Frequency-domain meshless solver for acoustic wave equation using a stable radial basis-finite difference (RBF-FD) algorithm with hybrid kernels

Mishra, Pankaj K.; Nath, Sankar Kumar; Fasshauer, Gregory E.; Sen, Mrinal K.

doi:10.1190/segam2017-17494511.1

Cited by 6 publications

(10 citation statements)

References 18 publications

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“…To obtain an explicit relation between k and kf we consider the plane wave with propagation angle θ and wave length �=2�/k. According to (25) we can take kf as a function which depends on k, θ the stencil S and ε, given by its square Note that the local truncation error (22) for ℒ=Δ is given just by The fictitious wavenumber is anisotropic, i.e., it depends on the wave propagation direction θ. As was the case in [27], our objective will be to choose the shape parameter ε such that the average phase error, over all angles of propagation, is minimum.…”

Section: Dispersion Analysismentioning

confidence: 99%

Optimal shape parameter for meshless solution of the 2D Helmholtz equation

Londoño-Arboleda

Montegranario-Riascos

2019

CT&F Cienc. Tecnol. Futuro

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The solution of the Helmholtz equation is a fundamental step in frequency domain seismic imaging. This paper deals with a numerical study of solutions for 2D Helmholtz equation using a Gaussian radial basis function-generated finite difference scheme (RBFFD). We analyze the behavior of the local truncation error in approximating partial derivatives of the 2D Helmholtz equation solutions when the shape parameter of RBF varies. For discretization, we performed, by means of a classical numerical dispersion analysis with plane waves, a minimization of the error function to obtain local and adaptive near optimal shape parameters according to the local wavelength of the required solution. In particular, the method is applied to obtain a simple and accurate solver by using stencils which seven nodes on hexagonal regular grids, wich mitigate pollution-effects. We validated numerically that the stability and isotropy are enhanced with respect to Cartesian grids. Our method is tested with standard case studies and velocity models, showing similar or better accuracy than finite difference and finite element methods. This is an efficient way for interacting with inverse and imaging problems such as Full Wave Inversion.

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Section: Dispersion Analysismentioning

confidence: 99%

Optimal shape parameter for meshless solution of the 2D Helmholtz equation

Londoño-Arboleda

Montegranario-Riascos

2019

CT&F Cienc. Tecnol. Futuro

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“…The RMS error was kept as the objective function in the particle swarm optimization. We sample the Franke's test function at various number of data points, i.e., [25,49,144,196,625,1296,2401,4096] and try to reconstruct it while finding the optimal parameter combination for the hybrid kernel. Figure 3 shows a typical particle swarm optimization procedure for this test when we try to reconstruct the Franke's test function Table 3 Results of the parameter optimization test for 2-D interpolation using hybrid kernel.…”

Section: Franke's Testmentioning

confidence: 99%

“…We take the same Franke's test function for this interpolation test. The optimization has been performed for various degrees of freedom, i.e., N = [25,49,144,196,625,1296,2401,4096]. The results have been tabulated in Table 4.…”

Section: The Objective Functionsmentioning

confidence: 99%

“…The objective function, which is frequently referred as the "cost function" in the RBF literature, is RMS error. In order to compare the convergence of linear reproduction test using the Gaussian kernel, the hybrid kernel and the hybrid kernel with polynomial augmentation, parameter optimization has been performed for various degrees of freedoms, i.e., [25,49,144,196,625,1296,2401,4096]. The RMS errors for various degrees of freedom using the hybrid kernel, the Gaussian kernel and the hybrid kernel with polynomial augmentation has been denoted as E H , E G , and E H+P respectively.…”

Section: Linear Reproductionmentioning

confidence: 99%

“…We have used the data for this numerical test from Table 3. The eigenvalues for various degrees of freedom, i.e., N = [25,49,144,196,625,1296,2401,4096] have been plotted together. For the hybrid kernel, the spectra has some negative eigenvalues for N = [25,49,144,196].…”

Section: Eigenvalue Analysis Of Interpolation Matricesmentioning

confidence: 99%

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Hybrid Gaussian-cubic radial basis functions for scattered data interpolation

et al. 2018

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Scattered data interpolation schemes using kriging and radial basis functions (RBFs) have the advantage of being meshless and dimensional independent; however, for the data sets having insufficient observations, RBFs have the advantage over geostatistical methods as the latter requires variogram study and statistical expertise. Moreover, RBFs can be used for scattered data interpolation with very good convergence, which makes them desirable for shape function interpolation in meshless methods for numerical solution of partial differential equations. For interpolation of large data sets, however, RBFs in their usual form, lead to solving an ill-conditioned system of equations, for which, a small error in the data can cause a significantly large error in the interpolated solution. In order to reduce this limitation, we propose a hybrid kernel by using the conventional Gaussian and a shape parameter independent cubic kernel. Global particle swarm optimization method has been used to analyze the optimal values of the shape parameter as well as the weight coefficients controlling the Gaussian and the cubic part in the hybridization. Through a series of numerical tests, we demonstrate that such hybridization stabilizes the interpolation scheme by yielding a far superior implementation compared to those obtained by using only the Gaussian or cubic kernels. The proposed kernel maintains the accuracy and stability at small shape parameter as well as relatively large degrees of freedom, which exhibit its potential for scattered data interpolation and intrigues its application in global as well as local meshless methods for numerical solution of PDEs.

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