Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: the 1D case
This is a fairly extensive body of work on numerical aspects around the quasi-reversibility method, which was introduced in the seminal monograph by \textit{R. Lattès} and \textit{J. L. Lions} [Méthode de quasi-réversibilité et applications. Paris: Dunod (1967; Zbl 0159.20803)]. Regularization of ill-posed problems is a wide field and the methods for finding stable approximate solutions are mostly rather generally applicable, like in the cases of Tikhonov and Lavrent'ev regularization as variants of variational regularization or in diverse forms of iterative regularization. But it is a piece of luck if one has an adapted and appropriate method for a specific problem class. This is just the case for regularizing the heat equation problem backward in time and similar problems of the wave equation when the quasi-reversibility method is applied. The singular perturbation of second order differential operators by higher (e.g. fourth) order differential operators delivers stabilization and convergence if the regularization parameter tends to zero, but unfortunately the numerical implementation by exploiting finite element techniques is much more complicated in such a case. In order to avoid exotic finite elements (e.g. of Hermitian type), which are not always available in numerical codes, the authors introduce and extend so-called mixed formulations for solving the inverse problems in the context of heat and wave equations. The focus is on the 1D case taking into account that this is a toy problem for a possible future extension to higher dimensions.NEWLINENEWLINE NEWLINE After a helpful introduction the Section 2 presents first the mixed formulation approach for the backward heat equation and then an adaption to the heat equation with lateral Cauchy data. In a third subsection of Section 2 the inverse obstacle problem is discussed, whereas Section 3 is devoted to ill-posed problems for the wave equation. Also for the wave equation with lateral data the solution is based on the presentation of a mixed formulation. The paper is completed by Section 4 on the discretization of the suggested mixed formulations and Section 5 presenting numerical case studies.
scientific article; zbMATH DE number 6552765
scientific article