Hybrid Gaussian-cubic radial basis functions for scattered data interpolation

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 stud...

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Published inComputational 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
LanguageEnglish
Published Cham Springer International Publishing 01.10.2018
Springer Nature B.V
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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.
AbstractList 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.
Author Mishra, Pankaj K.
Fasshauer, Gregory E.
Sen, Mrinal K.
Nath, Sankar K.
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  organization: Department of Geology and Geophysics, Indian Institute of Technology Kharagpur, Department of Mathematics, Hong Kong Baptist University
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  surname: Nath
  fullname: Nath, Sankar K.
  organization: Department of Geology and Geophysics, Indian Institute of Technology Kharagpur
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  givenname: Mrinal K.
  surname: Sen
  fullname: Sen, Mrinal K.
  organization: Jackson School of Geosciences, University of Texas at Austin
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  givenname: Gregory E.
  surname: Fasshauer
  fullname: Fasshauer, Gregory E.
  organization: Department of Applied Mathematics and Statistics, Colorado School of Mines
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Particle swarm optimization
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Snippet Scattered data interpolation schemes using kriging and radial basis functions (RBFs) have the advantage of being meshless and dimensional independent; however,...
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SubjectTerms 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
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Title Hybrid Gaussian-cubic radial basis functions for scattered data interpolation
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