Shape Parameter Of Radial Basis Function Research Articles

Efficient truncated randomized SVD for mesh-free kernel methods

This paper explores the utilization of randomized SVD (rSVD) in the context of kernel matrices arising from radial basis functions (RBFs) for the purpose of solving interpolation and Poisson problems. We propose a truncated version of rSVD, called trSVD, which yields a stable solution with a reduced condition number in comparison to the non-truncated variant, particularly when manipulating the scale or shape parameter of RBFs. Notably, trSVD exhibits exceptional proficiency in capturing the most significant singular values, enabling the extraction of critical information from the data. When compared to the conventional truncated SVD (tSVD), trSVD achieves comparable accuracy while demonstrating improved efficiency. Furthermore, we explore the potential of trSVD by employing scale parameter strategies, such as leave-one-out cross-validation and effective condition number. Then, we apply trSVD to solve a 2D Poisson equation, thereby showcasing its efficacy in handling partial differential equations. In summary, this study offers an efficient and accurate solver for RBF problems, demonstrating its practical applicability. The code implementation is provided to the scientific community for their access and reference.

Bayesian approach for radial kernel parameter tuning

In this paper we present a new fast and accurate method for Radial Basis Function (RBF) approximation, including interpolation as a special case, which enables us to effectively find the optimal value of the RBF shape parameter. In particular, we propose a statistical technique, called Bayesian optimization, that consists in modeling the error function with a Gaussian process, by which, through an iterative process, the optimal shape parameter is selected. The process is step by step self-updated resulting in a relevant decrease in search time with respect to the classical leave one out cross validation technique. Numerical results deriving from some test examples and an application to real data show the performance of the proposed method.

CURVATURE BASED CHARACTERIZATION OF RADIAL BASIS FUNCTIONS: APPLICATION TO INTERPOLATION

Choosing the scale or shape parameter of radial basis functions (RBFs) is a well-documented but still an open problem in kernel-based methods. It is common to tune it according to the applications, and it plays a crucial role both for the accuracy and stability of the method. In this paper, we first devise a direct relation between the shape parameter of RBFs and their curvature at each point. This leads to characterizing RBFs to scalable and unscalable ones. We prove that all scalable RBFs lie in the -class which means that their curvature at the point xj is proportional to, where cj is the corresponding spatially variable shape parameter at xj. Some of the most commonly used RBFs are then characterized and classified accordingly to their curvature. Then, the fundamental theory of plane curves helps us recover univariate functions from scattered data, by enforcing the exact and approximate solutions have the same curvature at the point where they meet. This leads to introducing curvature-based scaled RBFs with shape parameters depending on the function values and approximate curvature values of the function to be approximated. Several numerical experiments are devoted to show that the method performs better than the standard fixed-scale basis and some other shape parameter selection methods.

Radial basis function based finite element method: Formulation and applications

In the study, a method is proposed and developed for solid analysis that is a hybrid of the radial basis function and the finite element method (RBF-FEM). Based on the finite nodes, the method employs the radial basis functions to produce the shape functions with simplicity, especially when increasing the element node number. The formulation and applications of the method in the analysis of solids are examined. Several numerical examples are carried out to analyze the convergence, accuracy, and computational time cost. Its comeouts are then compared with that by the finite element method. The study also shows the factors that affect the accuracy of the method, such as the radial basis function types, radial basis function shape parameter, node number, and node position in an element. It is shown that the method quickly produces better results with large-node-number elements (seven and eight nodes) in comparison to the conventional numerical method. Discussion on the method and the extensibility of the approach are addressed in the conclusion.

An adaptive residual sub-sampling algorithm for kernel interpolation based on maximum likelihood estimations

In this paper we propose an enhanced version of the residual sub-sampling method (RSM) in Driscoll and Heryudono (2007) for adaptive interpolation by radial basis functions (RBFs). More precisely, we introduce in the context of sub-sampling methods a maximum profile likelihood estimation (MPLE) criterion for the optimal selection of the RBF shape parameter. This choice is completely automatic, provides highly reliable and accurate results for any RBFs, and, unlike the original RSM, guarantees that the RBF interpolant exists uniquely. The efficacy of this new method, called MPLE-RSM, is tested by numerical experiments on some 1D and 2D benchmark target functions.

