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

被引：51

作者：

Bakushinskii, Anatoly B. ^{[1
]}

Klibanov, Michael V. ^{[2
]}

Koshev, Nikolaj A. ^{[3
]}

机构：

[1] Russian Acad Sci, Inst Syst Anal, Fed Res Ctr Comp Sci & Control, 60 Oktober Anniversary Ave,9, Moscow 117312, Russia

[2] Univ North Carolina Charlotte, Dept Math & Stat, Charlotte, NC 28223 USA

[3] Univ Sao Paulo, Inst Computat Math, BR-13566590 Sao Carlos, SP, Brazil

来源：

NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS | 2017年 / 34卷

基金：

俄罗斯基础研究基金会;

关键词：

Global strict convexity; Existence of the minimizer; Carleman Weight Function; Ill-posed Cauchy problems; Quasilinear PDEs;

D O I：

10.1016/j.nonrwa.2016.08.008

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

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.

引用

页码：201 / 224

页数：24

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