Volume 15, Issue 2

We compare an inverse problem approach to parameter estimation with homogenization techniques for characterizing the electrical response of composite dielectric materials in the time domain. We first consider an homogenization method, based on the periodic unfolding method, to identify the dielectric response of a complex material with heterogeneous micro-structures which are described by spatially periodic parameters. We also consider electromagnetic interrogation problems for complex materials assuming multiple polarization mechanisms with distributions of parameters. An inverse problem formulation is devised to determine effective polarization parameters specific to the interrogation problem. We compare the results of these two approaches with the classical Maxwell-Garnett mixing model and a simplified model with a weighted average of parameters. Numerical results are presented for a specific example involving a mixture of ethanol and water (modeled with multiple Debye mechanisms). A comparison between each approach is made in the frequency domain (e.g., Cole-Cole diagrams), as well as in the time domain (e.g., plots of susceptibility kernels).

We present a strategy for choosing the regularization parameter (Lepskij-type balancing principle) for ill-posed problems in metric spaces with deterministic or stochastic noise. Additionally we improve the strategy in comparison to the previously used version for Hilbert spaces in some ways.

We suggest a variational model which allows to both denoise and preserve edges on images. Using duality turns the problem into a simpler one, that is analyzed and numerically solved by appropriate methods. Some numerical results for image restoration and edges reconstruction are eventually presented.

Computational results for the convexification algorithm of the authors are presented in the 2-Dimensional case. Convexification is a numerical method for some multidimensional coefficient inverse problems with rigorously guaranteed global convergence.

Inverse heat conduction problems (IHCPs) have been extensively studied over the last 50 years. They have numerous applications in many branches of science and technology. The problem consists in determining the temperature and heat flux at inaccessible parts of the boundary of a 2- or 3-dimensional body from corresponding data — called 'Cauchy data' — on accessible parts of the boundary. It is well known that IHCPs are severely illposed which means that small perturbations in the data may cause extremely large errors in the solution. In this contribution we first present the problem and show examples of calculations for 2-dimensional IHCP's where the direct problems are solved with the Finite Element package DEAL. As solution procedure we use Tikhonov's regularization in combination with the conjugate gradient method.

We present a novel mathematical algorithm for the characterization of non-conventional reservoirs which contain regions of low sensitivity. Our algorithm uses a level set representation of shapes describing different lithofacies in the reservoir. These shapes need to be reconstructed from the production data using a two-phase flow model. In order to deal with regions of low sensitivity, topological perturbations are applied successively during the reconstruction in these low sensitivity regions, and the level set technique will evolve the so created shapes following a gradient direction that minimizes the mismatch between the computed and the production data. New shapes created at wrong locations tend to disappear gradually, whereas those created at locations where a lithofacie is present tend to grow until they approximately match the correct boundaries. We will discuss different strategies and present numerical results which demonstrate and compare their performances for two realistic 2D test cases.