Radial Function Methods of Approximation Based on Using Harmonic Green's Functions (original) (raw)
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International Journal of Engineering Applied Sciences and Technology
Some radial basis functions with positive nonhalf-integer (PNH-RBFs) exponents have been used to develop numerical methods that are claimed to produce better numerical approximations when compared to methods constructed with radial basis functions having negative-integer/half-integer exponents referred to as standard RBFs. In this paper, we develop a numerical method for approximating the solution of steady state partial differential equations (PDEs) and a radial basis function method of lines (RBF-MOLs) for solving timedependent PDEs in two space dimensions using two PNH-RBFs. Two Poisson equations and a heat equation in two space dimensions were used as test problems to perform numerical experiments and compared with results from methods developed with the standard RBFs. From our results, all the radial basis function methods produced nearly the same accuracy regardless of the value of the exponent.
Radial Basis Function Methods for Solving Partial Differential Equations-A Review
Indian Journal of Science and Technology, 2016
Background/Objectives: The approximation using radial basis function (RBF) is an extremely powerful method to solve partial differential equations (PDEs). This paper presents different types of RBF methods to solve PDEs. Methods/ Statistical Analysis: Due to their meshfree nature, ease of implementation and independence of dimension, RBF methods are popular to solve PDEs. In this paper we examine different generalized RBF methods, including Kansa method, Hermite symmetric approach, localized and hybrid methods. We also discussed the preference of using meshfree methods like RBF over the mesh based methods. Findings: This paper presents a state-of-the-art review of the RBF methods. Some recent development of RBF approximation in solving PDEs is also discussed. The mathematical formulation of different RBF methods are discussed for better understanding. RBF methods have been actively developed over the years from global to local approximation and then to hybrid methods. Hybrid RBF methods help in reduction of computational cost and become very effective in solving large scale problems. Application/Improvements: RBF methods have been applied to various diverse fields like image processing, geo-modeling, pricing option and neural network etc.
A new class of oscillatory radial basis functions
Computers & Mathematics with …, 2006
Radial basis functions (RBFs) form a primary tool for multivariate interpolation, and they are also receiving increased attention for solving PDEs on irregular domains. Traditionally, only nonoscillatory radial functions have been considered. We find here that a certain class of oscillatory radial functions (including Gaussians as a special case) leads to nonsingular interpolants with intriguing features especially as they are scaled to become increasingly flat. This flat limit is important in that it generalizes traditional spectral methods to completely general node layouts. Interpolants based on the new radial functions appear immune to many or possibly all cases of divergence that in this limit can arise with other standard types of radial functions (such as multiquadrics and inverse multiquadratics).
Radial Basis Function Approximations: Comparison and Applications
Approximation of scattered data is often a task in many engineering problems. The Radial Basis Function (RBF) approximation is appropriate for large scattered (unordered) datasets in d-dimensional space. This approach is useful for a higher dimension d > 2, because the other methods require the conversion of a scattered dataset to an ordered dataset (i.e. a semi-regular mesh is obtained by using some tessellation techniques), which is computationally expensive. The RBF approximation is non-separable, as it is based on the distance between two points. This method leads to a solution of Linear System of Equations (LSE) Ac = h. In this paper several RBF approximation methods are briefly introduced and a comparison of those is made with respect to the stability and accuracy of computation. The proposed RBF approximation offers lower memory requirements and better quality of approximation.
Approximation of Green's functions
Zeitschrift f�r Physik, 1970
Special symmetries of the Green's functions of a non-relativistic many ferrnionsystem and conservation laws, expressible by hermitian generators, are formulated as relations for a Green's function operator. Approximations for the Green's functions, defined as partial summations of the perturbation expansion, and consistent with the symmetries and conservation laws are presented.
A New Radial Basis Function Approach Based on Hermite Expansion with Respect to the Shape Parameter
Mathematics, 2019
Owing to its high accuracy, the radial basis function (RBF) is gaining popularity in function interpolation and for solving partial differential equations (PDEs). The implementation of RBF methods is independent of the locations of the points and the dimensionality of the problems. However, the stability and accuracy of RBF methods depend significantly on the shape parameter, which is mainly affected by the basis function and the node distribution. If the shape parameter has a small value, then the RBF becomes accurate but unstable. Several approaches have been proposed in the literature to overcome the instability issue. Changing or expanding the radial basis function is one of the most commonly used approaches because it addresses the stability problem directly. However, the main issue with most of those approaches is that they require the optimization of additional parameters, such as the truncation order of the expansion, to obtain the desired accuracy. In this work, the Hermite...
Recovery of high order accuracy in radial basis function approximations of discontinuous problems
2010
Abstract Radial basis function (RBF) methods have been actively developed in the last decades. RBF methods are global methods which do not require the use of specialized points and that yield high order accuracy if the function is smooth enough. Like other global approximations, the accuracy of RBF approximations of discontinuous problems deteriorates due to the Gibbs phenomenon, even as more points are added.
A NEW RADIAL BASIS FUNCTION APPROXIMATION WITH REPRODUCTION
Approximation of scattered geometric data is often a task in many engineering problems. The Radial Basis Function (RBF) approximation is appropriate for large scattered (unordered) datasets in-dimensional space. This method is useful for a higher dimension ≥ 2, because the other methods require a conversion of a scattered dataset to a semi-regular mesh using some tessellation techniques, which is computationally expensive. The RBF approximation is non-separable, as it is based on a distance of two points. It leads to a solution of overdetermined Linear System of Equations (LSE). In this paper a new RBF approximation method is derived and presented. The presented approach is applicable for-dimensional cases in general.