TY - JOUR

T1 - Numerical solution of an ill-posed Cauchy problem for a quasilinear parabolic equation using a Carleman weight function

AU - Klibanov, Michael V.

AU - Koshev, Nikolaj A.

AU - Li, Jingzhi

AU - Yagola, Anatoly G.

PY - 2016/12/1

Y1 - 2016/12/1

N2 - 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.

AB - 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.

KW - Carleman weight function

KW - Ill-posed Cauchy problem

KW - numerical solution

KW - quasilinear parabolic PDE

UR - http://www.scopus.com/inward/record.url?scp=84999635661&partnerID=8YFLogxK

U2 - 10.1515/jiip-2016-0039

DO - 10.1515/jiip-2016-0039

M3 - Article

AN - SCOPUS:84999635661

VL - 24

SP - 761

EP - 776

JO - Journal of Inverse and Ill-Posed Problems

JF - Journal of Inverse and Ill-Posed Problems

SN - 0928-0219

IS - 6

ER -