The mystery of the shape parameter III

Luh, Lin-Tian

doi:10.1016/j.acha.2015.05.001

Cited by 21 publications

(21 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For both local and global RBF matrices, the condition number grows with β increasing number of elements and the shape parameter; however, the condition number increases at a faster rate with global systems. Luh [21][22][23] developed a shape parameter theory for CD-RBFs that is implemented with the extended precision feature of MATHEMATICA with extremely accurate results having max and RMS errors, O(10 -147 ), against problems with analytic solutions.…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Fully and sparsely supported radial basis functions

Kansa¹,

Holoborodko²

2020

Int. J. CMEM

View full text Add to dashboard Cite

The central idea of this paper is that computer mathematics is not identical to ideal mathematics because computer numbers only have finite precision. All functions, especially the positive definite transcendental functions, are truncated and the expansion coefficients have finite precision and all branching operations require time to complete. Of all the known methods used to obtain numerical solutions to integral and partial differential equations, the global continuously differential radial basis functions (RBFs) that are implemented on computers closely resemble many aspects of ideal mathematics. The global RBFs have the attributes required to obtain very accurate numerical results for a variety of partial differential and integral equations with smooth solutions. Without the need for extremely fine discretization, the global RBFs have their spline properties and exponential convergence rates. The resulting system of full equations can be executed very rapidly on graphical processing units and fieldprogrammable gate arrays because, with full systems, there is no branching and full systems solvers are very highly vectorized, optimizing the usage of very fast processors.

show abstract

Section: Discussionmentioning

confidence: 99%

“…Luh [21][22][23] made a significant achievement by developing a theory to find the optimal CD-RBF shape parameters by greatly expanding the original work of Madych and Nelson [24]. Luh considered factors such as the type CD-RBF spatial dimension and fill distance to construct the MN (Madych-Nelson) curve.…”

Section: Luh's Rbf Shape Parameter Theorymentioning

confidence: 99%

Fully and sparsely supported radial basis functions

Kansa¹,

Holoborodko²

2020

Int. J. CMEM

View full text Add to dashboard Cite

show abstract

“…The optimal choice of c is then the number minimizing M N (c). However, unlike [9], the range of c is the entire interval (0, ∞), rather than a proper subset of (0, ∞). We now begin our criteria.…”

Section: Criteria Of Choosing Cmentioning

confidence: 99%

“…

This is the fourth paper of our study of the shape parameter c contained in the famous multiquadrics (−1) ⌈β⌉ (c 2 + x 2 ) β , β > 0, and the inverse multiquadrics (c 2 + x 2 ) β , β < 0. The theoretical ground is the same as that of [10]. However we extend the space of interpolated functions to a more general one.

…”

mentioning

confidence: 99%

The mystery of the shape parameter IV

Luh¹

2014

Engineering Analysis with Boundary Elements

Self Cite

View full text Add to dashboard Cite

show abstract

“…From Madych's theoretical error estimation analysis, the suggested optimal c value is c opt~O (−lnλ/2ah) which indicates the c opt grows at the rate of h −1 (Madych, 1992). And also, Luh (2010) derived c opt ≈ (−1−β+n+4l )/2σ. However, these values are not the optimal ones because the matrix condition number increases exponentially as c increasing, while the optimal c typically cannot be reached with double precision computation.…”

Section: Introductionmentioning

confidence: 95%

Shape parameter selection for multi-quadrics function method in solving electromagnetic boundary value problems

Zhang

Guo

et al. 2016

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering

View full text Add to dashboard Cite

Purpose – The multi-quadrics (MQ) function is a kind of radial basis function. And the MQ method has been successfully adopted as a type of meshless method in solving electromagnetic boundary value problems. However, the accuracy of MQ interpolation or solving equations is severely influenced by shape parameter. Thus the purpose of this paper is to propose a case-independent shape parameter selection strategy from the aspect of coefficient matrix condition number analysis. Design/methodology/approach – The condition number of coefficient matrix is investigated. It is shown that the condition number is only a function of shape parameter and MQ node number, and is irrelevant to the interpolated function which means case-independent. The effective condition number which takes into account the interpolated function is introduced. Then, the relation between the relative root mean square error and condition number is analyzed. Three numerical experiments as transmission line, cable channel and grounding metal box model were carried out. Findings – In the numerical experiments, there is an approximate linear relationship between the logarithm of the condition number and shape parameter, an approximate quadratic relationship with node number. And the optimal shape parameter is corresponding to the early stage of condition number oscillation. Originality/value – This paper proposed a case-independent shape parameter selection strategy. For a finite precision computation, the upper limit of the condition number is predetermined. Therefore, the shape parameter can be chosen where condition number oscillates in early stage.

show abstract

The mystery of the shape parameter III

Cited by 21 publications

References 15 publications

Fully and sparsely supported radial basis functions

Fully and sparsely supported radial basis functions

The mystery of the shape parameter IV

Shape parameter selection for multi-quadrics function method in solving electromagnetic boundary value problems

Contact Info

Product

Resources

About