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

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.

On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates

“…We begin with an interior regularity result. (7) is such that for all ζ ∈ C ∞ 0 (Ω), ζu ε and ζλ ε belong to H 2 (Ω) and there exists a constant C > 0 which depends only on the geometry such that…”

“…Let us now establish a global regularity estimate (up to the boundary) in the restricted case when Γ ∩Γ = ∅ (see Figure 1 right). (7) is such that u ε and λ ε belong to H 2 (Ω) and there exists a constant C > 0 which depends only on the geometry such that…”

“…To prove such theorem, we need the following Lemma, which establishes an equivalent weak formulation to problem (1) and which is proved in [7]. Lemma 1.1.…”

“…For all f ∈ L 2 (Ω) and ε > 0, the problem (7) has a unique solution (u ε , λ ε ) ∈ V 0 ×Ṽ 0 . There exists a constant C which depends only on the geometry such that…”

An inverse space-dependent source problem for hyperbolic equations and the Lipschitz-like convergence of the quasi-reversibility method

“…For the numerical solutions of ISPs, the widely used approach is optimization. We draw the reader's attention to several important publications [12,22,23,24,29,46,48], in some of which good numerical results using optimal control were obtained. In particular, in [29] the ISP, which is similar with the one of the current paper, is considered, a numerical method is proposed and implemented.…”