Numerical solution of the Laplacian Cauchy problem by using a better postconditioning collocation Trefftz method

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

[2]  F. L. Bauer.  Optimally scaled matrices , 1963 .

[3]  A. Sluis.  Condition numbers and equilibration of matrices , 1969 .

[4]  Derek B. Ingham,et al.  An iterative boundary element method for solving numerically the Cauchy problem for the Laplace equation , 1997 .

[5]  Derek B. Ingham,et al.  An Iterative Boundary Element Method for the Solution of a Cauchy Steady State Heat Conduction Problem , 2000 .

[6]  Mohamed Jaoua,et al.  Solution of the Cauchy problem using iterated Tikhonov regularization , 2001 .

[7]  Weichung Yeih,et al.  On the Modified Tikhonov's Regularization Method for the Cauchy Problem of the Laplace Equation , 2001, Journal of Marine Science and Technology.

[8]  Chein-Shan Liu.  Cone of non-linear dynamical system and group preserving schemes , 2001 .

[9]  Fredrik Berntsson,et al.  Numerical solution of a Cauchy problem for the Laplace equation , 2001 .

[10]  A. Nachaoui,et al.  Convergence of an Alternating Method to Solve the Cauchy Problem for Poisson's Equation , 2002 .

[11]  Dinh Nho Hào,et al.  Stability results for the Cauchy problem for the Laplace equation in a strip , 2003 .

[12]  N. S. Mera,et al.  On the use of Genetic Algorithms for Solving Ill-Posed Problems , 2003 .

[13]  M. Jourhmane,et al.  An alternating method for an inverse Cauchy problem , 1999, Numerical Algorithms.

[14]  M. Essaouini,et al.  Numerical method for solving a class of nonlinear elliptic inverse problems , 2004 .

[15]  Marián Slodicka,et al.  A numerical approach for the determination of a missing boundary data in elliptic problems , 2004, Appl. Math. Comput..

[16]  Daniel Lesnic,et al.  Relaxation procedures for an iterative algorithm for solving the Cauchy problem for the Laplace equation , 2004 .

[17]  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..

[18]  R. Kress,et al.  A hybrid method for inverse boundary value problems in potential theory , 2005 .

[19]  Laurent Bourgeois,et al.  A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation , 2005 .

[20]  Chih-Wen Chang,et al.  Past Cone Dynamics and Backward Group Preserving Schemes for Backward Heat Conduction Problems , 2006 .

[21]  Laurent Bourgeois,et al.  Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation , 2006 .

[22]  Chein-Shan Liu,et al.  The Lie-Group Shooting Method for Nonlinear Two-Point Boundary Value Problems Exhibiting Multiple Solutions , 2006 .

[23]  A. Nachaoui,et al.  Convergence analysis for finite element approximation to an inverse Cauchy problem , 2006 .

[24]  Thouraya Baranger,et al.  Solving Cauchy problems by minimizing an energy-like functional , 2006 .

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

[26]  Chein-Shan Liu,et al.  An effectively modified direct Trefftz method for 2D potential problems considering the domain's characteristic length. , 2007 .

[27]  Chein-Shan Liu.  A Highly Accurate Solver for the Mixed-Boundary Potential Problem and Singular Problem in Arbitrary Plane Domain , 2007 .

[28]  A. Cheng,et al.  Trefftz, collocation, and other boundary methods—A comparison , 2007 .

[29]  Chein-Shan Liu.  A Modified Trefftz Method for Two-Dimensional Laplace Equation Considering the Domain's Characteristic Length , 2007 .

[30]  Chein-Shan Liu,et al.  Improving the Ill-conditioning of the Method of Fundamental Solutions for 2D Laplace Equation , 2008 .

[31]  Jianxin Deng,et al.  Design and development of SiC/(W,Ti)C gradient ceramic nozzle , 2008 .

[32]  Chein-Shan Liu.  A Highly Accurate MCTM for Inverse Cauchy Problems of Laplace Equation in Arbitrary Plane Domains , 2008 .

