Spectral analysis and preconditioning techniques for radial basis function collocation matrices

Cavoretto, Roberto; Rossi, Alessandra De; Donatelli, Marco; Serra‐Capizzano, Stefano

doi:10.1002/nla.774

Cited by 10 publications

(7 citation statements)

References 53 publications

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“…To do this, the use of data structures, like e.g. the kd-trees, or preconditioning techniques (see, e.g., [7]) might be of great utility.…”

Section: Discussionmentioning

confidence: 99%

A unified version of efficient partition of unity algorithms for meshless interpolation

Cavoretto¹

2012

AIP Conference Proceedings

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In this paper we give a unified presentation of efficient algorithms for meshless interpolation on a general domain K. In particular, here we focus on two specific cases: the former considers as interpolation domain the unit square, i.e. K = [0, 1] 2 ⊆ R 2 , whereas the latter is given by the unit two-sphere, namely K = S 2 ⊆ R 3 . Such algorithms are based on a partition of unity method, which is characterized by the use of radial or zonal basis functions as local interpolants. These procedures are implemented by using efficient searching techniques, exploiting suitable and optimal partitions of the domain in strips or spherical zones. Some numerical tests show accuracy and efficiency of the considered algorithms.

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“…To do this, the use of data structures, like e.g. the kd-trees, or preconditioning techniques (see, e.g., [7]) might be of great utility.…”

Section: Discussionmentioning

confidence: 99%

A unified version of efficient partition of unity algorithms for meshless interpolation

Cavoretto¹

2012

AIP Conference Proceedings

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“…Domain decomposition has been successfully to both CS-RBF and GS-RBF methods for PDEs. Domain decomposition can use either the overlapping or non-overlapping methods [13]- [19]. Assume the problem is continuous, and the computational domain is arbitrarily decomposed as a union of smaller subdomains containing an equal number of data centers (approximately).…”

Section: Treating Ill-conditingmentioning

confidence: 99%

Globally Versus Compactly Supported RBFS

Kansa

2018

Advances in Fluid Mechanics XII

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For many years, a debate has occurred whether radial basis functions having compact support (CS) or global support (GS) is best for engineering and scientific applications. CS RBFs converge as O(h (k+1)), h is the fill distance, and its systems of equations have many zeros. In contrast, GS RBFs converge as O((c/h)),  <1, c is the GS-RBF shape parameter. Previously, the barrier to exploiting the exponential convergence rate of GS-RBFs has been the ill-conditioning problem that is due to computer chip restrictions on the relatively large machine epsilon. Although computer chips with arbitrary precision are very rare presently, extended precision software has allowed the exploitation of the exponential convergence rates of GS-RBFs. When attempting modeling of higher dimension practical problems, previous methods such as domain decomposition, global optimization, pre-conditioning will need to be blended even on massively parallel computers.

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“…So, dealing with the matrix solvers could improve the condition of the RBF matrix. There are several studies that investigated and evaluated the preconditioning of the RBF matrix [3][4][5][6][7][8][9][10][11][12]. Kansa et al [11,12] used extended precision for the calculation in order to remove the computational singularity and enhance the condition of the interpolation matrix.…”

Section: Introductionmentioning

confidence: 99%

New Approach for Radial Basis Function Based on Partition of Unity of Taylor Series Expansion with Respect to Shape Parameter

2020

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Radial basis function (RBF) is gaining popularity in function interpolation as well as in solving partial differential equations thanks to its accuracy and simplicity. Besides, RBF methods have almost a spectral accuracy. Furthermore, the implementation of RBF-based methods is easy and does not depend on the location of the points and dimensionality of the problems. However, the stability and accuracy of RBF methods depend significantly on the shape parameter, which is primarily impacted by the basis function and the node distribution. At a small value of shape parameter, the RBF becomes more accurate, but unstable. Several approaches were followed in the open literature to overcome the instability issue. One of the approaches is optimizing the solver in order to improve the stability of ill-conditioned matrices. Another approach is based on searching for the optimal value of the shape parameter. Alternatively, modified bases are used to overcome instability. In the open literature, radial basis function using QR factorization (RBF-QR), stabilized expansion of Gaussian radial basis function (RBF-GA), rational radial basis function (RBF-RA), and Hermite-based RBFs are among the approaches used to change the basis. In this paper, the Taylor series is used to expand the RBF with respect to the shape parameter. Our analyses showed that the Taylor series alone is not sufficient to resolve the stability issue, especially away from the reference point of the expansion. Consequently, a new approach is proposed based on the partition of unity (PU) of RBF with respect to the shape parameter. The proposed approach is benchmarked. The method ensures that RBF has a weak dependency on the shape parameter, thereby providing a consistent accuracy for interpolation and derivative approximation. Several benchmarks are performed to assess the accuracy of the proposed approach. The novelty of the present approach is in providing a means to achieve a reasonable accuracy for RBF interpolation without the need to pinpoint a specific value for the shape parameter, which is the case for the original RBF interpolation.

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Spectral analysis and preconditioning techniques for radial basis function collocation matrices

Cited by 10 publications

References 53 publications

A unified version of efficient partition of unity algorithms for meshless interpolation

A unified version of efficient partition of unity algorithms for meshless interpolation

Globally Versus Compactly Supported RBFS

New Approach for Radial Basis Function Based on Partition of Unity of Taylor Series Expansion with Respect to Shape Parameter

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