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

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.

Recovering the initial condition of parabolic equations from lateral Cauchy data via the quasi-reversibility method

“…In fact, proving the "accuracy" of (18) when N → ∞ is extremely challenging and is out of the scope of this paper. However, we experience in many earlier works that the solution of (18), (19) and (20) well approximates Fourier coefficients of the function u(x, t), leading to good solutions of variety kinds of inverse problems, see [10,16,17,32]. Figure 1 displays the functions of u(x, t) and its approxi-…”

“…Assume that the set of admissible data H, defined in (22), is nonempty. Then, for all > 0, the functional J admits a unique minimizer satisfying (19) and (20). This minimizer is called the regularized solution to (18), (19) and (20).…”

“…The minimizer of J in H is called the regularized solution of (18), (19) and (20) obtained by the quasi-reversibility method.…”

“…We call it an approximate model. Solving it, together with the "over-determined" boundary conditions (19) and (20), for the Fourier coefficients (u n (x)) N n=1 of u(x, t), x ∈ Ω, t ∈ [0, T ], might not be rigorous. In fact, proving the "accuracy" of (18) when N → ∞ is extremely challenging and is out of the scope of this paper.…”

“…It is used to computed numerical solutions to ill-posed problems for partial differential equations. Due to its strength, since then, the quasi-reversibility method attracts the great attention of the scientific community see e.g., [19][20][21][22][23][24][25][26][27][28]. We refer the reader to [29] for a survey on this method.…”