Effective condition number for the selection of the RBF shape parameter with the fictitious point method

Based on the Uncertainty Principle of radial basis functions (RBFs), it is known that the condition number and the error cannot be both kept small at the same time. In contrast to the traditional condition number, the effective condition number provides a much better estimation of the actual condition number of the resultant matrix system. In this paper, motivated by the Uncertainty Principle of RBFs, we propose to apply the effective condition number as a numerical tool to determine a reasonably good shape parameter value in the context of the Kansa method coupled with the fictitious point method. Six examples for second and fourth order partial differential equations in 2D and 3D are presented to demonstrate the effectiveness of the proposed method.

RBFCUB: A numerical package for near-optimal meshless cubature on general polygons

In this paper we improve the cubature rules discussed in Sommariva and Vianello (2021) for the computation of integrals by radial basis functions (RBFs). More precisely, we introduce in the context of meshless cubature a leave-one-out cross validation criterion for the optimization of the RBF shape parameter. This choice allows us to get highly reliable and accurate results for any kind of both infinity and finite regularity RBF. The efficacy of this approximation scheme is tested by numerical experiments on complicated polygonal regions. The related Matlab software is provided to the scientific community in [1].

Two-step MPS-MFS ghost point method for solving partial differential equations

The fictitious boundary surrounding the domain is required in the implementation of the method of fundamental solutions (MFS). A similar approach of taking a portion of the centres of radial basis functions (RBFs) outside the domain in the context of the method of particular solutions (MPS) is proposed to further improve the performance of a two-step MPS-MFS method. Since the shape parameter of RBFs is problem dependent, several known procedures on how to determine a good shape parameter are introduced for solving various types of problems. The proposed method is not only highly accurate but also fairly stable due to the use of the particular solutions and fundamental solutions. To demonstrate the effectiveness of the proposed method, five numerical examples in highly complicate and irregular domains, which include second and fourth order elliptic partial differential equations in 2D and 3D, are presented.

A stable computation on local boundary-domain integral method for elliptic PDEs

Many local integral methods are based on an integral formulation over small and heavily overlapping stencils with local Radial Basis Functions (RBFs) interpolations. These functions have become an extremely effective tool for interpolation on scattered node sets, however the ill-conditioning of the interpolation matrix – when the RBF shape parameter tends to zero corresponding to best accuracy – is a major drawback. Several stabilizing methods have been developed to deal with this near flat RBFs in global approaches but there are not many applications to local integral methods. In this paper we present a new method called Stabilized Local Boundary Domain Integral Method (LBDIM-St) with a stable calculation of the local RBF approximation for small shape parameter that stabilizes the numerical error. We present accuracy results for some Partial Differential Equations (PDEs) such as Poisson, convection–diffusion, thermal boundary layer and an elliptic equation with variable coefficients.

Ghost point method using RBFs and polynomial basis functions

In this paper, we propose a new Kansa method with fictitious centre approach in which the radial basis function (RBF) approximation is augmented by polynomial basis functions. The proposed approach considerably improves not only the accuracy but also the stability of the recently proposed ghost point method using radial basis functions. The difficulty of selecting a good RBF shape parameter is no longer an issue. Two numerical examples are presented to show the effectiveness and the improvement of the proposed method over previous methods.

A fictitious points one–step MPS–MFS technique

The method of fundamental solutions (MFS) is a simple and efficient numerical technique for solving certain homogenous partial differential equations (PDEs) which can be extended to solving inhomogeneous equations through the method of particular solutions (MPS). In this paper, radial basis functions (RBFs) are considered as the basis functions for the construction of a particular solution of the inhomogeneous equation. A hybrid method coupling these two methods using both fundamental solutions and RBFs as basis functions has been effective for solving a large class of PDEs. In this paper, we propose an improved fictitious points method in which the centres of the RBFs are distributed inside and outside the physical domain of the problem and which considerably improves the performance of the MPS–MFS. We also describe various techniques to deal with the several parameters present in the proposed method, such as the location of the fictitious points, the source location in the MFS, and the estimation of a good value of the RBF shape parameter. Five numerical examples in 2D/3D and for second/fourth–order PDEs are presented and the performance of the proposed method is compared with that of the traditional MPS–MFS.

