www.mathworks.com/matlabcentral/fileexchange/29025-ordinary-kriging