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

Michael V. Klibanov, Nikolaj A. Koshev, Jingzhi Li, Anatoly G. Yagola

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)

Abstract

We solve numerically the side Cauchy problem for a 1-D parabolic equation. The initial condition is unknown. This is an ill-posed problem. The main difference with previous results is that our equation is quasilinear, whereas known publications on this topic work only with linear PDEs. The key idea is to minimize a weighted Tikhonov functional with the Carleman Weight Function (CWF) in it. Roughly, given a reasonable bounded set of any size in a reasonable Hilbert space, one can choose the parameter of the CWF in such a way that this functional becomes strictly convex on that set.

Original languageEnglish
Pages (from-to)761-776
Number of pages16
JournalJournal of Inverse and Ill-Posed Problems
Volume24
Issue number6
DOIs
Publication statusPublished - 1 Dec 2016
Externally publishedYes

Keywords

  • Carleman weight function
  • Ill-posed Cauchy problem
  • numerical solution
  • quasilinear parabolic PDE

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