Journal ArticleDOI
A survey of applications of the MFS to inverse problems
TLDR
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.Abstract:
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. The ease with which it can be implemented and its effectiveness have made it a very popular tool for the solution of a large variety of problems arising in science and engineering. In recent years, it has been used extensively for a particular class of such problems, namely inverse problems. In this study, in view of the growing interest in this area, we review the applications of the MFS to inverse and related problems, over the last decade.read more
Citations
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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
Singular boundary method for steady-state heat conduction in three dimensional general anisotropic media
TL;DR: The singular boundary method (SBM) as mentioned in this paper is a strong-form meshless boundary collocation method that uses the concept of the origin intensity factor to isolate the singularity of the fundamental solutions and overcomes the fictitious boundary issue.
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.
Journal ArticleDOI
Localized method of fundamental solutions for solving two-dimensional Laplace and biharmonic equations
TL;DR: The LMFS, the localized version of the MFS, is proposed for solving two-dimensional boundary value problems, governed by Laplace and biharmonic equations, in complicated domains for the first time.
References
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Book
Regularization of Inverse Problems
TL;DR: Inverse problems have been studied in this article, where Tikhonov regularization of nonlinear problems has been applied to weighted polynomial minimization problems, and the Conjugate Gradient Method has been used for numerical realization.
Journal ArticleDOI
The use of the L-curve in the regularization of discrete ill-posed problems
TL;DR: A unifying characterization of various regularization methods is given and it is shown that the measurement of “size” is dependent on the particular regularization method chosen, and a new method is proposed for choosing the regularization parameter based on the L-curve.
Book
Radial Basis Functions: Theory and Implementations
TL;DR: In this paper, a radial basis function approximation on infinite grids is proposed, based on the wavelet method with radial basis functions (WBFF) with compact support, which is a general method for approximation and interpolation.
Journal ArticleDOI
Inverse problems: A Bayesian perspective
TL;DR: The Bayesian approach to regularization is reviewed, developing a function space viewpoint on the subject, which allows for a full characterization of all possible solutions, and their relative probabilities, whilst simultaneously forcing significant modelling issues to be addressed in a clear and precise fashion.
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