A Joint Optimization Algorithm Based on the Optimal Shape Parameter–Gaussian Radial Basis Function Surrogate Model and Its Application

Sun, Jian; Gong, Dianxuan

doi:10.3390/math11143169

Cited by 7 publications

(2 citation statements)

References 32 publications

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“…Many other applications and tests of finite difference derivative approximations are provided in the literature [21,[32][33][34][35][36][37][38][39][40][41].…”

Section: Comparison Of the Real Solution With The Complex One By Fini...mentioning

confidence: 99%

Numerical Resolution of Differential Equations Using the Finite Difference Method in the Real and Complex Domain

Almeida Magalhães,

Brito,

Lamon

et al. 2024

Mathematics

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The paper expands the finite difference method to the complex plane, and thus obtains an improvement in the resolution of differential equations with an increase in numerical precision and a generalization in the mathematical modeling of problems. The article begins with a selection of the best techniques for obtaining finite difference coefficients for approximating derivatives in the real domain. Then, the calculation is expanded to the complex domain. The research expands forward, backward, and central difference approximations of the real case by a quadrant approximation in the complex plane, which facilitates the use in boundary conditions of differential equations. The article shows many real and complex finite difference equations with their respective order of error, intended to serve as a basis and reference, which have been tested in practical examples of solving differential equations used in engineering. Finally, a comparison is made between the real and complex techniques of finite difference methods applied in the Theory of Elasticity. As a surprising result, the article shows that the finite difference method has great advantages in numerical precision, diversity of formulas, and modeling generalities in the complex domain when compared to the real domain.

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“…Many other applications and tests of finite difference derivative approximations are provided in the literature [21,[32][33][34][35][36][37][38][39][40][41].…”

Section: Comparison Of the Real Solution With The Complex One By Fini...mentioning

confidence: 99%

Numerical Resolution of Differential Equations Using the Finite Difference Method in the Real and Complex Domain

Almeida Magalhães,

Brito,

Lamon

et al. 2024

Mathematics

View full text Add to dashboard Cite

show abstract

“…While in (Skala et al, 2020;Yaghouti & Ramezannezhad Azarboni, 2017), estimation of shape parameters are used on radial basis formulation method. An optimal algorithm for choosing a good shape parameter in image processing is developed and application of optimal shape parameter on gaussian radial basis function is proposed in (Sun et al, 2023).…”

Section: Introductionmentioning

confidence: 99%

An Optimal Multiquadric Variable Shape Parameter for Boundary Value Problems Using Particle Swarm Optimization

Bacar,

Rawhoudine

2024

JMR

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The multiquadric radial basis function method has been widely used to solve partial differential equations-based problems regarding its flexibility and meshfree characteristics. The accuracy and stability of this method are derived and based on the use of a free-shape parameter that sensibly controls the comportment of the technique. Significant improvements have already been reported and show that variable shape parameters conduct the method to handle problems with striking results compared to global-based techniques. Nevertheless, choosing a suitable set of shape parameters is still an open topic because of the complexity of the method when the number of collocation points increases. The current work proposes a variant particle swarm optimization based on local displacement with attractors to determine the multi-quadratic function's ``best'' optimal variable shape parameter in solving boundary value problems. Based on an initially random set of variable shape parameters, the proposed algorithm first performs and evaluates the errors between the expected exact solution and the approximate solution thoroughly. In the first stage, the particle swarm algorithm search for an optimal set of shape parameters that minimize the error and the conditioning number of the radial basis system matrix. In the second stage, the obtained optimal set of shape parameters is applied to solve the considered problem. In this way, when the number of collocation points increases, the first stage based on particle swarm optimization stabilizes the strategy. It proposes an ``acceptable'' set of shape parameters for the given problem. The proposed method is applied to a set of well-known boundary value problems in one and two-dimensional spaces and compared to other techniques published in the literature. The results show that the proposed method achieves more accurate solutions than recently proposed in the literature.

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Adaptive selection of shape parameters for MQRBF in arbitrary scattered data: enhancing finite difference solutions for complex PDEs

Sun,

Wang

2024

Comp. Appl. Math.

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A Joint Optimization Algorithm Based on the Optimal Shape Parameter–Gaussian Radial Basis Function Surrogate Model and Its Application

Cited by 7 publications

References 32 publications

Numerical Resolution of Differential Equations Using the Finite Difference Method in the Real and Complex Domain

Numerical Resolution of Differential Equations Using the Finite Difference Method in the Real and Complex Domain

An Optimal Multiquadric Variable Shape Parameter for Boundary Value Problems Using Particle Swarm Optimization

Adaptive selection of shape parameters for MQRBF in arbitrary scattered data: enhancing finite difference solutions for complex PDEs

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