Numerical solution of an ill-posed Cauchy problem for a quasilinear parabolic equation using a Carleman weight function

References

[1] Alifanov O. M., Inverse Heat Conduction Problems, Springer, New York, 1994. 10.1007/978-3-642-76436-3Search in Google Scholar

[2] Alifanov O. M., Artukhin E. A. and Rumyantcev S. V., Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems, Begell House, New York, 1995. 10.1615/978-1-56700-038-2.0Search in Google Scholar

[3] Andrieux S. and Baranger T. N., Solution of nonlinear Cauchy problem for hyperelastic solids, Inverse Problems 31 (2015), Article ID 115003. 10.1088/0266-5611/31/11/115003Search in Google Scholar

[4] Bastay G., Kozlov V. A. and Turesson B. O., Iterative methods for an inverse heat conduction problem, J. Inverse Ill-Posed Probl. 9 (2001), 375–388. 10.1515/jiip.2001.9.4.375Search in Google Scholar

[5] Becache E., Bourgeois L., Franceschini L. and Dardé J., Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case, Inverse Probl. Imaging 9 (2015), 971–1002. 10.3934/ipi.2015.9.971Search in Google Scholar

[6] Beilina L. and Klibanov M. V., Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012. 10.1007/978-1-4419-7805-9Search in Google Scholar

[7] Beilina L. and Klibanov M. V., Globally strongly convex cost functional for a coefficient inverse problem, Nonlinear Anal. Real World Appl. 22 (2015), 272–288. 10.1016/j.nonrwa.2014.09.015Search in Google Scholar

[8] Berntsson F., Kozlov V. A., Mpinganzima L. and Turesson B. O., An alternating iterative procedure for the Cauchy problem for the Helmholtz equation, Inverse Probl. Sci. Eng. 22 (2014), 45–62. 10.1080/17415977.2013.827181Search in Google Scholar

[9] Bourgeois L. and Dardé J., A quasi-reversibility approach to solve the inverse obstacle problem, Inverse Probl. Imaging 4 (2010), 351–377. 10.3934/ipi.2010.4.351Search in Google Scholar

[10] Bukhgeim A. L. and Klibanov M. V., Uniqueness in the large of a class of multidimensional inverse problems, Soviet Math. Dokl. 17 (1981), 244–247. Search in Google Scholar

[11] Dardé J., Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems, Inverse Probl. Imaging 10 (2016), 379–407. 10.3934/ipi.2016005Search in Google Scholar

[12] Eldén L., Numerical solution of the sideways heat equation by difference approximation in time, Inverse Problems 11 (1995), 913–923. 10.1088/0266-5611/11/4/017Search in Google Scholar

[13] Eldén L. and Berntsson F. and Regińska T., Wavelet and Fourier methods for solving the sideways heat equation, SIAM J. Sci. Comput. 21 (2000), 2187–2205. 10.1137/S1064827597331394Search in Google Scholar

[14] Hào D. N. and Lesnic D., The Cauchy problem for Laplace’s equation via the conjugate gradient method, IMA J. Appl. Math. 65 (2000), 199–217. 10.1093/imamat/65.2.199Search in Google Scholar

[15] Imanuvilov O. Y. and Yamamoto M., Global Lipschitz stability in an inverse hyperbolic problem by interior observations, Inverse Problems 17 (2001), 717–728. 10.1088/0266-5611/17/4/310Search in Google Scholar

[16] Imanuvilov O. Y. and Yamamoto M., Global uniqueness and stability in determining coefficients of wave equations, Comm. Partial Differential Equations 26 (2001), 1409–1425. 10.1081/PDE-100106139Search in Google Scholar

[17] Isakov V., Carleman estimates and applications to inverse problems, Milan J. Math. 72 (2004), 249–271. 10.1007/s00032-004-0033-6Search in Google Scholar

[18] Isakov V., Inverse Problems for Partial Differential Equations, 2nd ed., Springer, New York, 2006. Search in Google Scholar

