The MFS for numerical boundary identification in two-dimensional harmonic problems

In this study, we briefly review the applications of the method of fundamental solutions to inverse problems over the last decade. Subsequently, we consider the inverse geometric problem of identifying an unknown part of the boundary of a domain in which the Laplace equation is satisfied. Additional Cauchy data are provided on the known part of the boundary. The method of fundamental solutions is employed in conjunction with regularization in order to obtain a stable solution. Numerical results are presented and discussed.

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

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

[3]  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 .

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

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

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

[7]  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 .

[8]  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..

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

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

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

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

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

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

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

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

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

[18]  M. Golberg.  Boundary integral methods : numerical and mathematical aspects , 1999 .

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

[20]  D. Lesnic,et al.  BEM first-order regularisation method in linear elasticity for boundary identification , 2003 .

[21]  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..

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

[23]  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..

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

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

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

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

[28]  L. Marin.  Numerical boundary identification for Helmholtz-type equations , 2006 .

[29]  A. J. M. Ferreira.  Progress on meshless methods , 2009 .

[30]  M. Vogelius,et al.  A nonlinear elliptic boundary value problem related to corrosion modeling , 1998 .

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

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

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

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

[35]  V. Isakov.  On uniqueness of obstacles and boundary conditions from restricted dynamical and scattering data , 2008 .

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

[37]  Sergio Vessella,et al.  Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates , 2007, 0710.2192.

[38]  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..

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

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

[41]  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..

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

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

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

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

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

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

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

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

[50]  Alain J. Kassab,et al.  An inverse boundary element method/genetic algorithm based approach for retrieval of multi-dimensional heat transfer coefficients within film cooling holes/slots , 2005 .

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

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

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

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

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

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

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

[58]  M. Mciver,et al.  An inverse problem in electromagnetic crack detection , 1991 .

[59]  Fadil Santosa,et al.  Nondestructive evaluation of corrosion damage using electrostatic measurements , 1995 .

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

[61]  Daniel Lesnic,et al.  A Boundary Element Regularization Method for the Boundary Determination in Potential Corrosion Damage , 2002 .

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

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

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