Approximate global convergence and quasireversibility for a coefficient inverse problem with backscattering data. (English. Russian original) Zbl 1252.35282
J. Math. Sci., New York 181, No. 2, 126-163 (2012); translation from Probl. Mat. Anal. 62, 19-49 (2011).
Summary: A numerical method possessing the approximate global convergence property is developed for a 3-D coefficient inverse problem for hyperbolic partial differential equations with backscattering data resulting from a single measurement. An important part of this technique is the quasireversibility method. An approximate global convergence theorem is proved. Results of two numerical experiments are presented.
MSC:
| 35R30 | Inverse problems for PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
References:
| [1] | L. Beilina and M. V. Klibanov, ”A globally convergent numerical method for a coefficient inverse problem,” SIAM J. Sci. Comput. 31, No. 1, 478–509 (2008). · Zbl 1185.65175 · doi:10.1137/070711414 |
| [2] | L. Beilina and M. V. Klibanov, ”Synthesis of global convergence and adaptivity for a hyperbolic coefficient inverse problem in 3D,” J. Inverse Ill-Posed Probl. 18, No. 1, 85–132 (2010). · Zbl 1279.65116 |
| [3] | L. Beilina and M. V. Klibanov. ”A posteriori error estimates for the adaptivity technique for the Tikhonov functional and global convergence for a coefficient inverse problem,” Inverse Probl. 26, No. 4, ID 045012 (2010). · Zbl 1193.65165 |
| [4] | L. Beilina, M. V. Klibanov, and M. Yu. Kokurin, ”Adaptivity with relaxation for ill-posed problems and global convergence for a coefficient inverse problem” [in Russian], Probl. Mat. Anal. 46, 3–44 (2010); English transl.: J. Math. Sci. (New York) 167, No. 3, 279–325 (2010). · Zbl 1286.65147 |
| [5] | L. Beilina and M. V. Klibanov, ”Reconstruction of dielectrics from experimental data via a hybrid globally convergent/adaptive algorithm,” Inverse Probl. 26, ID 125009 (2010). · Zbl 1209.78010 |
| [6] | L. Beilina and M. V. Klibanov, ”The philosophy of the approximate global convergence for multidimensional coefficient inverse problems,” Complex Variables and Elliptic Equations [To appear] · Zbl 1236.65120 |
| [7] | L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York (2012). [To appear] · Zbl 1255.65168 |
| [8] | M. V. Klibanov, M. A. Fiddy, L. Beilina, N. Pantong, and J. Schenk, ”Picosecond scale experimental verification of a globally convergent algorithm for a coefficient inverse problem,” Inverse Probl. 26, No. 4, ID 045003 (2010). · Zbl 1190.78006 · doi:10.1088/0266-5611/26/4/045003 |
| [9] | M. V. Klibanov, A. B. Bakushinsky, and L. Beilina, ”Why a minimizer of the Tikhonov functional is closer to the exact solution than the first guess,” J. Inverse Ill-Posed Probl. 19, 83–105 (2011). · Zbl 1279.35113 |
| [10] | A. V. Kuzhuget and M. V. Klibanov, ”Global convergence for a 1-D inverse problem with application to imaging of land mines,” Appl. Anal. 89, 125-157 (2010). · Zbl 1205.65258 · doi:10.1080/00036810903481166 |
| [11] | A. V. Kuzhuget, L. Beilina, M. V. Klibanov et al. Blind Backscattering Experimental Data Collected in the Field and an Approximately Globally Convergent Inverse Algorithm, Preprint (2011). http://www.ma.utexas.edu/mp_arc · Zbl 1250.35185 |
| [12] | A. V. Kuzhuget, N. Pantong, and M. V. Klibanov, ”A globally convergent numerical method for a coefficient inverse problem with backscattering data,” Methods Appl. Anal. 18, 47–68 (2011). · Zbl 1279.65114 |
| [13] | L. Beilina and C. Johnson. ”Hybrid FEM/FDM method for an Inverse Scattering Problem,” In: Numerical Mathematics and Advanced Applications. ENUMATH 2001, pp. 545–556, Springer, Berlin (2003). · Zbl 1043.65111 |
| [14] | M. V. Klibanov, J. Su, N. Pantong, H. Shan, and H. Liu, ”A globally convergent numerical method for an inverse elliptic problem of optical tomography,” Appl. Anal. 89, 861–891 (2010). · Zbl 1191.65146 · doi:10.1080/00036811003649157 |
| [15] | N. Pantong, J. Su, H. Shan, M. V. Klibanov, and H. Liu, ”Globally accelerated reconstruction algorithm for diffusion tomography with continuous-wave source in an arbitrary convex shape domain,” J. Optical Soc. Am. A 26, 456–472 (2009). · doi:10.1364/JOSAA.26.000456 |
| [16] | H. Shan, M. V. Klibanov, J. Su, N. Pantong, and H. Liu, ”A globally accelerated numerical method for optical tomography with continuous wave source,” J. Inverse Ill-Posed Probl. 16 765-792 (2008). · Zbl 1152.34312 |
| [17] | J. Su, M. V. Klibanov, Y. Liu, Z. Lin, N. Pantong, and H. Liu, An Inverse Elliptic Problem of Medical Optics with Experimental Data. Preprint (2011). http://www.ma.utexas.edu/mp_arc |
| [18] | N. V. Alexeenko, V. A. Burov, and O. D. Rumyantseva, ”Solution of three-dimensional acoustical inverse problem: II. Modified Novikov algorithm,” Acous. Phys. 54, 407–419 (2008). · doi:10.1134/S1063771008030172 |
