Reconstruction of implicit curves and surfaces via RBF interpolation

Cuomo, Salvatore; Galletti, Ardelio; Giunta, Giulio; Marcellino, Livia

doi:10.1016/j.apnum.2016.10.016

Cited by 62 publications

(35 citation statements)

References 17 publications

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“…This proves the convergence ofū N to u is at least as rapid as that ofû N to u. By numerically computing the integral in (62) using the m N -point Gauss-Legendre rule with M uniform subdivisions similar to the integration scheme (32), we obtain the discrete iterated collocation solution as follows:…”

Section: Iterated Collocation Methodsmentioning

confidence: 80%

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Local multiquadric scheme for solving two‐dimensional weakly singular Hammerstein integral equations

Assari

Asadi‐Mehregan

2018

Int J Numerical Modelling

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The main purpose of this article is to present a numerical method for solving two-dimensional Fredholm-Hammerstein integral equations of the second kind with weakly singular kernels. The scheme utilizes locally supported (inverse) multiquadric functions constructed on scattered points as a basis in the discrete collocation method. The local (inverse) multiquadrics estimate a function in any dimensions via a small set of data instead of all points in the solution domain. The proposed method uses a special accurate quadrature formula based on the nonuniform Gauss-Legendre integration rule for approximating singular integrals appeared in the scheme. In comparison with the globally supported (inverse) multiquadric for the numerical solution of integral equations, the proposed method is stable and uses much less computer memory. Moreover, the algorithm of the presented approach is attractive and easy to implement on computers. Since the scheme does not require any mesh generations on the domain, it can be identified as a meshless method. The error analysis of the method is provided. The convergence accuracy of the new technique is examined over several two-dimensional Hammerstein integral equations and obtained results confirm the theoretical error estimates. KEYWORDS discrete collocation method, Hammerstein integral equation, local (inverse) multiquadric, meshless method, weakly singular kernel Int J Numer Model. 2019;32:e2488.wileyonlinelibrary.com/journal/jnm

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Section: Iterated Collocation Methodsmentioning

confidence: 80%

“…Finally, by numerically computing the integral in (38) similar to the integration scheme (32), we obtain the numerical solution of the integral equation 20 bŷ…”

Section: Solution Of Integral Equationsmentioning

confidence: 99%

Local multiquadric scheme for solving two‐dimensional weakly singular Hammerstein integral equations

Assari

Asadi‐Mehregan

2018

Int J Numerical Modelling

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“…Due to the infinite support of the classical RBFs, any interpolation routine would behave as a field problem, without localization characteristics, returning full matrices for inversion. There are, of course, several numerical techniques to deal with issues of interpolation and approximation with RBFs, 13 but these are usually rather expensive. 14 These problems sparked the proposition of functions with local domain.…”

Section: A Review Of Rbfsmentioning

confidence: 99%

Two new classes of compactly supported radial basis functions for approximation of discrete and continuous data

Menandro

2019

Engineering Reports

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Radial basis functions (RBFs), first proposed for interpolation of scattered data, have gotten the scientific community interest in the past two decades, with applications ranging from interpolation with the dual reciprocity approach of the boundary element method to mesh‐free finite element applications. The use of compactly supported RBFs (CSRBFs) has spread due to their localization properties. However, the mathematical derivation of continuous polynomial functions for different continuity requirements can be quite cumbersome, and although a few classes of functions have already been proposed, there is still room for nonpolynomial trial functions. In this paper, two new classes of CSRBFs are presented for which the continuity requirements are guaranteed. To justify the novelty claim, a view of RBF literature is conducted. The first function class proposed consists of a combination of different inverse polynomial functions, similar to inverse quadric functions, and is thus called inverse class. The second class is called the rational class, in which the functions are obtained as a ratio of two polynomial functions. The proposed functions are used on an approximation software, which takes advantage of their simplicity. The results demonstrate the accuracy and convergence of the proposed functions when compared to some referenced CSRBFs.

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“…Many scholars have done a lot of work on RBF research. Skala [13] used CSRBF to analyze big datasets, Cuomo et al [2,3] studied the reconstruction of implicit curves and surfaces by RBF interpolation. Kedward et al [8] used multiscale RBF interpolation to study mesh deformation.…”

Section: Introductionmentioning

confidence: 99%

Adaptive RBF Interpolation for Estimating Missing Values in Geographical Data

Gao

Mei

Cuomo

et al. 2020

Lecture Notes in Computer Science

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The quality of datasets is a critical issue in big data mining. More interesting things could be mined from datasets with higher quality. The existence of missing values in geographical data would worsen the quality of big datasets. To improve the data quality, the missing values are generally needed to be estimated using various machine learning algorithms or mathematical methods such as approximations and interpolations. In this paper, we propose an adaptive Radial Basis Function (RBF) interpolation algorithm for estimating missing values in geographical data. In the proposed method, the samples with known values are considered as the data points, while the samples with missing values are considered as the interpolated points. For each interpolated point, first, a local set of data points are adaptively determined. Then, the missing value of the interpolated point is imputed via interpolating using the RBF interpolation based on the local set of data points. Moreover, the shape factors of the RBF are also adaptively determined by considering the distribution of the local set of data points. To evaluate the performance of the proposed method, we compare our method with the commonly used k Nearest Neighbors (kNN) interpolation and Adaptive Inverse Distance Weighted (AIDW) methods, and conduct three groups of benchmark experiments. Experimental results indicate that the proposed method outperforms the kNN interpolation and AIDW in terms of accuracy, but worse than the kNN interpolation and AIDW in terms of efficiency.

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Reconstruction of implicit curves and surfaces via RBF interpolation

Cited by 62 publications

References 17 publications

Local multiquadric scheme for solving two‐dimensional weakly singular Hammerstein integral equations

Local multiquadric scheme for solving two‐dimensional weakly singular Hammerstein integral equations

Two new classes of compactly supported radial basis functions for approximation of discrete and continuous data

Adaptive RBF Interpolation for Estimating Missing Values in Geographical Data

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