Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case
Eliane Bécache - Laboratoire POEMS, ENSTA ParisTech, 828, Boulevard des Maréchaux, 91762, Palaiseau Cedex, 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.
Received: November 2014; Revised: June 2015; Available Online: October 2015.
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