Finding a good shape parameter of RBF to solve PDEs based on the particle swarm optimization algorithm

The present study aims at integrating the Particle Swarm Optimization (PSO) algorithm with Kansa’s method based on meshless collocation methods in order to determine a good shape parameter of Radial Basis Function (RBF) for solving partial differential equations (PDEs). For this purpose, we use a two-staged experimental design. While in the first stage, PSO algorithm was used to determine an optimal shape parameter for the related RBFs, in the second stage, we employed Kansa’s method to estimate the RMS error for specifying approximate solutions. To study the performance of the proposed algorithm, we offer numerical results for two examples of partial differential equations and show the effectiveness of the proposed method. Numerical results demonstrated the performance superiority of the new algorithm model. The findings also indicated that the evolutionary algorithm model is more effective than the golden section search algorithm in finding a good shape parameter of RBF.

Application of meshless local Petrov Galerkin method (MLPG5) for EIT forward problem

This paper deals with the application of Meshless Local Petrov Galerkin method(MLPG) for solving the forward problem associated with Electrical Impedance Tomography(EIT). Out of six variants of MLPG, here we have used MLPG5 to solve the forward problem. Use of Heaviside function as test function eliminates the domain integral term from the weak formulation. Radial point interpolation method(RPIM) is used and the resulting shape functions possess Kronecker delta function property. Hence no modification is required in the weak form to impose essential boundary conditions. Aforementioned advantages have motivated us to apply this meshless method for EIT forward problem. Numerical accuracy of this method is determined by finding the relative error between numerical solution and exact solution. There are several factors that can affect the accuracy such as local sub domain size(αsub), influence domain size(αinf), nodal density, type and shape of radial basis functions used. A detailed parametric study is carried out in this paper. Radial basis functions(RBF) such as Multiquadrics(MQ), Inverse Multiquadrics(INVMQ) and Gaussian are used for function interpolation along with linear polynomial. Shape parameter of RBFs is also varied to study the effect on accuracy. The solution obtained from the meshless method and analytical method are compared with solutions from Finite Element Method using linear triangular elements. Relative error is found to be minimum when αsub lies in the range of 0.6–0.7 for MQ and INVMQ and αinf > 5 since the boundary voltage profile should be the same when current is injected into the circular medium from any angle under homogeneous conditions. For Gaussian function, minimum error is observed for αsub in the interval 0.5–0.6 and αinf > 5. Gaussian has shown the lowest error compared to MQ and INVMQ. Forward problem in EIT usually deals with nonhomogeneous medium and therefore we need to extend this study to nested domains. Since it is very difficult to get exact solution for nonhomogeneous medium, same conductivity value is given for both background(outer domain) and the inhomogeneity(inner domain) with some additional coupling conditions to be satisfied at the interface. Resulting boundary potentials are matched with the exact solution of homogeneous medium. MQ is used for MLPG5 implementation and error is calculated for different values of αsub and αinf. Minimum error is obtained at αsub = 0.6 for all values of αinf. It is better to select αinf > 5 to ensure the boundary profiles to overlap in all current injection cycles.

Curvilinear Mesh Adaptation Using Radial Basis Function Interpolation and Smoothing

We present a new iterative technique based on radial basis function (RBF) interpolation and smoothing for the generation and smoothing of curvilinear meshes from straight-sided or other curvilinear meshes. Our technique approximates the coordinate deformation maps in both the interior and boundary of the curvilinear output mesh by using only scattered nodes on the boundary of the input mesh as data sites in an interpolation problem. Our technique produces high-quality meshes in the deformed domain even when the deformation maps are singular due to a new iterative algorithm based on modification of the RBF shape parameter. Due to the use of RBF interpolation, our technique is applicable to both 2D and 3D curvilinear mesh generation without significant modification.

A radial basis function method for fractional Darboux problems

In this paper, a radial basis function (RBF) collocation known as Kansa’s method has been extended to solve fractional Darboux problems. The fractional derivatives are described in the Caputo sense. Integration of radial functions that appears due to fractional derivatives have been dealt using Gauss–Jacobi quadrature method. The equation has been linearized using successive approximation. A few test problems have been solved and compared with available solutions. The effect of RBF shape parameter on accuracy and convergence has also been discussed.

