A numerical study of variable coefficient elliptic Cauchy problem via projection method

References

• Andrieux , S. , Baranger , T. N. and Ben Abda , A. 2006 . Solving Cauchy problem by minimizing an energy-like functional . Inverse Problems , 22 : 115 – 133 .
   Web of Science ®Google Scholar
• Azaïez , M. , Belgacem , F. B. and Fekih , H. E. 2006 . On Cauchy's problem: II. Completion, regularization and approximation . Inverse Problems , 22 : 1307 – 1336 .
   Web of Science ®Google Scholar
• Baskar , S. and Prasad , P. 2006 . Formulation of the problem of sonic boom by a maneuvering aerofoil as a one-parameter family of Cauchy problem . Proc. Indian Acad. Sci. (Math Sci.) , 116 ( 1 ) : 97 – 119 .
   Web of Science ®Google Scholar
• Belgacem , F. B. 2007 . Why is the Cauchy problem severely ill-posed? . Inverse Problems , 23 : 823 – 836 .
   Web of Science ®Google Scholar
• Belgacem , F. B. , Du , D. T. and Jelassi , F. 2011 . Extended-domain-Lavrentiev's regularization for the Cauchy problem . Inverse Problems , 27 : 27 045005
   Google Scholar
• Blum , J. 1989 . Numerical Simulation and Optimal Control in Plasma Physics with Application to Tokamaks , Paris : Modern Applied Mathematics, Wiley/Gauthier-Villars .
   Google Scholar
• Bourgeois , L. and Darde , J. 2010 . A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data . Inverse Problems , 26 : 21 095016
   Web of Science ®Google Scholar
• Bruckner , G. and Pereverzev , S. V. 2003 . Self-regularization of projection methods with a posteriori discretization level choice for severely ill-posed problems . Inverse Problems , 19 : 147 – 156 .
   Web of Science ®Google Scholar
• Cao , H. and Perverzev , S. V. 2007 . The balancing principle for the regularization of elliptic Cauchy problems . Inverse Problems , 23 : 1943 – 1961 .
   Web of Science ®Google Scholar
• Cao , H. , Klibanov , M. V. and Pereverzev , S. V. 2009 . A Carleman estimate and the balancing principle in the quasi reversibility method for solving the Cauchy problem for the Laplace equation . Inverse Problems , 25 : 21 035005
   Web of Science ®Google Scholar
• Cimetière , A. , Delvare , F. , Jaoua , M. and Pons , F. 2001 . Solution of the Cauchy problem using iterated Tikhonov regularization . Inverse Problems , 17 : 553 – 570 .
   Web of Science ®Google Scholar
• Dhanumjya , P. and Gupta , H. S. An ill-posed Cauchy problem for second order elliptic equation appearing in a sonic boom . Available at http://math.iisc.ernet.in/~eprints/preprints08.htm.
   Google Scholar
• Engl , H. and Leitão , A. 2001 . A mann iterative regularization method for elliptic Cauchy problems . Numer. Funct. Anal. Optim. , 22 : 861 – 884 .
   Web of Science ®Google Scholar
• Engl , H. , Hanke , M. and Neubauer , A. 1996 . Regularization of Inverse Problem , Dordrecht : Kluwer .
   Google Scholar
• Essaouini , M. , Nachaoui , A. and Ei Hajji , S. 2004 . Reconstruction of boundary data for a class of nonlinear inverse problems . J. Inverse Ill-Posed Problems , 12 ( 4 ) : 369 – 385 .
   Google Scholar
• Essaouini , M. , Nachaoui , A. and Ei Hajji , S. 2004 . Numerical method for solving a class of nonlinear elliptic inverse problems . J. Comput. Appl. Math. , 162 : 165 – 181 .
   Web of Science ®Google Scholar
• Fasino , D. and Inglese , G. 1999 . An inverse Robin problem for Laplace's equation: Theoretical results and numerical methods . Inverse problems , 15 : 41 – 48 .
   Web of Science ®Google Scholar
• Grisvard , P. 1985 . Elliptic Problems in Nonsmooth Domains , Boston : Pitman Advanced Publishing Program .
   Google Scholar
• Groetsch , C. W. and Neubauer , A. 1988 . Convergence of a general projection method for an operator equation of the first kind . Houston J. Math. , 14 : 201 – 208 .
   Web of Science ®Google Scholar
• Hadamard , J. 1953 . Lectures on Cauchy's Problem in Linear Partial Differential Equation , New York : Dover .
   Google Scholar
• Hao , D. N. and Lesnic , D. 2000 . The Cauchy problem for Laplace's equation via the conjugate gradient method . IMA J. Appl. Math. , 65 : 199 – 217 .
   Web of Science ®Google Scholar
• Hon , Y. C. and Wei , T. 2001 . Backus–Gilbert algorithm for the Cauchy problems of the Laplace equation . Inverse Problems , 17 : 261 – 271 .
   Web of Science ®Google Scholar
• Isakov , V. 1998 . Inverse Problems for Partial Differential Equations , New York : Springer .
   Google Scholar
• Kaltenbacher , B. 2000 . Regularization by projection with a posteriori discretization level choice for linear and nonlinear ill-posed problems . Inverse Problems , 16 : 1523 – 1539 .
   Web of Science ®Google Scholar
• Kohn , R. V. and Vogelius , M. S. 1985 . Determining conductivity by boundary measurements II. Interior results . Commun. Pure Appl. Math. , 38 : 643 – 667 .
   Web of Science ®Google Scholar
• Kügler , P. and Leitão , A. 2003 . Mean value iterations for nonlinear elliptic Cauchy problems . Numer. Math. , 96 : 269 – 293 .
   Web of Science ®Google Scholar
• Lazarov , R. D. , Lu , S. and Pereverzev , S. V. 2007 . On the balancing principle for some problems of numerical analysis . Numer. Math. , 106 : 659 – 689 .
   Web of Science ®Google Scholar
• Lions , J. L. and Magenes , E. 1972 . Non-homogeneous Boundary Value Problems and Applications , Vol. 1 , New York : Springer .
   Google Scholar
• Luecke , G. R and Hickey , K. R . 1985 . Convergence of approximate solutions of an operator equation . Houston J. Math. , 11 : 345 – 354 .
   Web of Science ®Google Scholar
• Mghazli , Z. 1992 . Regularity of an elliptic problem with mixed Dirichlet–Robin boundary conditions in a polygonal domain . Calcolo , 29 ( 3–4 ) : 241 – 267 .
   Google Scholar
• Natterer , F. 1977 . Regularisierung schlecht gestellter Probleme durch Projektionsverfahren . Numer. Math. , 28 : 329 – 341 . (German).
   Web of Science ®Google Scholar
• Pereverzev , S. V. and Schock , E. 2005 . On the adaptive selection of the parameter in regularization of ill-posed problem . SIAM J. Numer. Anal. , 43 : 2060 – 2076 .
   Web of Science ®Google Scholar
• Raymond , J. P. Optimal Control of Partial Differential Equations Available at http://www.math.univ-toulouse.fr/~raymond/book-ficus.pdf.
   Google Scholar
• Seidman , T. I. 1980 . Nonconvergence results for the application of least-squares estimation to ill-posed problems . J. Optim. Theory Appl. , 30 : 535 – 547 .
   Web of Science ®Google Scholar
• Takeuchi , T. and Yamamoto , M. 2008 . Tikhonov regularization by a reproducing kernel Hilbert space for the Cauchy problem for an elliptic equation . SIAM J. Sci. Comput. , 31 : 112 – 142 .
   Web of Science ®Google Scholar