Band 25, Heft 6

In order to obtain regularized approximations for the solution q of the parameter identification problem -∇.(q⁢∇⁡u)=f{-\nabla.(q\nabla u)=f} in Ω along with the Neumann boundary condition q⁢∂⁡u∂⁡ν=g{q\frac{\partial u}{\partial\nu}=g} on ∂⁡Ω{\partial\Omega}, which is an ill-posed problem, we consider its weak formulation as a linear operator equation with operator as a function of the data u∈W1,∞⁢(Ω){u\in W^{1,\infty}(\Omega)}, and then apply the Tikhonov regularization and a finite-dimensional approximation procedure when the data is noisy. Here, Ω is a bounded domain in ℝd{{\mathbb{R}}^{d}} with Lipschitz boundary, f∈L2⁢(Ω){f\in L^{2}(\Omega)} and g∈H-1/2⁢(∂⁡Ω){g\in H^{-1/2}(\partial\Omega)}. This approach is akin to the equation error method of Al-Jamal and Gockenback (2012) wherein error estimates are obtained in terms of a quotient norm, whereas our procedure facilitates to obtain error estimates in terms of the regularization parameters and data errors with respect to the norms of the spaces under consideration. In order to obtain error estimates when the noisy data belongs to L2⁢(Ω){L^{2}(\Omega)} instead of W1,∞⁢(Ω){W^{1,\infty}(\Omega)}, we shall make use of a smoothing procedure using the Clement operator under additional assumptions of Ω and u .

We investigate a quantitative thermoacoustic tomography process. We aim to recover the electric susceptibility and the conductivity of a medium when the sources are in the microwaves range. We focus on the case where the source signal has a slow time-varying envelope. We present the direct problem coupling equations for the electric field, the temperature variation and the pressure (to be measured via sensors). Then we give a variational formulation of the inverse problem which takes into account the entire electromagnetic, thermal and acoustic coupled system, and perform the formal computation of the optimality system.

In this work the following problem is discussed: Unknown are two objects, a probability density (corresponding to a distribution, or to a probability measure) f and its support supp⁡{f}=[L,U]{\operatorname{supp}\{f\}=[L,U]} which is assumed to be bounded/compact. Available is the sequence of all integer order moments of f and the goal is to recover the support [L,U]{[L,U]} or both the support [L,U]{[L,U]} and the density f . This paper provides the sequences involving integer order moments that approximate the compact support of a probability measure. The proposed sequences are based on the ratios of subsequent moments and they converge faster than the previously suggested sequences representing the n -th roots of n -th moments. As a result, the approximation of a probability density function with unknown support is proposed. The convergence of newly defined approximations of the boundaries and the densities are demonstrated through various examples, including the case of noisy data.

A probabilistic collocation based polynomial chaos expansion method is developed to solve stochastic boundary value problems with random coefficients and randomly distributed initial data. In this paper we deal with two different boundary value problems with random data: the Darcy equation with random lognormally distributed hydraulic conductivity, and a diffusion equation with absorption, with random distribution of the initial concentration under periodic boundary conditions. Special attention is paid to the extension of the probabilistic collocation method to input data with arbitrary correlation functions defined both analytically and through measurements. We construct the relevant Karhunen–Loève expansion from a special randomized singular value decomposition of the correlation matrix, which makes possible to treat problems of high dimension. We show that the unknown statistical characteristics of the random input data can be recovered from the correlation analysis of the solution field.

A final and maybe the simplest formulation of the enclosure method applied to inverse obstacle problems governed by partial differential equations in a spatial domain with an outer boundary over a finite time interval is presented. The method employs only a single pair of quite natural Neumann data prescribed on the outer boundary and the corresponding Dirichlet data on the same boundary of the solution of the governing equation over a finite time interval, that is, a single point on the graph of the so-called response operator . It is shown that the methods enables us to extract the distance of a given point outside the domain to an embedded unknown obstacle, that is, the maximum sphere centered at the point whose exterior encloses the unknown obstacle. To make the explanation of the idea clear only an inverse obstacle problem governed by the wave equation is considered.

In this paper, we analyze the use of simple models for solving the inverse problem in electrocardiography (IPE), which aims at recovering the heart condition from a set of remote voltages measurements. Specifically, we consider here the problem of estimating the shape, size and location of cardiac ischemic regions. The forward problem to generate the data (voltage measurements) is formulated by using the Luo–Rudy model, which provides a detailed description of the electrical behavior of cardiac cells. As for the inversion process, we use the two-current phenomenological model. The inversion procedure also incorporates a semi-automatic stage to characterize the conduction properties of the cardiac tissue. The ischemic regions are modeled by using standard level set techniques. Numerical results show that the algorithm is capable of estimating the position, size and shape of cardiac ischemic regions from noisy voltage measurements, for both 2D and 3D geometries. Our inverse procedure is benchmarked against zero-order Tikhonov regularization. This work is a proof of principle demonstrating the possibility of using simple models in the IPE towards realistic situations.

An inverse problem to determine a kernel in an evolutionary integral equation occurring in modeling of subdiffusion is considered. The existence, uniqueness and stability of a solution of the inverse problem are proved in an abstract setting. The results are global in time.

In this work, we consider inverse nodal problems of the Sturm–Liouville equation with nonlocal integral conditions at two end-points. We prove that a dense subset of nodal points uniquely determine the potential function of the Sturm–Liouville equation up to a constant.

MUltiple SIgnal Classification (MUSIC) is a well-known non-iterative location detection algorithm for small, perfectly conducting cracks in inverse scattering problems. However, when the applied wavenumbers are unknown, inaccurate locations of targets are extracted by MUSIC with inappropriate wavenumbers, a fact that has been confirmed by numerical simulations. To date, the reason behind this phenomenon has not been theoretically investigated. Motivated by this fact, we identify the structure of MUSIC-type imaging functionals with inappropriate wavenumbers by establishing a relationship with Bessel functions of order zero of the first kind. This result explains the reasons for inaccurate results. Various results of numerical simulations with noisy data support the identified structure of MUSIC.

Inverse problems of spectral analysis are studied for differential pencils on A-graphs. The uniqueness of recovering operators from their spectra is proved, and a constructive procedure for the solution of this class of inverse problems is provided.

This paper investigates the inverse scattering problem for a discrete Dirac system on the entire line with coefficients that stabilize to zero in one direction. We develop an algorithm for solving the inverse problem of reconstruction of coefficients. We derive a necessary and a sufficient condition on the scattering data so that the inverse problem is uniquely solvable.