[33]  Chein-Shan Liu,et al.  A modified collocation Trefftz method for the inverse Cauchy problem of Laplace equation , 2008 .

[34]  A. Cheng,et al.  Trefftz and Collocation Methods , 2008 .

[35]  C. Fu,et al.  Fourier regularization method for solving a Cauchy problem for the Laplace equation , 2008 .

[36]  Chein-Shan Liu.  A highly accurate collocation Trefftz method for solving the Laplace equation in the doubly connected domains , 2008 .

[37]  Chu-Li Fu,et al.  Two regularization methods for a Cauchy problem for the Laplace equation , 2008 .

[38]  Chein-Shan Liu.  A Highly Accurate MCTM for Direct and Inverse Problems of Biharmonic Equation in Arbitrary Plane Domains , 2008 .

[39]  T. Lu,et al.  Stability analysis of Trefftz methods for the stick-slip problem☆ , 2009 .

[40]  Chein-Shan Liu,et al.  A Novel Method for Solving the Cauchy Problem of Laplace Equation Using the Fictitious Time Integration Method , 2009 .

[41]  Chu-Li Fu,et al.  The Fourier regularization for solving the Cauchy problem for the Helmholtz equation , 2009 .

[42]  Zhuo-Jia Fu,et al.  BOUNDARY PARTICLE METHOD FOR INVERSE CAUCHY PROBLEMS OF INHOMOGENEOUS HELMHOLTZ EQUATIONS , 2009 .

[43]  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..

[44]  Chein-Shan Liu,et al.  Applications of the modified Trefftz method for the Laplace equation , 2009 .

[45]  S. Atluri,et al.  On Solving the Ill-Conditioned System Ax=b: General-Purpose Conditioners Obtained From the Boundary-Collocation Solution of the Laplace Equation, Using Trefftz Expansions With Multiple Length Scales , 2009 .

[46]  S. Atluri,et al.  On Solving the Direct/Inverse Cauchy Problems of Laplace Equation in a Multiply Connected Domain, Using the Generalized Multiple-Source-Point Boundary-Collocation Trefftz Method &Characteristic Lengths , 2010 .

[48]  Satya N. Atluri,et al.  An Iterative and Adaptive Lie-Group Method for Solving the Calderón Inverse Problem , 2010 .

[49]  Yimin Wei,et al.  Effective condition number and its applications , 2010, Computing.

[50]  Chung-Lun Kuo,et al.  A Spring-Damping Regularization and a Novel Lie-Group Integration Method for Nonlinear Inverse Cauchy Problems , 2011 .

[51]  Chein-Shan Liu.  An Analytical Method for the Inverse Cauchy Problem of Laplace Equation in a Rectangular Plate , 2011 .

[52]  Ji Lin,et al.  A new investigation into regularization techniques for the method of fundamental solutions , 2011, Math. Comput. Simul..

[53]  Chein-Shan Liu,et al.  The Spring-Damping Regularization Method and the Lie-Group Shooting Method for Inverse Cauchy Problems , 2011 .

[54]  Walter Gautschi,et al.  Optimally scaled and optimally conditioned Vandermonde and Vandermonde-like matrices , 2011 .

[55]  Saeid Abbasbandy,et al.  Group preserving scheme for the Cauchy problem of the Laplace equation , 2011 .

[56]  Weichung Yeih,et al.  Numerical simulation of the two-dimensional sloshing problem using a multi-scaling Trefftz method , 2012 .

[57]  Chein-Shan Liu.  An equilibrated method of fundamental solutions to choose the best source points for the Laplace equation , 2012 .

[58]  Chein-Shan Liu,et al.  Optimally scaled vector regularization method to solve ill-posed linear problems , 2012, Appl. Math. Comput..

[59]  Chein-Shan Liu,et al.  A novel mixed group preserving scheme for the inverse Cauchy problem of elliptic equations in annular domains , 2012 .

[60]  Chein-Shan Liu,et al.  Optimally Generalized Regularization Methods for Solving Linear Inverse Problems , 2012 .