Object identification problems: numerical analysis

We aim at the problem of identification of geometric objects (defects, obstacles, scatterers) and reconstruction of ist geometric and physical parameters from boundary measurements. This is an important problem of high actual interest both from a theoretical and practical point of view. Object identification has numerous applications in the engineering and biomedical sciences in the context of non-destructive testing with acoustic, elastic, electromagnetic waves. From a mathematical point of view, object identification is an inverse problem, which belongs to the field of shape and topology optimization. We consider arbitrary shapes and on a-priori unknown boundary conditions of the test objects. This is certainly a challenging task in the framework of forward and inverse problems. In the proposal we focus primarily on ist numerical issues within an optimization context. In a broad scope, our goals in the project are split into core parts as follows: 1. For forward problems related to acoustic and wave phenomena we aim to get highly accurate interpolation of the variational solution within generalized finite elements. For this task we will elaborate quasilinear methods based on a Petrov-Galerkin enrichment. 2. We aim at topological sensitivity (topological derivatives) of objective functions with respect to geometric variables of test objects of arbitrary shapes and unknown boundary conditions. For this purpose we will utilize asymptotic methods of singular perturbation theory endowed with variational techniques. 3. For inverse problems, we aim to obtain high resolution properties of the object reconstruction and imaging with respect to discretization, varying and noisy data. This difficult task will be approached by means of optimality conditions based imaging functions and using the geometric concept of level sets. All our results in the project will be supported by rigorous estimates and numerical experiments. As the result we shall derive efficient numerical algorithms for use in inverse scattering, tomography, and other potential industrial applications.

Object identification problems: numerical analysis. The project addressed to the problem of identification of geometric objects (obstacles, inhomogeneities, cracks) and reconstruction of its geometric and physical parameters from optimization of costs (the energy, least-square misfit of observation data, boundary measurements, far-field pattern, and alike). This is an important problem of high actual interest both from a theoretical and practical point of view. The following research was conducted in the project: (i) For forward problems related to acoustic and wave phenomena, we developed a novel approach for interpolation of variational solutions of partial differential equations based on the Petrov-Galerkin enrichment in the context of generalized finite element methods. Our investigation resulted in the numerical schemes of high precision and low costs, which are of high importance for numerical optimization purposes. (ii) The optimization approach was applied to a range of problems governed by elliptic and parabolic partial differential equations. Their specialty concerned the presence of unilateral constraints and geometric singularities like cracks, inclusions, micro-particles in a pore space. Thus, we proved well-posedness of variational solutions for multi-phase diffusion systems stemming from electrokinetics and generalized solutions of nonlinear equations arising due to implicit constitutive theories in mechanics. (iii) Related to the underlying identification problems, the cost functions were investigated for its shape-topological sensitivity when varying geometric and physical variables. Forced by unilateral constraints, a class of geometry-dependent Lagrangians was analyzed with respect to the property of shape differentiability. For these purposes we extended the asymptotic methods of shape and topology perturbations, periodic homogenization, by endowing them with variational techniques. (iv) We investigated the object imaging in inverse problems for sensitivity and stability properties. With the help of imaging functions derived from the shape-topological analysis and zero-order (derivative-free) optimality conditions we brought the problems in a form suitable for robust and highly accurate computation with respect to discretization, varying and noisy data, which was demonstrated in numerical tests of identification of the geometric objects. All these results were supported by rigorous a-priori and residual error estimates. We stress that the issues (i)-(iv) were broader than just for the underlying forward and inverse problems. The overall results obtained under the project advanced our understanding important for the theory of forward and inverse problems, overdetermined and free-boundary problems, shape-topological control, nonlinear and numerical optimization; as well as for destructive and non-destructive testing, acoustic scattering, with transdisciplinary applications in engineering and biomedical sciences. During the 5-year project period (1.4.2014-31.3.2019) one PhD thesis was completed, we report on 26 publications (more 7 are submitted or in preparation), editing 1 book, participation in 68 meetings (39 presentations, 5 plenary, 6 co-organizations), 25 outcome (9 invited lectures) and 7 income visits, see https://static.uni-graz.at/fileadmin/_Persoenliche_Webseite/kovtunenko_victor/fwf-p26147/