[19] Kabanikhin S. I., Inverse and Ill-Posed Problems, De Gruyter, Berlin, 2012. 10.1515/9783110224016Search in Google Scholar

[20] Kabanikhin S. I. and Karchevsky A. L., Optimizational method for solving the Cauchy problem for an elliptic equation, J. Inverse Ill-Posed Probl. 3 (1955), 21–46. 10.1515/jiip.1995.3.1.21Search in Google Scholar

[21] Karchevsky A. L., Marchuk I. V. and Kabov O. A., Calculation of the heat flux near the liquid–gas–solid contact line, Appl. Math. Model. 40 (2013), 1029–1037. 10.1016/j.apm.2015.06.018Search in Google Scholar

[22] Klibanov M. V., Global convexity in a three-dimensional inverse acoustic problem, SIAM J. Math. Anal. 28 (1997), 1371–1388. 10.1137/S0036141096297364Search in Google Scholar

[23] Klibanov M. V., Carleman estimates for global uniqueness, stability and numerical methods for coe cient inverse problems, J. Inverse Ill-Posed Probl. 21 (2013), 477–560. 10.1515/jip-2012-0072Search in Google Scholar

[24] Klibanov M. V., Carleman estimates for the regularization of ill-posed Cauchy problems, Appl. Numer. Math. 94 (2015), 46–74. 10.1016/j.apnum.2015.02.003Search in Google Scholar

[25] Klibanov M. V., Carleman weight functions for solving ill-posed Cauchy problems for quasilinear PDEs, Inverse Problems 31 (2015), Article ID 125007. 10.1088/0266-5611/31/12/125007Search in Google Scholar

[26] Klibanov M. V. and Kamburg V. G., Globally strictly convex cost functional for an inverse parabolic problem, Math. Methods Appl. Sci. 39 (2016), 930–940. 10.1002/mma.3531Search in Google Scholar

[27] Klibanov M. V. and Thành N. T., Recovering of dielectric constants of explosives via a globally strictly convex cost functional, SIAM J. Appl. Math. 75 (2015), 518–537. 10.1137/140981198Search in Google Scholar

[28] Klibanov M. V. and Timonov A., Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, 2004. 10.1515/9783110915549Search in Google Scholar

[29] Ladyzhenskaya O. A., Solonnikov V. A. and Ural’ceva N. N., Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, 1968. 10.1090/mmono/023Search in Google Scholar

[30] Lattes R. and Lions J.-L., The Method of Quasireversibility: Applications to Partial Differential Equations, Elsevier, New York, 1969. Search in Google Scholar

[31] Lavrentiev M. M., Romanov V. G. and Shishatskii S. P., Ill-Posed Problems of Mathematical Physics and Analysis, American Mathematical Society, Providence, 1986. 10.1090/mmono/064Search in Google Scholar

[32] Rischette R., Baranger T. N. and Debit N., Numerical analysis of an energy-like minimization method to solve a parabolic Cauchy problem with noisy data, J. Comput. Appl. Math. 271 (2014), 206–222. 10.1016/j.cam.2014.03.024Search in Google Scholar

[33] Tikhonov A. N., Goncharsky A. V., Stepanov V. V. and Yagola A. G., Numerical Methods for the Solution of Ill-Posed Problems, Kluwer, London, 1995. 10.1007/978-94-015-8480-7Search in Google Scholar

[34] Titarenko V. N. and Yagola A. G., Cauchy problems for Laplace equation on compact sets, Inverse Probl. Sci. Eng. 10 (2002), 235–254. 10.1080/10682760290007462Search in Google Scholar

[35] Triggiani R. and Zhang Z., Global uniqueness and stability in determining the electric potential coefficient of an inverse problem for Schrödinger equations on Riemannian manifolds, J. Inverse Ill-Posed Probl 23 (2015), 587–609. 10.1515/jiip-2014-0003Search in Google Scholar

[36] Yamamoto M., Carleman estimates for parabolic equations and applications. Topical review, Inverse Problems 25 (2009), Article ID 123013. 10.1088/0266-5611/25/12/123013Search in Google Scholar