A survey of applications of the MFS to inverse problems

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.

[1]  Michael A. Golberg,et al.  The method of fundamental solutions for Poisson's equation , 1995 .

[2]  M. Golberg,et al.  Discrete projection methods for integral equations , 1996 .

[3]  B. Tomas Johansson,et al.  A method of fundamental solutions for transient heat conduction , 2008 .

[4]  On the Determination of a Robin Boundary Coefficient in an Elastic Cavity Using the MFS , 2009 .

[5]  Dario Fasino,et al.  An inverse Robin problem for Laplace's equation: theoretical results and numerical methods , 1999 .

[6]  D. Lesnic,et al.  Detection of cavities using the method of fundamental solutions , 2009 .

[7]  V. D. Kupradze,et al.  The method of functional equations for the approximate solution of certain boundary value problems , 1964 .

[8]  N. S. Mera,et al.  A three-dimensional boundary determination problem in potential corrosion damage , 2005 .

[9]  Graeme Fairweather,et al.  The method of fundamental solutions for elliptic boundary value problems , 1998, Adv. Comput. Math..

[10]  Dayong Zhou,et al.  The method of fundamental solutions for solving a Cauchy problem of Laplace's equation in a multi-connected domain , 2008 .

[11]  Elena Beretta,et al.  Stable determination of boundaries from Cauchy data , 1999 .

[12]  Boundary element method for the Cauchy problem in linear elasticity , 2001 .

[13]  Andrzej Piotr Zielinski,et al.  Boundary Collocation Techniques and Their Application in Engineering , 2009 .

[14]  Y. Rudy,et al.  Application of the Method of Fundamental Solutions to Potential-based Inverse Electrocardiography , 2006, Annals of Biomedical Engineering.

[15]  Daniel Lesnic,et al.  A method of fundamental solutions for the one-dimensional inverse Stefan problem , 2011 .

[16]  Y. Hon,et al.  A Meshless Computational Method For Solving Inverse Heat Conduction Problems , 2002 .

[17]  W. Chen,et al.  A meshless, integration-free, and boundary-only RBF technique , 2002, ArXiv.

[18]  T. Wei,et al.  Convergence analysis for the Cauchy problem of Laplace’s equation by a regularized method of fundamental solutions , 2010, Adv. Comput. Math..

[19]  A. Karageorghis and D. Lesnic.  The Pressure-StreamFunction MFS Formulation for the Detection of an Obstacle Immersed in a Two-Dimensional Stokes Flow , 2010 .

[20]  Bangti Jin,et al.  Boundary knot method for some inverse problems associated with the Helmholtz equation , 2005 .

[21]  Andreas Poullikkas,et al.  The Method of Fundamental Solutions for Stokes flows with a free surface , 1998 .

[22]  Andreas Karageorghis,et al.  Regularized MFS-Based Boundary Identification in Two-Dimensional Helmholtz-Type Equations , 2009 .

[23]  Božidar Šarler,et al.  Solution of potential flow problems by the modified method of fundamental solutions: Formulations with the single layer and the double layer fundamental solutions , 2009 .

[24]  A. Tikhonov,et al.  Nonlinear Ill-Posed Problems , 1997 .

[25]  Liviu Marin,et al.  A meshless method for the numerical solution of the Cauchy problem associated with three-dimensional Helmholtz-type equations , 2005, Appl. Math. Comput..

[26]  Y. Hon,et al.  Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators , 2007 .

[27]  Andrew M. Stuart,et al.  Inverse problems: A Bayesian perspective , 2010, Acta Numerica.

[28]  Bangti Jin,et al.  The method of fundamental solutions for inverse source problems associated with the steady‐state heat conduction , 2007 .

[29]  D. L. Young,et al.  The method of fundamental solutions and condition number analysis for inverse problems of Laplace equation , 2008, Comput. Math. Appl..

[30]  L. Marin.  A relaxation method of an alternating iterative MFS algorithm for the Cauchy problem associated with the two‐dimensional modified Helmholtz equation , 2012 .

[31]  L. Marin.  Stable MFS Solution to Singular Direct and Inverse Problems Associated with the Laplace Equation Subjected to Noisy Data , 2008 .

