Liviu Marin

We investigate the reconstruction of the solution to the Navier system of linear elastostatics in a doubly connected domain from the knowledge of both the displacements (Dirichlet data) and the tractions (Neumann data), i.e. Cauchy data, on an accessible part of the boundary. An iterative algorithm is provided to generate a stable numerical approximation to the solution of this inverse ill-posed problem by extending the findings of Baravdish et al. (2018) and Marin and Lesnic (2005) to linear elastostatics and Cauchy data from the corresponding fractional Sobolev spaces, respectively. The convergence of the proposed algorithm is proved by reformulating the Cauchy problem as an operator equation in appropriate quotient spaces and, equivalently, a corresponding fixed point problem for an affine operator. This iterative algorithm has a self-regularising character for exact data and is regularised/stabilised via the discrepancy principle (Morozov, 1966) together with Nemirovskii's rule Nemirovskii, 1987) for perturbed data. Numerical experiments are carried out for three-dimensional isotropic linear materials, using the finite element method, in order to confirm the feasibility of this approach.

A comprehensive convergence and stability analysis of some probabilistic numerical methods designed to solve Cauchy-type inverse problems is performed in this study. Such inverse problems aim at solving an elliptic partial differential equation (PDE) or a system of elliptic PDEs in a bounded Euclidean domain, subject to incomplete boundary and/or internal conditions, and are usually severely ill-posed. In a very recent paper \cite{CiGrMaI}, a probabilistic numerical framework has been developed by the authors, wherein such inverse problems could be analysed thoroughly by simulating the spectrum of some corresponding direct problem and its singular value decomposition based on stochastic representations and Monte Carlo simulations. Herein a full probabilistic error analysis of the aforementioned methods is provided, whereas the convergence of the corresponding approximations is proved and explicit error bounds are provided. This is achieved by employing tools from several areas such as spectral theory, regularity theory for elliptic measures, stochastic representations, and concentration inequalities.

We study the reconstruction of an unknown/inaccesible smooth inner boundary from the knowledge of the Dirichlet condition (temperature) on the entire boundary of a doubly connected domain occupied by a two-dimensional homogeneous anisotropic solid and an additional Neumann condition (normal heat flux) on the known, accessible, and smooth outer boundary in the framework of steady-state heat conduction with heat sources. This inverse geometric problem is approached through an operator that maps an admissible inner boundary belonging to the space of 2π−periodic and twice continuously differentiable functions into the Neumann data on the outer boundary which is assumed to be continuous. We prove that this operator is differentiable and hence a gradient-based method that employs the anisotropic single layer representation of the solution to an appropriate Dirichlet problem for the two-dimensional anisotropic heat conduction, is developed for approximating the unknown inner boundary. Numerical results are presented for both exact and perturbed Neumann data on the outer boundary and show the convergence, stability, and robustness of the proposed method.

We study the recovery of the missing discontinuous/non-smooth thermal boundary conditions on an inaccessible portion of the boundary of the domain occupied by a solid from Cauchy data prescribed on the remaining boundary assumed to be accessible, in case of stationary anisotropic heat conduction with non-smooth/discontinuous conductivity coefficients. This inverse boundary value problem is ill-posed and, therefore, should be regularized. Consequently, a stabilising method is developed based on a priori knowledge on the solution to this inverse problem and the smoothing feature of the direct problems involved. The original problem is transformed into a control one which reduces to solving an appropriate minimisation problem in a suitable function space. The latter problem is tackled by employing an appropriate variational method which yields a gradient-type iterative algorithm that consists of two direct problems and their corresponding adjoint ones. This approach yields an algorithm designed to approximate specifically merely L²–boundary data in the context of a non-smooth/discontinuous anisotropic conductivity tensor, hence both the notion of solution to the direct problems involved and the convergence analysis of the approximate solutions generated by the algorithm proposed require special attention. The numerical implementation is realised for two-dimensional homogeneous anisotropic solids using the finite element method, whilst regularization is achieved by terminating the iteration according to two stopping criteria.