Journal ArticleDOI
Adaptive multilayer method of fundamental solutions using a weighted greedy QR decomposition for the Laplace equation
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TLDR
A weighted greedy QR decomposition (GQRD) is proposed to choose significant source points by introducing a weighting parameter and it is concluded that the numerical solution tends to be more accurate when the average degree of approximation is larger and that the proposed method can yield more accurate solutions with a less number of source points than the conventional GQRD.About:
This article is published in Journal of Computational Physics.The article was published on 2012-08-01. It has received 25 citations till now. The article focuses on the topics: Method of fundamental solutions & Laplace's equation.read more
Citations
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On choosing the location of the sources in the MFS
TL;DR: This work investigates the satisfactory location for the sources outside the closure of the domain of the problem under consideration by means of a leave-one-out cross validation algorithm and obtains locations of the sources which lead to highly accurate results, at a relatively low cost.
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An overview of the method of fundamental solutions—Solvability, uniqueness, convergence, and stability
TL;DR: In this article, the authors give an overview of the MFS as a heuristic numerical method, which has the flexibility of using various forms of fundamental solutions, singular, hypersingular or nonsingular, mixing with general solutions and particular solutions, for different purposes.
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Optimal sources in the MFS by minimizing a new merit function: Energy gap functional
TL;DR: A new merit function is derived, namely the energy gap functional, whose minimum leads to the optimal distribution of source points whose minimum can improve the accuracy of the MFS for solving the mixed boundary value problem as well as the Cauchy problem of the Laplace equation.
Journal ArticleDOI
Fundamental kernel-based method for backward space-time fractional diffusion problem
Fangfang Dou,Y.C. Hon +1 more
TL;DR: An efficient and accurate numerical scheme for solving a backward space-time fractional diffusion problem (BSTFDP) and combines the standard Tikhonov regularization technique with the generalized cross validation (GCV) method for an optimal placement of the source points in the use of fundamental solutions.
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The MFS versus the Trefftz method for the Laplace equation in 3D
TL;DR: In this article, the Trefftz method with cylindrical basis functions was used to solve the Laplace equation with non-harmonic boundary conditions in complicated irregular domains in 3D.
References
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Journal ArticleDOI
The Approximate Solution of Elliptic Boundary-Value Problems by Fundamental Solutions
Rudolf Mathon,R. L. Johnston +1 more
TL;DR: In this paper, the boundary conditions of an elliptic equation are approximated by using fundamental solutions with singularities located outside the region of interest as trial functions, and a highly adaptive though nonlinear approximation is achieved employing only a small number of trial functions.
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Some comments on the ill-conditioning of the method of fundamental solutions
TL;DR: It is found that Gaussian elimination can be used reliably to solve the MFS equations and the use of the singular value decomposition shows no improvement overGaussian elimination provided that the boundary condition is non-noisy.
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Results on meshless collocation techniques
TL;DR: In this paper, the authors prove asymptotic feasibility for a generalized variant using separated trial and test spaces, and a greedy variation of this technique is provided, allowing a fully adaptive matrix-free and data-dependent meshless selection of the test and trial spaces.
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Method of fundamental solutions: singular value decomposition analysis
TL;DR: In this paper, an alternative solution procedure based on the singular value decomposition of the coefficient matrix is suggested and it is shown that the numerical results are extremely accurate (often within machine precision) and relatively independent of the location of the source points.