Multiquadrics without the Shape Parameter for Solving Partial Differential Equations

Abstract

In this article, we present multiquadric radial basis functions (RBFs), including multiquadric (MQ) and inverse multiquadric (IMQ) functions, without the shape parameter for solving partial differential equations using the fictitious source collocation scheme. Different from the conventional collocation method that assigns the RBF at each center point coinciding with an interior point, we separated the center points from the interior points, in which the center points were regarded as the fictitious sources collocated outside the domain. The interior, boundary, and source points were therefore collocated within, on, and outside the domain, respectively. Since the radial distance between the interior point and the source point was always greater than zero, the MQ and IMQ RBFs and their derivatives in the governing equation were smooth and globally infinitely differentiable. Accordingly, the shape parameter was no longer required in the MQ and IMQ RBFs. Numerical examples with the domain in symmetry and asymmetry are presented to verify the accuracy and robustness of the proposed method. The results demonstrated that the proposed method using MQ RBFs without the shape parameter acquires more accurate results than the conventional RBF collocation method with the optimum shape parameter. Additionally, it was found that the locations of the fictitious sources were not sensitive to the accuracy.

Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs

“…To compare the proposed PRBF with the conventional PRBF, we firstly consider a steady-state groundwater flow problem [33]:…”

“…The fictitious sources were placed on the external domain by utilizing (25). The fictitious source boundaries were defined as [33,34] ∂Ω s = (x s j , y s j ) x s j = ηρ s j cos θ s j , y s j = ηρ s j sin θ s j (25) where ∂Ω s denotes the boundaries of fictitious sources. Figure 3 illustrates the MAE values of the conventional and proposed PRBFs with different orders (values of k).…”

“…The adaptive greedy algorithm prefers exterior centers for large scaling parameters. Similar to the idea of exterior centers, Ku et al presented a fictitious source collocation scheme where the centers are replaced by fictitious sources collocated outside the domain as well [33,34]. However, different from the greedy algorithm, the role of fictitious sources is emphasized on the prevention of singularity without the shape parameter for the MQ.…”

An RBF-Based h-Adaptive Cartesian Grid Refinement Method for Arbitrary Single/Multi-Body Hull Modeling and Reconstruction

“…Frank [10] compared the scattered data interpolation algorithms and pointed out that the MQ-RBF has a better fitting result. It can be used to solve partial differential equations [33] and has been widely used in geologic modeling [6]. It has also been adopted in our previous works [17][18][19].…”

Forecasting Grain Production and Static Capacity of Warehouses Using the Natural Neighbor and Multiquadric Equations

“…(asymmetric) data, whereas multiquadric equations are used to interpolate qualitative variables(Tsidaev, 2016;Yamamoto et al, 2018).Recent studies have shown the wide application and advantages of spatial interpolation methods using the Natural Neighbor (NN) interpolator and multiquadric equations(Belinha et al, 2017;Ku et al, 2020). The spatial interpolation methods NN and the radial basis function, also known as the study of multiquadric equations, allow the knowledge of the spatial behavior of variables at non-sampled locations, facilitating the decisionmaking(Sampedro et al, 2019;Millan et al, 2020).…”