An improved radial basis-pseudospectral method with hybrid Gaussian-cubic kernels

Abstract

While pseudospectral (PS) methods can feature very high accuracy, they tend to be severely limited in terms of geometric flexibility. Application of global radial basis functions overcomes this, however at the expense of problematic conditioning (1) in their most accurate flat basis function regime, and (2) when problem sizes are scaled up to become of practical interest. The present study considers a strategy to improve on these two issues by means of using hybrid radial basis functions that combine cubic splines with Gaussian kernels. The parameters, controlling Gaussian and cubic kernels in the hybrid RBF, are selected using global particle swarm optimization. The proposed approach has been tested with radial basis-pseudospectral method for numerical approximation of Poisson, Helmholtz, and Transport equation. It was observed that the proposed approach significantly reduces the ill-conditioning problem in the RBF-PS method, at the same time, it preserves the stability and accuracy for very small shape parameters. The eigenvalue spectra of the coefficient matrices in the improved algorithm were found to be stable even at large degrees of freedom, which mimic those obtained in pseudospectral approach. Also, numerical experiments suggest that the hybrid kernel performs significantly better than both pure Gaussian and pure cubic kernels.

Hybrid Gaussian-cubic radial basis functions for scattered data interpolation

“…6. Future work shall involve the application of the proposed hybrid kernels in local RBF interpolations such as RBF-FD and for stable meshless schemes for numerical solution of PDEs [25,26]. 7.…”

A local hybrid kernel meshless method for numerical solutions of two‐dimensional fractional cable equation in neuronal dynamics

“…Hybrid Gaussian‐cubic radial basis function (HGC‐RBF) was initially introduced in [55] and then in [56, 57] they were used successfully for solving PDEs. As stated in [55–57], combination of the cubic kernel in the Gaussian kernel decreases the condition number and increases accuracy, remarkably. Moreover, such a hybridization of the kernels makes the used algorithm well‐posed and stabilizes interpolation.…”

“…The HGC‐RBF was first developed in [55] and has been shown to be stabilized for scattered data interpolation problems. Later in works [56, 57] it is shown that the HGC‐RBF is a plausible approach for numerical treatment of PDEs. The HGC‐RBF is defined as: where ε is shape parameter corresponds to Gaussian RBF and α , β are coefficients which govern contribution of the Gaussian and cubic kernels in the hybrid kernel.…”

Toward Automated Early Detection of Risks for a CO2 Plume Containment From Permanent Seismic Monitoring Data

“…(2017) reviewed existing approaches to ROM for CO 2 flow simulations. We split them into three groups: Simplified physics/idealized subsurface methods are derived from the first physical principles but neglect some of complicating factors, such as spatial heterogeneity of the reservoir properties (e.g., radial displacement formulation by P. K. Mishra, Nath, et al., 2017) and/or capillary effects (e.g., vertical equilibrium by Cowton et al., 2018). The effective reservoir parameters are usually derived from high‐fidelity reservoir simulations. No physics/complex subsurface or proxy‐models (e.g., Zubarev, 2009) derive a regression between few integral characteristics of reservoir performance and petrophysical parameters based on high‐fidelity reservoir simulations.…”

“…Efficient data assimilation algorithms and ROM techniques are standard tools for production optimization in petroleum industry and hydrology. S. Mishra, Nath, et al (2017) reviewed existing approaches to ROM for CO 2 flow simulations. We split them into three groups:…”