[32]  L. Marin.  An alternating iterative MFS algorithm for the Cauchy problem for the modified Helmholtz equation , 2010 .

[33]  B. Tomas Johansson,et al.  Relaxation procedures for an iterative MFS algorithm for the stable reconstruction of elastic fields from Cauchy data in two-dimensional isotropic linear elasticity , 2010 .

[34]  L. Marin.  Reconstruction of Boundary Data in Two-Dimensional Isotropic Linear Elasticity from Cauchy Data Using an Iterative MFS Algorithm , 2010 .

[35]  Nicholas Zabaras,et al.  Hierarchical Bayesian models for inverse problems in heat conduction , 2005 .

[36]  J. Desanto.  Mathematical and numerical aspects of wave propagation , 1998 .

[37]  A. Bogomolny.  Fundamental Solutions Method for Elliptic Boundary Value Problems , 1985 .

[38]  C. Fan,et al.  The time-marching method of fundamental solutions for wave equations , 2009 .

[39]  G. Fairweather,et al.  THE SIMPLE LAYER POTENTIAL METHOD OF FUNDAMENTAL SOLUTIONS FOR CERTAIN BIHARMONIC PROBLEMS , 1989 .

[40]  G. Fairweather,et al.  The method of fundamental solutions for problems in potential flow , 1984 .

[41]  Derek B. Ingham,et al.  The method of fundamental solutions for a biharmonic inverse boundary determination problem , 2008 .

[42]  B. Johansson,et al.  A relaxation method of an alternating iterative algorithm for the Cauchy problem in linear isotropic elasticity , 2010 .

[43]  T. Wei,et al.  An adaptive greedy technique for inverse boundary determination problem , 2010, J. Comput. Phys..

[44]  Marcelo J. Colaço,et al.  Estimation of the heat transfer coefficient by means of the method of fundamental solutions , 2008 .

[45]  V. Morozov.  On the solution of functional equations by the method of regularization , 1966 .

[46]  Eric L. Miller,et al.  Simultaneous multiple regularization parameter selection by means of the L-hypersurface with applications to linear inverse problems posed in the wavelet transform domain , 1998, Optics & Photonics.

[47]  Andreas Poullikkas,et al.  The numerical solution of three-dimensional Signorini problems with the method of fundamental solutions , 2001 .

[48]  Y. C. Hon,et al.  A computational method for inverse free boundary determination problem , 2008 .

[49]  Wen Chen,et al.  New Insights in Boundary-only and Domain-type RBF Methods , 2002, ArXiv.

[50]  L. Marin.  An iterative MFS algorithm for the Cauchy problem associated with the Laplace equation , 2009 .

[51]  J. Hadamard,et al.  Lectures on Cauchy's Problem in Linear Partial Differential Equations , 1924 .

[52]  Liviu Marin.  Boundary Reconstruction in Two-Dimensional Functionally Graded Materials Using a Regularized MFS , 2009 .

[53]  M. Arab,et al.  The method of fundamental solutions for the inverse space-dependent heat source problem , 2009 .

[54]  Daniel Lesnic,et al.  The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity , 2004 .

[55]  D. Lesnic,et al.  The method of fundamental solutions for the inverse conductivity problem , 2010 .

[56]  Y. Hon,et al.  A MESHLESS SCHEME FOR SOLVING INVERSE PROBLEMS OF LAPLACE EQUATION , 2003 .

[57]  J. Telles,et al.  Third degree polynomial transformation for boundary element integrals: Further improvements , 1994 .

[58]  D. L. Young,et al.  The method of fundamental solutions for inverse 2D Stokes problems , 2005 .

[59]  Yiorgos Sokratis Smyrlis,et al.  Applicability and applications of the method of fundamental solutions , 2009, Math. Comput..

[60]  M. Li,et al.  A DISCREPANCY PRINCIPLE FOR THE SOURCE POINTS LOCATION IN USING THE MFS FOR SOLVING THE BHCP , 2009 .

[61]  Y. Hon,et al.  A fundamental solution method for inverse heat conduction problem , 2004 .

[62]  Carlos J. S. Alves,et al.  The direct method of fundamental solutions and the inverse Kirsch- Kress method for the reconstruction of elastic inclusions or cavities , 2009 .

