Carleman weight functions for a globally convergent numerical method for ill-posed Cauchy problems for some quasilinear PDEs

• Published in: Nonlinear analysis: real world applications Vol. 34; pp. 201 - 224
• Main Authors: Bakushinskii, Anatoly B., Klibanov, Michael V., Koshev, Nikolaj A.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 01.04.2017; Elsevier BV
• Subjects: Carleman Weight Function; Cauchy problems; Documents; Existence of the minimizer; Functionals; Global strict convexity; Ill posed problems; Ill-posed Cauchy problems; Inverse problems; Mathematical analysis; Numerical analysis; Parabolic differential equations; Quasilinear PDEs; Weighting functions; Carleman Weight Function; Ill-posed Cauchy problems; Global strict convexity; Existence of the minimizer; Quasilinear PDEs
• ISSN: 1468-1218; 1878-5719
• DOI: 10.1016/j.nonrwa.2016.08.008

In a series of publications of the second author, including some with coauthors, globally strictly convex Tikhonov-like functionals were constructed for some nonlinear ill-posed problems. The main element of such a functional is the presence of the Carleman Weight Function. Compared with previous publications, the main novelty of this paper is that the existence of the regularized solution (i.e. the minimizer) is proved rather than assumed. The method works for both ill-posed Cauchy problems for some quasilinear PDEs of the second order and for some Coefficient Inverse Problems. However, to simplify the presentation, we focus here only on ill-posed Cauchy problems. Along with the theory, numerical results are presented for the case of a 1-D quasilinear parabolic PDE with the lateral Cauchy data given on one edge of the interval (0, 1).