Journal
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
Volume 34, Issue -, Pages 201-224Publisher
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.nonrwa.2016.08.008
Keywords
Global strict convexity; Existence of the minimizer; Carleman Weight Function; Ill-posed Cauchy problems; Quasilinear PDEs
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Funding
- Russian Foundation for Basic Research [15-01-00026, 16-01-00039]
- US Army Research Laboratory
- US Army Research Office [W911NF-15-1-0233]
- Office of Naval Research [N00014-15-1-2330]
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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). (C) 2016 Elsevier Ltd. All rights reserved.
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