A Radial Basis Function based Frames Strategy for Bypassing the Runge Phenomenon

Similarly to polynomials, smooth radial basis function (RBF) interpolants converge exponentially fast to analytic functions on a one dimensional bounded domain but are also vulnerable to the Runge phenomenon [R. Platte, IMA J. Numer. Anal. 31 (2014), pp. 1578--1597]. A common topic in the study of RBFs has been to find and match an optimal node sets and an RBF shape parameter; the location of the nodes is critical to prevent the Runge phenomenon from occur ring, and small variations in the shape parameter can radically change the interpolation accuracy. However, even a finely tuned combination of shape parameter and node distribution leads to numerical unstability in finite precision arithmetic as the number of nodes increases. In [D. Huybrechs, SIAM J. Numer. Anal. 47 (2010), pp. 4326--4355], [B. Adcock, D. Huybrechs, and J. Martin-Vaquero, Found. Comput. Math., 14 (2014), pp. 635--687.], Huybrechs et al. showed that it is always possible, using the Fourier extensions method, to stably approximate an ana...

The localized method of approximated particular solutions for near-singular two- and three-dimensional problems

In this paper, the localized method of approximate particular solutions (LMAPS) using radial basis functions (RBFs) has been simplified and applied to near-singular elliptic problems in two- and three-dimensional spaces. The leave-one-out cross validation (LOOCV) is used in LMAPS to search for a good shape parameter of multiquadric RBF. The main advantage of the method is that a small number of neighboring nodes can be chosen for each influence domain in the discretization to achieve high accuracy. This is especially efficient for three-dimension problems. There is no need to apply adaptivity on node distribution near the region containing spikes of the forcing terms. To examine the performance and limitations of the method, we deliberately push the spike of the forcing term to be extremely large and still obtain excellent results. LMAPS is far superior than the compactly supported RBF (Chen et al. 2003) for such elliptic boundary value problems.

The use of radial basis functions (RBFs) collocation and RBF-QR methods for solving the coupled nonlinear sine-Gordon equations

Radial basis function (RBF) approximation is an extremely powerful tool for solving various types of partial differential equations, since the method is meshless and can be spectrally accurate. A perceived practical obstacle is that the interpolation matrix becomes increasingly ill-conditioned as the RBF shape parameter becomes small, corresponding to flat RBFs. In this paper, the authors solve the one and two-dimensional time-dependent coupled sine-Gordon equations using RBFs collocation and RBF-QR methods and show how one can overcome the ill-conditioning of coefficient matrix for the small shape parameters using RBF-QR method. The main aim of the current paper is to show that the meshless techniques based on the collocation methods are also suitable for solving the system of coupled nonlinear equations especially sine-Gordon equation. Several test problems are employed and results of numerical experiments are presented and also are compared with analytical solutions. The obtained results confirm the acceptable accuracy of the new methods.

Radial basis function method for a multidimensional linear elliptic equation with nonlocal boundary conditions

The development of numerical methods for the solution of partial differential equations (PDEs) with nonlocal boundary conditions is important, since such type of problems arise as mathematical models of various real-world processes. We use radial basis function (RBF) collocation technique for the solution of a multidimensional linear elliptic equation with classical Dirichlet boundary condition and nonlocal integral conditions. RBF-based meshless methods are easily implemented and efficient, especially for multidimensional problems formulated on complexly shaped domains. In this paper, properties of the method are investigated by studying two- and three-dimensional test examples with manufactured solutions. We analyze the influence of the RBF shape parameter and the distribution of the nodes on the accuracy of the method as well as the influence of nonlocal conditions on the conditioning of the collocation matrix.

Comparison study of Gauss, MQ and TPS for interpolation application

There are mainly two parts in this paper. In the first part, we briefly introduce the basic theory of the radial basis function (RBF) and its application in scattered data interpolation, and investigate the difference among three kernel functions: multi-quadric, Gauss and thin plate spline, in detail. According to the multiple examples, we make a lot of experiments, and develop a system in the C++ programme language and the OpenGL open graphics library to make simulation experiments. Finally, we analyse and summarise the different characteristics of the each kernel function in the process of interpolation. In addition, on the basis of the well-known interpolation theory and the analysis of the experimental data, we list several examples for interpolation comparisons. In the second part, the distance of the current marked points down to its most neighbouring marked points is used as shape parameter. Combining with other shape parameter for analysis and comparisons, the results represent the different influences of RBF shape parameter on the curve and surface construction. Our experiment results obtained in this article have theoretical guide values in graphic interpolation modelling.