| [19] | J. Bikowski, K. Knudsen, and J. L. Mueller, ”Direct numerical reconstruction of conductivities in three dimensions using scattering transforms,” Inverse Probl. 27 (2011). ID 015002. · Zbl 1232.78007 · doi:10.1088/0266-5611/27/1/015002 |
| [20] | V. A. Burov, S. A. Morozov, and O. D. Rumyantseva, ”Reconstruction of fine-scale structure of acoustical scatterers on large-scale contrast background,” Acous. Imaging 26, 231-238 (2002). · doi:10.1007/978-1-4419-8606-1_30 |
| [21] | V. A. Burov, N. V. Alekseenko, and O. D. Rumyantseva, ”Multifrequency generalization of the Novikov algorithm for the two-dimensional inverse scattering problem,” Acous. Phys. 55, 843–856 (2009). · doi:10.1134/S1063771009060190 |
| [22] | M. DeAngelo and J. L. Mueller, ”2-D bar reconstructions of human chest and tank using an improved approximation to the scattering transform,” Physiol. Measurement 31, 221-232 (2010). · doi:10.1088/0967-3334/31/2/008 |
| [23] | D. Isaacson, J. L. Mueller, J. C. Newell, and S. Siltanen, ”Imaging cardiac activity by the D-bar methods for electrical impedance tomography,” Physiol. Measurements 27, 43–50 (2006). |
| [24] | R. G. Novikov, ”Multidimensional inverse spectral problem for the equation{\(\Delta\)}{\(\psi\)}+(v(x)Eu(x)){\(\psi\)}=0,” Funkt. Anal. Prilozh. 2284, 11-22 (1988). |
| [25] | R. G. Novikov, ”The inverse scattering problem on a fixed energy level for the twodimensional Schrödinger operator,” J. Funct. Anal. 103, 409-463 (1992). · Zbl 0762.35077 · doi:10.1016/0022-1236(92)90127-5 |
| [26] | R. G. Novikov, ”The bar approach to approximate inverse scattering at fixed energy in three dimensions,” Int. Math. Res. Pap. 6, 287-349 (2005). · Zbl 1284.35464 · doi:10.1155/IMRP.2005.287 |
| [27] | R. G. Novikov, ”An effectivization of the global reconstruction in the Gel’fand–Calderon inverse problem in three dimensions,” Contemp. Math. 494, 161-184 (2009). · Zbl 1183.35271 · doi:10.1090/conm/494/09649 |
| [28] | A. B. Bakushinsky and M. Yu. Kokurin, Iterative Methods for Approximate Solutions of Inverse Problems, Springer, Dordrecht (2004). · Zbl 1070.65038 |
| [29] | B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, de Gruiter, New York (2008). · Zbl 1145.65037 |
| [30] | M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht (2004). · Zbl 1069.65106 |
| [31] | R. Lattes and J.-L. Lions, The Method of Quasireversibility: Applications to Partial Differential Equations, Elsevier, New York (1969). |
| [32] | L. Bourgeois and J. Darde, ”About stability and regularization of ill-posed elliptic Cauchy problems: the case of Lipschitz domains,” Appl. Anal. 89, 1745-1768 (2010). · Zbl 1206.35252 · doi:10.1080/00036810903393809 |
| [33] | L. Bourgeois and J. Darde, ”A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data,” Inverse Probl. 26 (2010). ID 095016. · Zbl 1200.35315 · doi:10.1088/0266-5611/26/9/095016 |
| [34] | H. Cao, M. V. Klibanov and S. V. Pereverzev, ”A Carleman estimate and the balancing principle in the quasi-reversibility method for solving the Cauchy problem for the Laplace equation,” Inverse Probl. 25 (2009). ID 35005. · Zbl 1169.35385 · doi:10.1088/0266-5611/25/3/035005 |
| [35] | C. Clason and M. V. Klibanov, ”The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium,” SIAM J. Sci. Comp. 30, 1-23 (2007). · Zbl 1159.65346 · doi:10.1137/06066970X |
| [36] | M. V. Klibanov and F. Santosa, ”A computational quasi-reversibility method for Cauchy problems for Laplace’s equation,” SIAM J. Appl. Math. 51, 1653-1675 (1991). · Zbl 0769.35005 · doi:10.1137/0151085 |
| [37] | D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin (1983). · Zbl 0562.35001 |
| [38] | H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Boston (2000). · Zbl 0859.65054 |
| [39] | M. M. Lavrentiev, V. G. Romanov, and S. P. Shishatskii, Ill-Posed Problems of Mathematical Physics and Analysis, Am. Math. Soc., Providence, RI (1986). |
| [40] | A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov and A. G. Yagola, Numerical Methods for Solutions of Ill-Posed Problems, Kluwer, London (1995). · Zbl 0831.65059 |
| [41] | M. V. Klibanov, ”Inverse problems and Carleman estimates,” Inverse Probl. 8, No. 4, 575–596 (1992). · Zbl 0755.35151 · doi:10.1088/0266-5611/8/4/009 |
| [42] | A. Friedman, Partial Differential Equations of Parabolic Type, Prentice Hall, Inc., Englewood Cliffs, N.J. (1964). · Zbl 0144.34903 |
| [43] | H. Bateman and A. Erdelyi, Tables of Integral Transforms. Vol. 1, McGrawHill, New York (1954). |
| [44] | O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uralceva, Linear and Quasilinear Equations Of Parabolic Type, Am. Math. Soc., Providence, RI (1968). |
| [45] | V. P. Mikhailov, Partial Differential Equations, M., Mir Publishers (1978). |
| [46] | Tasbles of Dielectric Constants. http://www.asiinstr.com/technical/Dielectric%20Constants.htm |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
