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A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems

Laurent Bourgeois 1, * Arnaud Recoquillay 2 
* Corresponding author
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : This paper is dedicated to a new way of presenting the Tikhonov regularization in the form of a mixed formulation. Such formulation is well adapted to the regularization of linear ill-posed partial differential equations because when it comes to discretization, the mixed formulation enables us to use some standard finite elements. As an application of our theory, we consider an inverse obstacle problem in an acoustic waveguide. In order to solve it we use the so-called “exterior approach”, which couples the mixed formulation of Tikhonov regularization and a level set method. Some 2d numerical experiments show the feasibility of our approach.
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Laurent Bourgeois, Arnaud Recoquillay. A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems. ESAIM: Mathematical Modelling and Numerical Analysis, 2018, 52 (1), pp.123-145. ⟨10.1051/m2an/2018008⟩. ⟨hal-02922747⟩

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