The shape parameter in the shifted surface spline III

Luh, Lin-Tian

doi:10.1016/j.enganabound.2012.05.004

Cited by 10 publications

(22 citation statements)

References 19 publications

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“…We emphasize that our approach for choosing c optimally is reliable only when the parameter δ is small enough. This phenomenon can also be seen in the experiment of [12]. Note that in Figures 1-5, the value c minimizing the MN curve moves rapidly to 120 and remains there when δ is small.…”

Section: Methodssupporting

confidence: 67%

The mystery of the shape parameter III

Luh

2016

Applied and Computational Harmonic Analysis

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This is a continuation of our earlier study of the shape parameter c contained in the famous multiquadrics (−1) ⌈ β 2 ⌉ (c 2 + x 2 ) β 2 , β > 0, and the inverse multiquadrics (c 2 + x 2 ) β , β < 0. In the previous two papers we presented criteria for the optimal choice of c, based on the exponentialtype error bound. In this paper a new set of criteria is developed, based on the improved exponentialtype error bound. This results in much sharper error estimates when c is chosen appropriately, with the same size of fill distance. What is important is that the optimal value of c can be successfully predicted without any search when fill distance is of reasonable size, making it practically useful. The drawback is that the distribution of the data points is not purely scattered. However it seems to be harmless.

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Section: Methodssupporting

confidence: 67%

The mystery of the shape parameter III

Luh

2016

Applied and Computational Harmonic Analysis

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“…To the author's knowledge, there are three kinds of error bound for shifted-surface-spline interpolation:algebraic type, exponential type, and improved exponential type. Among them the algebraic type shows nothing about the effect of c. The exponential type works well for this purpose, as can be seen in [5] and [6]. However the improved exponential type works better as will be seen in this article.…”

Section: Introductionmentioning

confidence: 75%

“…However, since δ can be theoretically arbitrarily small, c 0 can be very close to zero, theoretically. Now, on the right side of either (5) or (6) , and for f ∈ E σ ,…”

Section: Criteria Of Choosing Cmentioning

confidence: 99%

“…The constant ω is much smaller than the one appears in [5] and [6] where the error bound is O(ω 1 d ) only. Moreover, the constant ω here only mildly depends on the dimension n. It means that for high-dimensional problems the criteria of choosing the shape parameter presented in this paper are much better than those of [5] and [6]. The drawback is that the distribution of data points must be slightly controlled.…”

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confidence: 98%

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Evenly spaced data points and radial basis functions

Luh¹

2011

Boundary Elements and Other Mesh Reduction Methods XXXIII

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The purpose of this article is to introduce a kind of data setting to handle radial basis functions. Traditionally the meshless method RBF uses scattered data setting to do interpolations. This approach faces two hard problems. First, the optimal choice of the shape parameters contained in smooth radial functions are not easy to find. Second, the crucial constant ω in the exponential-type error bound, which is O(ω 1 d ), is too large, making this error bound meaningful only when the fill distance d is extremely small. However, in the evenly spaced data setting, an error bound of the form O(is established where ω is much sharper than that of the former one. What's important is that whenever this error bound is adopted, the optimal choice of the shape parameter can always be found with the fill distance d of reasonable size.We express the effect of the shape parameter c by explicitly defined functions and present concrete criteria of the optimal choice of c, which do not require too many data points.

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“…For the choice of the free parameter c in [10], it was observed that a stable and high accurate solution can be obtained when c 1:8 for one-dimensional problems. It is remarked here that a theoretical justification on the choice of this free parameter c in MQ was recently given by Luh [14]. In this paper, we simply select this parameter as c ¼ 1=N 1 and the number of polynomial terms is Q ¼ 6 for all examples in this section.…”

Section: Numerical Examplesmentioning

confidence: 97%

Finite integration method for solving multi-dimensional partial differential equations

Li¹,

Chen²,

Hon³

et al. 2015

Applied Mathematical Modelling

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a b s t r a c tBased on the recently developed Finite Integration Method (FIM) for solving one-dimensional ordinary and partial differential equations, this paper extends the technique to higher dimensional partial differential equations. The main idea is to extend the first order finite integration matrices constructed by using either Ordinary Linear Approach (OLA) (uniform distribution of nodes) or Radial Basis Function (RBF) interpolation (uniform/random distributions of nodes) to higher order integration matrices. Using standard time integration techniques, such as Laplace transform, we have shown that the FIM is capable for solving time-dependent partial differential equations. Illustrative numerical examples are given in two-dimension to compare the FIM (FIM-OLA and FIM-RBF) with the finite difference method and point collocation method to demonstrate its superior accuracy and efficiency.

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The shape parameter in the shifted surface spline III

Cited by 10 publications

References 19 publications

The mystery of the shape parameter III

The mystery of the shape parameter III

Evenly spaced data points and radial basis functions

Finite integration method for solving multi-dimensional partial differential equations

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