Adaptive Gaussian radial basis function methods for initial value problems: Construction and comparison with adaptive multiquadric radial basis function methods

Gu, Jiaxi; Jung, Jae-Hun

doi:10.1016/j.cam.2020.113036

Cited by 12 publications

(12 citation statements)

References 10 publications

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“…The conjecture stated in [3] that there exists at least one shape parameter for the consistency to any order of accuracy and it is main reason why RBF approximations yields spectral accuracy. We do not provide much details here, interested readers can look at [3,4]. We have derived modified Euler method (3.4), however, these have some technical difficulties.…”

Section: Consistency Of Adaptive Imq-rbf Euler Methodsmentioning

confidence: 99%

“…In [3,4], the adaptive RBF solvers have been developed for solving ODEs such as Euler's method, the midpoint method, and the Adams methods by replacing the polynomial basis with multi-quadratic and Gaussian RBFs respectively. The critical concept of adaptive RBF solvers is that the radial basis functions are defined with a free (shape) parameter which can be adjusted according to the local conditions of the solution.…”

Section: Introductionmentioning

confidence: 99%

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Construction and comparative study of Euler method with adaptive IQ and IMQ-RBFs

Rathan¹,

Deepit²

2021

Preprint

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The fundamental purpose of the present work is to constitute an enhanced Euler method with adaptive inverse-quadratic and inverse-multi-quadratic radial basis function (RBF) interpolation technique to solve initial value problems. These enhanced methods improve the local convergence of numerical solutions by utilizing a free parameter of radial basis functions. Consistency, convergence and stability analysis are provided to support our claims for each method. Numerical results show that the accuracy and rate of convergence of each proposed method are the same as the original Euler method or improved by making the local truncation error vanish; thus, the adaptive methods are optimal.

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Section: Consistency Of Adaptive Imq-rbf Euler Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Construction and comparative study of Euler method with adaptive IQ and IMQ-RBFs

Rathan¹,

Deepit²

2021

Preprint

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“…, g n b ; n i denotes the number of interior points; and n b denotes the number of boundary points. The undetermined coefficients can be obtained by solving a system of simultaneous linear equations in matrix form as shown in (13). Accordingly, the solution of u(x) can be obtained using the function approximation of (8).…”

Section: The Fictitious Sourcesmentioning

confidence: 99%

“…Function approximation using the interpolation method of the radial basis function (RBF) was proposed in 1971 by Hardy [11], who used the multiquadric (MQ) RBF for scattered data interpolation to solve problems. In addition to the MQ RBF, several RBFs may be found, such as the inverse multiquadric (IMQ), Gaussian, and polyharmonic spline (PS) functions [12][13][14]. Among them, the PS and the MQ RBFs may provide much more accurate solutions than other RBFs [15,16].…”

Section: Introductionmentioning

confidence: 99%

Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs

Liu

Hong

et al. 2021

Mathematics

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In this article, a novel infinitely smooth polyharmonic radial basis function (PRBF) collocation method for solving elliptic partial differential equations (PDEs) is presented. The PRBF with natural logarithm is a piecewise smooth function in the conventional radial basis function collocation method for solving governing equations. We converted the piecewise smooth PRBF into an infinitely smooth PRBF using source points collocated outside the domain to ensure that the radial distance was always greater than zero to avoid the singularity of the conventional PRBF. Accordingly, the PRBF and its derivatives in the governing PDEs were always continuous. The seismic wave propagation problem, groundwater flow problem, unsaturated flow problem, and groundwater contamination problem were investigated to reveal the robustness of the proposed PRBF. Comparisons of the conventional PRBF with the proposed method were carried out as well. The results illustrate that the proposed approach could provide more accurate solutions for solving PDEs than the conventional PRBF, even with the optimal order. Furthermore, we also demonstrated that techniques designed to deal with the singularity in the original piecewise smooth PRBF are no longer required.

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“…e main conclusion of recent studies regarding the increase of the form factor in the multiquadric functions is that, as the approximation function generates flatter and softer curves, it becomes also more insensitive to the Euclidean distance between the center point and any known sampling point, and the elements of the interpolation matrix become almost equal, which makes the matrix ill-conditioned or even singular in the limit situation. Other recent studies have proposed automatic curve fitting methodologies based on RBF [22] and adaptive methods for eigenvalue problems [23], initial value problems [24], and noise reduction problems [25], among other applications.…”

Section: Introductionmentioning

confidence: 99%

Use of Multiquadric Functions for Multivariable Representation of the Aerodynamic Coefficients of Airfoils

Ribeiro

Albuquerque

Gamboa

et al. 2021

Mathematical Problems in Engineering

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Given an array (or matrix) of values for a function of one or more variables, it is often desired to find a value between two given points. Multivariable interpolation and approximation by radial basis functions are important subjects in approximation theory that have many applications in Science and Engineering fields. During the last decades, radial basis functions (RBFs) have found increasingly widespread use for functional approximation of scattered data. This research work aims at benchmarking two different approaches: an approximation by radial basis functions and a piecewise linear multivariable interpolation in terms of their effectiveness and efficiency in order to conclude about the advantages and disadvantages of each approach in approximating the aerodynamic coefficients of airfoils. The main focus of this article is to study the main factors that affect the accuracy of the multiquadric functions, including the location and quantity of centers and the choice of the form factor. It also benchmarks them against piecewise linear multivariable interpolation regarding their precision throughout the selected domain and the computational cost required to accomplish a given amount of solutions associated with the aerodynamic coefficients of lift, drag and pitching moment. The approximation functions are applied to two different multidimensional cases: two independent variables, where the aerodynamic coefficients depend on the Reynolds number (Re) and the angle-of-attack (α), and four independent variables, where the aerodynamic coefficients depend on Re, α, flap chord ratio (cflap), and flap deflection (δflap).

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Adaptive Gaussian radial basis function methods for initial value problems: Construction and comparison with adaptive multiquadric radial basis function methods

Cited by 12 publications

References 10 publications

Construction and comparative study of Euler method with adaptive IQ and IMQ-RBFs

Construction and comparative study of Euler method with adaptive IQ and IMQ-RBFs

Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs

Use of Multiquadric Functions for Multivariable Representation of the Aerodynamic Coefficients of Airfoils

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