[63]  R. Mathon,et al.  The Approximate Solution of Elliptic Boundary-Value Problems by Fundamental Solutions , 1977 .

[64]  Xin Li,et al.  Trefftz Methods for Time Dependent Partial Differential Equations , 2004 .

[65]  G. Wahba.  Practical Approximate Solutions to Linear Operator Equations When the Data are Noisy , 1977 .

[66]  T. Wei,et al.  An inverse boundary problem for one-dimensional heat equation with a multilayer domain ☆ , 2009 .

[67]  A. Tikhonov,et al.  Numerical Methods for the Solution of Ill-Posed Problems , 1995 .

[68]  T. Wei,et al.  Identification of a moving boundary for a heat conduction problem in a multilayer medium , 2010 .

[69]  Liang Yan,et al.  A meshless method for solving an inverse spacewise-dependent heat source problem , 2009, J. Comput. Phys..

[70]  Nuno F. M. Martins,et al.  An iterative MFS approach for the detection of immersed obstacles , 2008 .

[71]  B. Šarler,et al.  Solution of a two-dimensional bubble shape in potential flow by the method of fundamental solutions , 2006 .

[72]  B. Tomas Johansson,et al.  A method of fundamental solutions for transient heat conduction in layered materials , 2009 .

[73]  L. Marin.  Stable Boundary and Internal Data Reconstruction in Two-Dimensional Anisotropic Heat Conduction Cauchy Problems Using Relaxation Procedures for an Iterative MFS Algorithm , 2010 .

[74]  D. L. Young,et al.  Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points , 2008, J. Comput. Phys..

[75]  Liviu Marin,et al.  Relaxation procedures for an iterative MFS algorithm for two-dimensional steady-state isotropic heat conduction Cauchy problems , 2011 .

[76]  C. Fu,et al.  The method of fundamental solutions for the inverse heat source problem. , 2008 .

[77]  Derek B. Ingham,et al.  The method of fundamental solutions for detection of cavities in EIT , 2009 .

[78]  C. S. Chen,et al.  A new method of fundamental solutions applied to nonhomogeneous elliptic problems , 2005, Adv. Comput. Math..

[79]  V. D. Kupradze,et al.  A method for the approximate solution of limiting problems in mathematical physics , 1964 .

[80]  Andreas Karageorghis.  The method of fundamental solutions for the solution of steady-state free boundary problems , 1992 .

[81]  Y. Hon,et al.  Conditional stability estimation for an inverse boundary problem with non-smooth boundary in ℛ³ , 2001 .

[82]  Bangti Jin,et al.  The method of fundamental solutions for inverse boundary value problems associated with the steady‐state heat conduction in anisotropic media , 2006 .

[83]  ProblemsPer Christian HansenDepartment.  The L-curve and its use in the numerical treatment of inverse problems , 2000 .

[84]  Dianne P. O'Leary,et al.  The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems , 1993, SIAM J. Sci. Comput..

[85]  L. Marin.  A meshless method for the stable solution of singular inverse problems for two-dimensional Helmholtz-type equations , 2010 .

[86]  Jan Adam Kołodziej,et al.  Application of the method of fundamental solutions and radial basis functions for inverse heat source problem in case of steady-state , 2010 .

[87]  Liang Yan,et al.  Reconstruction of the corrosion boundary for the Laplace equation by using a boundary collocation method , 2009, Math. Comput. Simul..

[88]  Graeme Fairweather,et al.  The method of fundamental solutions for scattering and radiation problems , 2003 .

[89]  E. Kita,et al.  Trefftz method: an overview , 1995 .

[90]  Liviu Marin,et al.  A meshless method for solving the cauchy problem in three-dimensional elastostatics , 2005 .

[91]  Graeme Fairweather,et al.  The method of fundamental solutions for axisymmetric potential problems , 1999 .

[92]  L. Marin.  The Method of Fundamental Solutions for Inverse Problems Associated with the Steady-State Heat Conduction in the Presence of Sources , 2008 .

[93]  T. Ohe,et al.  Uniqueness and convergence of numerical solution of the cauchy problem for the laplace equation by a charge simulation method , 2004 .

[94]  L. Yan,et al.  Reconstruction of part of a boundary for the Laplace equation by using a regularized method of fundamental solutions , 2009 .

