Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations

Kansa, Edward J.; Hon, Y.C.

doi:10.1016/s0898-1221(00)00071-7

Cited by 377 publications

(201 citation statements)

References 21 publications

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“…[20][21][22][23][24][25]). However, there is still a lack of mathematical theories for specifying optimal values of the free parameters of the RBF.…”

Section: Introductionmentioning

confidence: 99%

A Cartesian grid technique based on one‐dimensional integrated radial basis function networks for natural convection in concentric annuli

Mai‐Duy

Le-Cao

Tran-Cong

2007

Numerical Methods in Fluids

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This paper reports a radial-basis-function (RBF)-based Cartesian grid technique for the simulation of two-dimensional buoyancy-driven flow in concentric annuli. The continuity and momentum equations are represented in the equivalent stream function formulation that reduces the number of equations from three to one, but involves higher-order derivatives. The present technique uses a Cartesian grid to discretize the problem domain. Along a grid line, one-dimensional integrated RBF networks (1D-IRBFNs) are employed to represent the field variables. The capability of 1D-IRBFNs to handle unstructured points with accuracy is exploited to describe non-rectangular boundaries in a Cartesian grid, while the method's ability to avoid the reduction of convergence rate caused by differentiation is instrumental in improving the quality of the approximation of high-order derivatives. The method is applied to simulate thermally-driven flows in annuli between two circular cylinders and between an outer square cylinder and an inner circular cylinder. High Rayleighnumber solutions are achieved and they are in good agreement with previously published numerical data.

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“…[20][21][22][23][24][25]). However, there is still a lack of mathematical theories for specifying optimal values of the free parameters of the RBF.…”

Section: Introductionmentioning

confidence: 99%

A Cartesian grid technique based on one‐dimensional integrated radial basis function networks for natural convection in concentric annuli

Mai‐Duy

Le-Cao

Tran-Cong

2007

Numerical Methods in Fluids

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“…However, despite the success of RBF methods in many scientific and engi- 25 neering applications, their accuracy is dependent on a user defined parameter, namely the RBF width or the shape parameter. In this work, it is denoted by β.…”

Section: Introductionmentioning

confidence: 99%

“…Fornberg and Wright [11] proposed the Contour-Padé algorithm which can stably compute the whole region of the shape parameter on the complex plane. Many different approaches to enhance the stability of DRBF methods have been proposed, for example [23,25,26,27,28,29,30,31,32] and their references therein. For IRBF approaches, Sarra [16] studied the case of global flat IRBFs.…”

Section: Introductionmentioning

confidence: 99%

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A numerical study of compact approximations based on flat integrated radial basis functions for second-order differential equations

Tien

Mai‐Duy

Tran

et al. 2016

Computers & Mathematics with Applications

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In this paper, we propose a simple but effective preconditioning technique to improve the numerical stability of Integrated Radial Basis Function (IRBF) methods. The proposed preconditioner is simply the inverse of a well-conditioned matrix that is constructed using non-flat IRBFs. Much larger values of the free shape parameter of IRBFs can thus be employed and better accuracy for smooth solution problems can be achieved. Furthermore, to improve the accuracy of local IRBF methods, we propose a new stencil, namely Combined Compact IRBF (CCIRBF), in which (i) the starting point is the fourth-order derivative; and (ii) nodal values of first-and second-order derivatives at side nodes of the stencil are included in the computation of first-and second-order derivatives at the middle node in a natural way. The proposed stencil can be employed in uniform/nonuniform Cartesian grids. The preconditioning technique in combination with the CCIRBF scheme employed with large values of the shape parameter are tested with elliptic equations and then applied to simulate several fluid flow problems governed by Poisson, Burgers, convectiondiffusion, and Navier-Stokes equations. Highly accurate and stable solutions are obtained. In some cases, the preconditioned schemes are shown to be several orders of magnitude more accurate than those without preconditioning.

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“…Since then, it has received a great deal of attention from the research community. A great number of publications are available in the literature, e.g., for a convergence proof and error bound [13], the numerical solution of potential problems [14][15][16][17][18], high-order differential equations [19,20], Kirchhoff plate bending problems [21] and viscous-fluid-flow problems [22,23]. In a standard RBFN-based method, all dependent variables are first approximated by global RBFNs, and the governing equations are then discretized in the strong form by point collocation.…”

Section: Introductionmentioning

confidence: 99%

Computing non-Newtonian fluid flow with radial basis function networks

Mai‐Duy

Tanner

2005

Int. J. Numer. Meth. Fluids

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Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations

Cited by 377 publications

References 21 publications

A Cartesian grid technique based on one‐dimensional integrated radial basis function networks for natural convection in concentric annuli

A Cartesian grid technique based on one‐dimensional integrated radial basis function networks for natural convection in concentric annuli

A numerical study of compact approximations based on flat integrated radial basis functions for second-order differential equations

Computing non-Newtonian fluid flow with radial basis function networks

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