An inverse backward problem for degenerate two-dimensional parabolic equation

This paper deals with the determination of an initial condition in the degenerate two-dimensional parabolic equation \[\partial_{t}u-\mathrm{div}\left(a(x,y)I_2\nabla u\right)=f,\quad (x,y)\in\Omega,\; t\in(0,T),\] where \(\Omega\) is an open, bounded subset of \(\mathbb{R}^2\), \(a \in C^1(\bar{\Omega})\) with \(a\geqslant 0\) everywhere, and \(f\in L^{2}(\Omega \times (0,T))\), with initial and boundary conditions \[u(x,y,0)=u_0(x,y), \quad u\mid_{\partial\Omega}=0,\] from final observations. This inverse problem is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. To show the convergence of the descent method, we prove the Lipschitz continuity of the gradient of the Tikhonov functional. Also we present some numerical experiments to show the performance and stability of the proposed approach.

An Inverse Source Problem for Singular Parabolic Equations with Interior Degeneracy

The main purpose of this work is to study an inverse source problem for degenerate/singular parabolic equations with degeneracy and singularity occurring in the interior of the spatial domain. Using Carleman estimates, we prove a Lipschitz stability estimate for the source term provided that additional measurement data are given on a suitable interior subdomain. For the numerical solution, the reconstruction is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. The Fréchet differentiability of the Tikhonov functional and the Lipschitz continuity of the Fréchet gradient are proved. These properties allow us to apply gradient methods for numerical solution of the considered inverse source problem.

Domain Decomposition Methods for Recovering Robin Coefficients in Elliptic and Parabolic Systems

We shall derive and propose several efficient domain decomposition methods for solving the nonlinear inverse problem of identifying the Robin coefficients in elliptic and parabolic systems. The highly ill-posed inverse problems are transformed into output least-squares nonlinear and non-convex minimizations with classical Tikhonov regularization. The Levenberg–Marquardt method is applied to transform the non-convex minimizations into convex minimizations, which will be solved by several efficient domain decomposition methods. The methods are completely local and the local minimizers have explicit expressions within the subdomains. Several numerical experiments are presented to show the accuracy and efficiency of the methods; in particular, the convergence seems nearly optimal in the sense that the iteration number of the methods is nearly independent of mesh sizes.