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Inverse Problems and Imaging (IPI)
 

Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case

Pages: 971 - 1002, Volume 9, Issue 4, November 2015      doi:10.3934/ipi.2015.9.971

 
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Eliane Bécache - Laboratoire POEMS, ENSTA ParisTech, 828, Boulevard des Maréchaux, 91762, Palaiseau Cedex, France (email)
Laurent Bourgeois - Laboratoire POEMS, ENSTA ParisTech, 828, Boulevard des Maréchaux, 91762, Palaiseau Cedex, France (email)
Lucas Franceschini - Laboratoire POEMS, ENSTA ParisTech, 828, Boulevard des Maréchaux, 91762, Palaiseau Cedex, France (email)
Jérémi Dardé - Institut de Mathématiques, Université de Toulouse, 118, Route de Narbonne, F-31062 Toulouse Cedex 9, France (email)

Abstract: In this paper we address some ill-posed problems involving the heat or the wave equation in one dimension, in particular the backward heat equation and the heat/wave equation with lateral Cauchy data. The main objective is to introduce some variational mixed formulations of quasi-reversibility which enable us to solve these ill-posed problems by using some classical Lagrange finite elements. The inverse obstacle problems with initial condition and lateral Cauchy data for heat/wave equation are also considered, by using an elementary level set method combined with the quasi-reversibility method. Some numerical experiments are presented to illustrate the feasibility for our strategy in all those situations.

Keywords:  Backward heat equation, heat/wave equation with lateral Cauchy data, inverse obstacle problem, quasi-reversibility method, let-set method, finite element method, mixed formulation.
Mathematics Subject Classification:  Primary: 35R25, 35R30, 35R35; Secondary: 65M60.

Received: November 2014;      Revised: June 2015;      Available Online: October 2015.

 References