A new version of the quasi-reversibility method for the thermoacoustic tomography and a coefficient inverse problem

References

• Clason , C and Klibanov , MV . 2007 . The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium . SIAM J. Sci. Comp. , 30 : 1 – 23 .
   Web of Science ®Google Scholar
• Kazemi , M and Klibanov , MV . 1993 . Stability estimates for ill-posed problems involving hyperboilc equations and inequalities . Appl. Anal. , 50 : 93 – 102 .
   Google Scholar
• Klibanov , MV and Malinsky , J . 1991 . Newton-Kantorovich method for 3-dimensional inverse scattering problem and stability of the hyperbolic Cauchy problem with time-dependent data . Inverse Prob. , 7 : 577 – 595 .
   Web of Science ®Google Scholar
• Klibanov , MV and Timonov , A . 2004 . Carleman Estimates for Coefficient Inverse Problems and Numerical Applications , Utrecht : VSP .
   Google Scholar
• Klibanov , MV and Rakesh , MV . 1992 . Numerical solution of a timelike Cauchy problem for the wave equation . Math. Methods Appl. Sci. , 15 : 559 – 570 .
   Web of Science ®Google Scholar
• Klibanov , MV . 2005 . Lipschitz stability for hyperbolic inequalities in octants with the lateral Cauchy data and refocusing in time reversal . J. Inverse Ill-Posed Prob. , 13 : 353 – 363 .
   Google Scholar
• Fat Ho , Lop . 1986 . Observabilité frontieré de l’équation des ondes . C.R. Acad. Sci. 302 , 12 : 443 – 446 .
   Google Scholar
• Romanov , VG . 2006 . Carleman estimates for second order hyperbolic equations . Sib. Math. J. , 22 : 135 – 151 .
   Google Scholar
• Romanov , VG . 2006 . Stability estimates in inverse problems for hyperbolic equations . Milan J. Math. , 74 : 357 – 385 .
   Google Scholar
• Triggiani , R and Yao , PF . 2006 . Carleman estimates with no lower order terms for general Riemann wave equations. Global uniqueness and stability in one shot . Appl. Meth. Optim. , 46 : 334 – 375 .
   Google Scholar
• Isakov , V . 2006 . Inverse Prob. for Partial Differential Equations , New York : Springer .
   Google Scholar
• Kabanikhin , SI , Bektemesov , MA and Nechaev , DV . 2005 . Numerical solution of the 2D thermoacoustic problem . J. Inverse Ill-Posed Prob. , 13 : 265 – 276 .
   Google Scholar
• Klibanov , MV , Kabanikhin , SI and Nechaev , DV . 2006 . Numerical solution of the problem of computational time reversal in a quadrant . Waves Random Complex Media , 16 : 473 – 494 .
   Web of Science ®Google Scholar
• Lattes , R and Lions , J-L . 1969 . Applications to Partial Differential Equations , New York : Elsevier .
   Google Scholar
• Klibanov , MV and Santosa , F . 1991 . A computational quasi-reversibility method for Cauchy problems for Laplace's equation . SIAM J. Appl. Math. , 31 : 1653 – 1675 .
   Google Scholar
• Burgeois , L . 2005 . A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation . Inverse Prob. , 21 : 1087 – 1104 .
   Google Scholar
• Burgeois , L . 2006 . Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation . Inverse Prob. , 22 : 413 – 430 .
   Google Scholar
• Klibanov , MV and Danilaev , PG . 1990 . On the solution of coefficient inverse problems by the method of quasi-inversion . Soviet Math. Doklady , 41 : 83 – 87 .
   Google Scholar
• Kuchment , P and Kunyansky , L . 2007 . Mathematics of thermoacoustic tomography preprint, Available at arXiv: 0704.0286v2 [math.AP], 21 October 2007
   Google Scholar
• Xu , M , Feng , D and Wang , LV . 2003 . Time-domain reconstruction algorithms and numerical simulations for thermoacoustic tomography in various geometries . IEEE Trans. Biomed. Eng. , 50 : 1086 – 1099 .
   PubMed Web of Science ®Google Scholar
• Agranovsky , M and Kuchment , P . 2007 . Uniqueness of reconstruction and an inversion procedure for thermoacoustic tomography with variable sound speed . Inverse Prob. , 23 : 2089 – 2102 .
   Google Scholar
• Finch , D , Patch , S and Rakesh , MV . 2004 . Determining a function from its mean values over a family of spheres . SIAM J. Math. Anal. , 35 : 1213 – 1240 .
   Web of Science ®Google Scholar
• Kunyansky , L . 2007 . Explicit inversion formulae for the spherical mean Radon transform . Inverse Prob. , 23 : 373 – 383 .
   Web of Science ®Google Scholar
• Lavrentév , MM , Romanov , VG and Shishatskii , SP . 1986 . Ill-Posed Problems of Mathematicl Physics and Analysis , Providence, RI : AMS .
   Google Scholar
• Romanov , VG . 1974 . Integral Geometry and Inverse Problems for Hyperbolic Equations , New York : Springer .
   Google Scholar