Selection of an Interval for Variable Shape Parameter in Approximation by Radial Basis Functions

Biazar, Jafar; Hosami, Mohammad

doi:10.1155/2016/1397849

Cited by 10 publications

(6 citation statements)

References 20 publications

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“…Step 4. Apply the initial conditions, U (0) , V (0) , to get and from (30), (31), (33), (34), (35), (36), (37), (38), (40), (41), (42), and (43).…”

Section: Computing Setup and Algorithmmentioning

confidence: 99%

“…Step 6. Construct the collocation matricesΑ andΒ in the forms expressed in (30), (31), (33), (34), (35), (36), (37), (38), (40), (41), (42), and (43) using the solutions previously obtained in Step 5.…”

Section: Computing Setup and Algorithmmentioning

confidence: 99%

“…Later in 2003, Lee et al [39] suggested that the final numerical solutions obtained are found to be less affected by the method when the approximation is applied locally rather than globally. A rather recent work is the selection of an interval for variable shape parameter by Biazar and Hosami in 2016 [40], where a novel algorithm for determining and interval was proposed.…”

Section: Low-to Moderate-reynolds Number Casesmentioning

confidence: 99%

See 2 more Smart Citations

Numerical Solution to Coupled Burgers’ Equations by Gaussian-Based Hermite Collocation Scheme

Chuathong

Kaennakham

2018

Journal of Applied Mathematics

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One of the most challenging PDE forms in fluid dynamics namely Burgers equations is solved numerically in this work. Its transient, nonlinear, and coupling structure are carefully treated. The Hermite type of collocation mesh-free method is applied to the spatial terms and the 4th-order Runge Kutta is adopted to discretize the governing equations in time. The method is applied in conjunction with the Gaussian radial basis function. The effect of viscous force at high Reynolds number up to 1,300 is investigated using the method. For the purpose of validation, a conventional global collocation scheme (also known as “Kansa” method) is applied parallelly. Solutions obtained are validated against the exact solution and also with some other numerical works available in literature when possible.

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“…Step 4. Apply the initial conditions, U (0) , V (0) , to get and from (30), (31), (33), (34), (35), (36), (37), (38), (40), (41), (42), and (43).…”

Section: Computing Setup and Algorithmmentioning

confidence: 99%

Section: Computing Setup and Algorithmmentioning

confidence: 99%

Section: Low-to Moderate-reynolds Number Casesmentioning

confidence: 99%

See 1 more Smart Citation

Numerical Solution to Coupled Burgers’ Equations by Gaussian-Based Hermite Collocation Scheme

Chuathong

Kaennakham

2018

Journal of Applied Mathematics

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“…Recently in 2016 Biazar and Hosami [41] present an algorithm that suggested to determine a valid interval in variable shape parameter. Esmaeilbeigi and Hosseini [40] presented a new approach based on the GA to find a good shape parameter in the resolution of partial differential equations by the Kansa method from where the results obtained show that the proposed algorithm based on the Genetic optimization is efficient and provides a reasonable shape parameter with acceptable solution precision.…”

Section: Introductionmentioning

confidence: 99%

RBF collocation path-following approach: optimal choice for shape parameter based on genetic algorithm

Saffah¹,

Hassouna²,

Timesli³

et al. 2021

Math. Model. Comput.

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This paper presents a new method to solve a challenging problem and a topic of current research namely the selection of optimal shape parameters for the Radial Basis Function (RBF) collocation methods in both interpolation and nonlinear Partial Differential Equations (PDEs) problems. To this intent, a compromise must be made to achieve the conflict between accuracy and stability referred to as the trade-off or uncertainty principle. The use of genetic algorithm and path-following continuation allows us on the one hand to avoid the local optimum issue associated with RBF interpolation matrices, which are inherently ill-conditioned and on the other side, to map the original optimization problem of defining a shape parameter into a root-finding problem. Our computational experiments applied on nonlinear problems in structural calculations using our proposed adaptive algorithm based on genetic optimization with automatic selection of the shape parameter can yield more accuracy and a good precision compared to the same state of the art algorithm from literature with a fixed and given shape parameter and Finite Element Method (FEM).

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“…The studies on the shape parameter can be categorized as modifying the shape parameter [13], using variable shape parameter [14][15][16][17], and finding optimal shape parameter [18][19][20][21][22][23][24][25]. Despite the success of identifying the optimal value of the shape parameter, the proposed methods are neither universal nor consistent.…”

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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Selection of an Interval for Variable Shape Parameter in Approximation by Radial Basis Functions

Cited by 10 publications

References 20 publications

Numerical Solution to Coupled Burgers’ Equations by Gaussian-Based Hermite Collocation Scheme

Numerical Solution to Coupled Burgers’ Equations by Gaussian-Based Hermite Collocation Scheme

RBF collocation path-following approach: optimal choice for shape parameter based on genetic algorithm

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

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