A trustable shape parameter in the kernel-based collocation method with application to pricing financial options

2021 ◽  
Vol 126 ◽  
pp. 108-117
Author(s):  
Mohammad Shirzadi ◽  
Mehdi Dehghan ◽  
Ali Foroush Bastani
Keyword(s):  
Collocation Method ◽  
Shape Parameter ◽  
Financial Options

A novel local radial basis function collocation method for multi-dimensional piezoelectric problems

2021 ◽  
pp. 1045389X2110572
Author(s):  
Amir Noorizadegan ◽  
Der Liang Young ◽  
Chuin-Shan Chen
Keyword(s):  
Radial Basis Function ◽  
Collocation Method ◽  
Basis Function ◽  
Shape Parameter ◽  
Piezoelectric Medium ◽  
The Finite Element Method ◽  
Mechanical Field ◽  
Radial Basis ◽  
2D And 3D

The local radial basis function collocation method (LRBFCM), a strong-form formulation of the meshless numerical method, is proposed for solving piezoelectric medium problems. The proposed numerical algorithm is based on the local Kansa method using variable shape parameter. We introduce a novel technique for the determination of shape parameter in the LRBFCM, which leads to greater accuracy, and simplicity. The implemented algorithm is first verified with a 2D Poisson equation. Then, we employed LRBFCM in a numerical simulation for 2D and 3D piezoelectric problems involving mutual coupling of the electric field and elastodynamic equations for mechanical field. The presented meshless method is verified using corresponding results obtained from the finite element method and moving least squares meshless local Petrov–Galerkin method. In particular, the 2D piezoelectric problem is verified with an exact solution.

Shape Parameter Optimization of Inverse Multiquadrics Radial Basis Function for Vibration of Laminated Composite Plates

2014 ◽  
Vol 1082 ◽  
pp. 433-436
Author(s):  
Ying Tao Chen ◽  
Song Xiang ◽  
Wei Ping Zhao
Keyword(s):  
Radial Basis Function ◽  
Collocation Method ◽  
Basis Function ◽  
Shape Parameter ◽  
Natural Frequencies ◽  
Laminated Composite ◽  
Composite Plates ◽  
Laminated Composite Plates ◽  
Simply Supported ◽  
Radial Basis

Free vibration of simply laminated composite plates is studied by the global collocation method based on inverse multiquadrics radial basis function. The choice of shape parameter of radial basis function has the important effect on the accuracy of meshless radial basis function collocation method. Genetic algorithm is used to optimize the shape parameter of inverse multiquadrics radial basis function. The natural frequencies of simply supported laminated composite plates are calculated using the inverse multiquadrics radial basis function with optimal shape parameter and compared with the analytical solutions.

scholarly journals Multiquadrics without the Shape Parameter for Solving Partial Differential Equations

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1813 ◽  
Author(s):  
Cheng-Yu Ku ◽  
Chih-Yu Liu ◽  
Jing-En Xiao ◽  
Shih-Meng Hsu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Collocation Method ◽  
Interior Point ◽  
Shape Parameter ◽  
Radial Distance ◽  
Optimum Shape ◽  
Partial Differential ◽  
Collocation Scheme ◽  
Interior Points

In this article, we present multiquadric radial basis functions (RBFs), including multiquadric (MQ) and inverse multiquadric (IMQ) functions, without the shape parameter for solving partial differential equations using the fictitious source collocation scheme. Different from the conventional collocation method that assigns the RBF at each center point coinciding with an interior point, we separated the center points from the interior points, in which the center points were regarded as the fictitious sources collocated outside the domain. The interior, boundary, and source points were therefore collocated within, on, and outside the domain, respectively. Since the radial distance between the interior point and the source point was always greater than zero, the MQ and IMQ RBFs and their derivatives in the governing equation were smooth and globally infinitely differentiable. Accordingly, the shape parameter was no longer required in the MQ and IMQ RBFs. Numerical examples with the domain in symmetry and asymmetry are presented to verify the accuracy and robustness of the proposed method. The results demonstrated that the proposed method using MQ RBFs without the shape parameter acquires more accurate results than the conventional RBF collocation method with the optimum shape parameter. Additionally, it was found that the locations of the fictitious sources were not sensitive to the accuracy.

Generalized Multiquadrics with Optimal Shape Parameter and Exponent for Deflection and Stress of Functionally Graded Plates

2014 ◽  
Vol 709 ◽  
pp. 121-124 ◽  
Author(s):  
Ying Tao Chen ◽  
Song Xiang ◽  
Wei Ping Zhao
Keyword(s):  
Genetic Algorithm ◽  
Radial Basis Function ◽  
Collocation Method ◽  
Basis Function ◽  
Shape Parameter ◽  
Optimal Shape ◽  
Functionally Graded ◽  
Functionally Graded Plates ◽  
Radial Basis ◽  
Meshless Collocation

Deflection and stress of simply functionally graded plates are calculated by the meshless collocation method based on generalized multiquadrics radial basis function. The generalized multiquadric radial basis function has the shape parameter c and exponent which have the important effect in the accuracy of the approximation. The deflection and stress of simply functionally graded plates are calculated using the generalized multiquadrics with optimal shape parameter and exponent which is optimized by the genetic algorithm.

scholarly journals A Collocation Method Using Radial Polynomials for Solving Partial Differential Equations

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1419 ◽  
Author(s):  
Cheng-Yu Ku ◽  
Jing-En Xiao
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Collocation Method ◽  
Shape Parameter ◽  
Taylor Series Expansion ◽  
Three Dimensions ◽  
Partial Differential ◽  
In Series ◽  
Even Order ◽  
Kansa Method

In this article, a collocation method using radial polynomials (RPs) based on the multiquadric (MQ) radial basis function (RBF) for solving partial differential equations (PDEs) is proposed. The new global RPs include only even order radial terms formulated from the binomial series using the Taylor series expansion of the MQ RBF. Similar to the MQ RBF, the RPs is infinitely smooth and differentiable. The proposed RPs may be regarded as the equivalent expression of the MQ RBF in series form in which no any extra shape parameter is required. Accordingly, the challenging task for finding the optimal shape parameter in the Kansa method is avoided. Several numerical implementations, including problems in two and three dimensions, are conducted to demonstrate the accuracy and robustness of the proposed method. The results depict that the method may find solutions with high accuracy, while the radial polynomial terms is greater than 6. Finally, our method may obtain more accurate results than the Kansa method.

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