Journal ArticleDOI
On the choice of source points in the method of fundamental solutions
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The choice of source points are discussed and a choice along the discrete normal direction is proposed and a direct connection between the approximation based on radial basis functions (RBF) and the MFS approximation in a higher dimension is presented.Abstract:
The method of fundamental solutions (MFS) may be seen as one of the simplest methods for solving boundary value problems for some linear partial differential equations (PDEs). It is a meshfree method that may present remarkable results with a small computational effort. The meshfree feature is particularly attractive when we need to change the shape of the domain, which occurs, for instance, in shape optimization and inverse problems. The MFS may be viewed as a Trefftz method, where the approximations have the advantage of verifying the linear PDE, and therefore we may bound the inner error from the boundary error, in well-posed problems. A main counterpart for these global numerical methods, that avoid meshes, are the associated linear systems with dense and ill conditioned matrices. In these methods a sort of uncertainty principle occurs—we cannot get both accurate results and good conditioning—one of the two is lost. A specific feature of the MFS is some freedom in choosing the source points. This might lead to excellent results, but it may also lead to poor results, or even to impossible approximations. In this work we will discuss the choice of source points and propose a choice along the discrete normal direction (following [Alves CJS, Antunes PRS. The method of fundamental solutions applied to the calculation of eigenfrequencies and eigenmodes of 2D simply connected shapes. Comput Mater Continua 2005;2(4):251–66]), with a possible local criterion to define the distance to the boundary. We will also address some extensions that connect the asymptotic MFS to other methods by choosing the sources on a circle/sphere far from the boundary. We also present a direct connection between the approximation based on radial basis functions (RBF) and the MFS approximation in a higher dimension. This increase in dimension was somehow already present in a previous work [Alves CJS, Chen CS. A new method of fundamental solutions applied to non-homogeneous elliptic problems. Adv Comput Math 2005;23:125–42], where the frequency was used as the extra dimension. The free parameters in RBF inverse multiquadrics 2D approximation correspond in fact to the source point distance to the boundary plane in a Laplace 3D setting. Some numerical simulations are presented to illustrate theoretical issues.read more
Citations
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Journal ArticleDOI
A survey of applications of the MFS to inverse problems
TL;DR: The method of fundamental solutions (MFS) is a relatively new method for the numerical solution of boundary value problems and initial/boundary value problems governed by certain partial differential equations as discussed by the authors.
Journal ArticleDOI
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.
Journal ArticleDOI
Original article: A new investigation into regularization techniques for the method of fundamental solutions
Ji Lin,Wen Chen,Fuzhang Wang +2 more
TL;DR: Numerical results show that the damped singular value decomposition under the parameter choice of the generalized cross-validation performs the best in terms of the MFS stability analysis.
Journal ArticleDOI
An equilibrated method of fundamental solutions to choose the best source points for the Laplace equation
TL;DR: In this article, the authors proposed a new method to choose the best source points by using the MFS with multiple lengths R k for the distribution of source points, which are solved from an uncoupled system of nonlinear algebraic equations.
Journal ArticleDOI
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.
References
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Inverse Acoustic and Electromagnetic Scattering Theory
David Colton,Rainer Kress +1 more
TL;DR: Inverse Medium Problem (IMP) as discussed by the authors is a generalization of the Helmholtz Equation for direct acoustical obstacle scattering in an Inhomogeneous Medium (IMM).
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TL;DR: Inverse Boundary Value Problems (IBV) as discussed by the authors, the heat equation is replaced by the Tikhonov regularization and regularization by Discretization (TBD) method.
Approximation scheme with applications to computational fluid-dynamics-- i surface approximations and partial derivative estimates
TL;DR: In this article, the authors presented an enhanced multiquadrics (MQ) scheme for spatial approximations, which is a true scattered data, grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions.
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Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates
TL;DR: In this article, the authors presented a powerful, enhanced multiquadrics (MQ) scheme developed for spatial approximations, which is a true scattered data, grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions.
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