Multiquadric and its shape parameter—A numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation

Cheng, Alexander H.‐D.

doi:10.1016/j.enganabound.2011.07.008

Cited by 159 publications

(71 citation statements)

References 85 publications

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“…The parameter c can greatly affect the accuracy of the approximation, but the determination of an optimal c is still a research problem (see, for example, Cheng [8]). In a recent paper by Godinho and Tadeu [9], the definition of this parameter is performed by choosing the value that allows minimizing the residual of the PDE throughout the analysis domain.…”

Section: Formulation Of the Kansa's Methods For Interior Problemsmentioning

confidence: 99%

Iterative coupling between the MFS and Kansa’s method for acoustic problems

Godinho¹,

Soares²,

Pereira³

et al. 2013

Boundary Elements and Other Mesh Reduction Methods XXXV

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In the present work, a numerical frequency-domain model based on the joint use of two distinct meshless methods (the Method of Fundamental Solutions and Kansa's Method) is discussed. In this context, the MFS is used to model the homogeneous part of the propagation domain, while the Kansa's Method is applied to model the presence of possible heterogeneities. For Kansa's Method, the MQ RBF is used, and the optimal value of its free parameter is computed by minimizing the residual of the PDE throughout the subdomain. The coupling between the two parts of the propagation domain is performed iteratively, allowing totally independent spatial discretizations to be used for each of the sub-domains of the model. Given this strategy, the use of matching collocation points at common surfaces is not necessary. To improve the behavior of the iterative process, an optimized algorithm, based on the use of a varying relaxation parameter, is used to speed up and/or to ensure the convergence of the iterative coupling. A set of numerical results is here presented to illustrate the behavior of the proposed strategy in terms of convergence.

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Section: Formulation Of the Kansa's Methods For Interior Problemsmentioning

confidence: 99%

Iterative coupling between the MFS and Kansa’s method for acoustic problems

Godinho¹,

Soares²,

Pereira³

et al. 2013

Boundary Elements and Other Mesh Reduction Methods XXXV

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“…It was found [5,6,8] that the shape parameter has a significant influence on accuracy. A larger value of this parameter theoretically should make the solution more accurate but leads to an ill-conditioned system, which may not be accurately solved.…”

Section: Adopting "Leave-one-out" Algorithm For Boundary-value-problemmentioning

confidence: 99%

“…Among them, one can distinguish methods that apply interpolant composed of radial basis functions (RBF) and use the collocation technique in order to discretize a differential problem -the Kansa method and RBF-based pseudospectral method (RBF-PS). There are many papers devoted to these methods [3][4][5][6][7]. The problem of choosing a respective value of the shape parameter for RBF is the issue which appears in almost all of these papers.…”

Section: Introductionmentioning

confidence: 99%

The adaptation of the cross validation aproach for RBF-based collocation methods

Krowiak

Podgórski²

2017

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The adaptation of the cross validation aproach for rbf-based collocation methods Adaptacja podejscia krzyżowego sprawdzania do metod bazujących na funkcjach rbf Abstract The paper shows the adaptation of the cross validation approach, known from interpolation problems, for estimating the value of a shape parameter for radial basis functions. The latter are involved in two collocation techniques used on an unstructured grid to find approximate solution of differential equations. To obtain accurate results, the shape parameter should be chosen as a result of a trade-off between accuracy and conditioning of the system. The cross validation approach called "leave one out" takes these aspects into consideration. The numerical examples that summarize the investigations show the usefulness of the approach.Keywords: radial basis functions, shape parameter, cross-validation approach Streszczenie W artykule pokazano adaptację algorytmu krzyżowego sprawdzania, znanego z zagadnień statystyki i interpolacji, do wyznaczenia wartości współczynnika kształtu w radialnych funkcjach bazowych. Funkcje te użyto w dwóch typach technik kolokacyjnych stosowanych na nieregularnej siatce do przybliżonego rozwiązywania równań różniczkowych. Aby otrzymać rezultaty o odpowiedniej dokładności, współczynnik kształtu powinien być dobrany na bazie kompromisu pomiędzy dokładnością a uwarunkowaniem układu równań. Przedstawiony algorytm, zwany "leave one out", bierze te aspekty pod uwagę. Podsumowaniem artykułu są numeryczne testy, które pokazują użyteczność tego podejścia.Słowa kluczowe: radialne funkcje bazowe, współczynnik kształtu, algorytm cross-validation

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“…An important problem is determining the optimal value for the shape parameter. Several articles have been dedicated to introducing different algorithms to compute a constant value, as an optimal or a good value, for the shape parameter [6][7][8][9][10][11][12][13][14][15]. Kansa and Carlson showed that instead of a fixed shape parameter variable shape parameters are also useful [16].…”

Section: Introductionmentioning

confidence: 99%

Selection of an Interval for Variable Shape Parameter in Approximation by Radial Basis Functions

Biazar

Hosami

2016

Advances in Numerical Analysis

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In radial basis function approximation, the shape parameter can be variable. The values of the variable shape parameter strategies are selected from an interval which is usually determined by trial and error. As yet there is not any algorithm for determining an appropriate interval, although there are some recipes for optimal values. In this paper, a novel algorithm for determining an interval is proposed. Different variable shape parameter strategies are examined. The results show that the determined interval significantly improved the accuracy and is suitable enough to count on in variable shape parameter strategies.

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Multiquadric and its shape parameter—A numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation

Cited by 159 publications

References 85 publications

Iterative coupling between the MFS and Kansa’s method for acoustic problems

Iterative coupling between the MFS and Kansa’s method for acoustic problems

The adaptation of the cross validation aproach for RBF-based collocation methods

Selection of an Interval for Variable Shape Parameter in Approximation by Radial Basis Functions

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