Shape parameter estimation in RBF function approximation

Karageorghis, Andréas; Tryfonos, P.

doi:10.2495/cmem-v7-n3-246-259

Cited by 5 publications

(6 citation statements)

References 21 publications

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“…However, the selection of the shape parameter α seems to be critical, and optimal choice is still an open question [8][9][10]. Additionally, RBFs find applications in solving partial differential equations (PDEs) and surface reconstruction of acquired data [21,22,[35][36][37][38][39].…”

Section: Discussionmentioning

confidence: 99%

“…The final interpolated value is determined by a linear interpolation covering the neighbor cells [7]. It should be noted that the radial basis functions have the shape parameter α, which influences the precision of the final interpolation and the choice of the value might be critical [8][9][10].…”

Section: Id Rbf Function Id Rbf Functionmentioning

confidence: 99%

“…It should be noted that the α shape parameter has to be set reasonably. Note, that there are several suboptimal shape parameters [8,9,34].…”

Section: Rbf Interpolationmentioning

confidence: 99%

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Meshfree Interpolation of Multidimensional Time-Varying Scattered Data

Skala,

Mourycova

2023

Computers

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Interpolating and approximating scattered scalar and vector data is fundamental in resolving numerous engineering challenges. These methodologies predominantly rely on establishing a triangulated structure within the data domain, typically constrained to the dimensions of 2D or 3D. Subsequently, an interpolation or approximation technique is employed to yield a smooth and coherent outcome. This contribution introduces a meshless methodology founded upon radial basis functions (RBFs). This approach exhibits a nearly dimensionless character, facilitating the interpolation of data evolving over time. Specifically, it enables the interpolation of dispersed spatio-temporally varying data, allowing for interpolation within the space-time domain devoid of the conventional “time-frames”. Meshless methodologies tailored for scattered spatio-temporal data hold applicability across a spectrum of domains, encompassing the interpolation, approximation, and assessment of data originating from various sources, such as buoys, sensor networks, tsunami monitoring instruments, chemical and radiation detectors, vessel and submarine detection systems, weather forecasting models, as well as the compression and visualization of 3D vector fields, among others.

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Section: Discussionmentioning

confidence: 99%

Section: Id Rbf Function Id Rbf Functionmentioning

confidence: 99%

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Meshfree Interpolation of Multidimensional Time-Varying Scattered Data

Skala,

Mourycova

2023

Computers

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“…Consider a given data set, X ¼ x j È É N j¼1 ; containing N distinct interpolation points, suppose we have an unknown function u, collated at the same given points in the domain Ω. Any radial kernel interpolant ũ, containing a shape parameter ε, can be written as (Karageorghis & Tryfonos, 2019)…”

Section: The Shape Parametermentioning

confidence: 99%

A Study of the Effects of the Shape Parameter and Type of Data Points Locations on the Accuracy of the Hermite-Based Symmetric Approach Using Positive Definite Radial Kernels

Nengem,

Haruna

2023

JDSIS

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This article considers two globally supported and positive radial kernel and three different patterns of data point locations on the same computational domain. The research aimed to study the effects of the shape parameter and type of data points locations on the accurate performance of an Hermite-Based Symmetric Approach. A two-dimensional Helmholtz equation and the two-dimensional Poisson equation were used as a test functions. The problems were first solved on the three different types of data point locations using linear Laguerre-Gaussians and then linear Matern. In each case the graph of the error against the shape parameter was drawn to enables easy identification of the optimal value of the shape parameter.

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“…The sensitivity of the shape parameter was also investigated and optimal values were empirically formulated 37 . The idea of including the shape parameter in the vector of the RBF unknowns as a varying value with each center on the grid was explored 38 . The major problem with finding an optimal shape parameter is that it adds excessive calculations to the approximation and solution procedure.…”

Section: Introductionmentioning

confidence: 99%

A shape parameter insensitive CRBFs‐collocation for solving nonlinear optimal control problems

Seleit,

Elgohary

2023

Optim Control Appl Methods

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In this work, we introduce a Coupled Radial Basis Functions collocation (CRBFs‐Coll) approach for solving optimal control problems. CRBFs are real‐valued Radial Basis Functions (RBFs) augmented with a conical spline. They show insensitivity to the shape parameter resulting in robust behavior in function approximation. The method is applied to classical nonlinear optimal control problems: Zermelo's problem, a Duffing oscillator with various boundary conditions, and a nonlinear pendulum on a cart problem. The nonlinear optimal control problem is solved by deriving the necessary conditions for optimality followed by collocation using CRBFs as basis functions for approximating the two‐point boundary value problem (TPBVP). The resulting system of nonlinear algebraic equations (NAEs) is then solved using a standard solver. Unlike existing methods that rely on nodal distributions that are denser at the boundaries to obtain a solution, the present CRBFs‐Coll approach is capable of solving the optimal control problem on uniform nodes without the need for interpolation. The present CRBFs‐Coll approach is shown to be simple to implement and provides accurate results over an equidistant nodal distribution while maintaining a continuous representation of the control and states.

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Shape parameter estimation in RBF function approximation

Cited by 5 publications

References 21 publications

Meshfree Interpolation of Multidimensional Time-Varying Scattered Data

Meshfree Interpolation of Multidimensional Time-Varying Scattered Data

A Study of the Effects of the Shape Parameter and Type of Data Points Locations on the Accuracy of the Hermite-Based Symmetric Approach Using Positive Definite Radial Kernels

A shape parameter insensitive CRBFs‐collocation for solving nonlinear optimal control problems

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