[95]  Leevan Ling,et al.  Optimality of the method of fundamental solutions , 2011 .

[96]  B. Tomas Johansson,et al.  A comparative study on applying the method of fundamental solutions to the backward heat conduction problem , 2011, Math. Comput. Model..

[97]  C. Alves,et al.  Density results using Stokeslets and a method of fundamental solutions for the Stokes equations , 2004 .

[98]  L. Marin.  An Alternating Iterative MFS Algorithm for the Cauchy Problem in Two-Dimensional Anisotropic Heat Conduction , 2009 .

[99]  Martin D. Buhmann,et al.  Radial Basis Functions: Theory and Implementations: Preface , 2003 .

[100]  Daniel Lesnic,et al.  The MFS for numerical boundary identification in two-dimensional harmonic problems , 2011 .

[101]  Vladimir Maz’ya,et al.  An iterative method for solving the Cauchy problem for elliptic equations , 1991 .

[102]  C. S. Chen,et al.  Some comments on the use of radial basis functions in the dual reciprocity method , 1998 .

[103]  Daniel Lesnic,et al.  The method of fundamental solutions for free surface Stefan problems. , 2009 .

[104]  Liviu Marin,et al.  Regularized method of fundamental solutions for boundary identification in two-dimensional isotropic linear elasticity , 2010 .

[105]  Carlos J. S. Alves,et al.  Full identification of acoustic sources with multiple frequencies and boundary measurements , 2009 .

[106]  Liviu Marin,et al.  Boundary reconstruction in two-dimensional steady state anisotropic heat conduction using a regularized meshless method , 2010 .

[107]  B. Meng,et al.  A method of fundamental solutions for inverse heat conduction problems in an anisotropic medium. , 2007 .

[108]  T. Wei,et al.  Two regularization methods for the Cauchy problems of the Helmholtz equation , 2010 .

[109]  Nuno F. M. Martins,et al.  Recovering the source term in a linear diffusion problem by the method of fundamental solutions , 2008 .

[110]  Andreas Poullikkas,et al.  The method of fundamental solutions for Signorini problems , 1998 .

[111]  Y. Hon,et al.  The method of fundamental solution for solving multidimensional inverse heat conduction problems , 2005 .

[112]  Y. Hon,et al.  A numerical computation for inverse boundary determination problem , 2000 .

[113]  Daniel Lesnic,et al.  Application of the MFS to inverse obstacle scattering problems , 2011 .

[114]  P. Hansen.  Discrete Inverse Problems: Insight and Algorithms , 2010 .

[115]  Carlos J. S. Alves,et al.  On the choice of source points in the method of fundamental solutions , 2009 .

[116]  D. L. Young,et al.  The Method of Fundamental Solutions for One-Dimensional Wave Equations , 2009 .

[117]  D. Lesnic,et al.  THE METHOD OF FUNDAMENTAL SOLUTIONS FOR AN INVERSE INTERNAL BOUNDARY VALUE PROBLEM FOR THE BIHARMONIC EQUATION , 2009 .

[118]  Daniel Lesnic,et al.  The method of fundamental solutions for inverse boundary value problems associated with the two-dimensional biharmonic equation , 2005, Math. Comput. Model..

[119]  T. Wei,et al.  Reconstruction of a moving boundary from Cauchy data in one-dimensional heat equation , 2009 .

[120]  Piotr Gorzelańczyk,et al.  Some remarks concerning the shape of the source contour with application of the method of fundamental solutions to elastic torsion of prismatic rods , 2008 .

[121]  Bangti Jin,et al.  A meshless method for some inverse problems associated with the Helmholtz equation , 2006 .

[122]  Daniel Lesnic,et al.  The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations , 2005 .

[123]  A. Karageorghis.  A Practical Algorithm for Determining the Optimal Pseudo-Boundary in the Method of Fundamental Solutions , 2009 .

[124]  N. S. Mera.  The method of fundamental solutions for the backward heat conduction problem , 2005 .

[125]  Liviu Marin,et al.  Numerical solution of the Cauchy problem for steady-state heat transfer in two-dimensional functionally graded materials , 2005 .

[126]  Bangti Jin,et al.  Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation , 2005 .