Volume 26, Issue 4

We obtain rate of convergence estimates for approximations delivered by Tikhonov’s scheme of solving ill-posed nonconvex optimization problems in a Hilbert space. The problems under investigation involve minimization of twice Frechet differentiable functionals given with errors on a closed convex set having a nonempty interior and smooth boundary. Assuming that the desired solution satisfies an appropriate source condition which includes the second derivative of the cost functional and depends on the geometry of constraints near the solution, we establish accuracy estimates in terms of the error level. Both the a priori and a posteriori parameter choice rules are analyzed.

An inverse coefficient problem for time-dependent wave equation in three dimensions is under consideration. We are looking for a spatially varying coefficient of this equation knowing special time integrals of the wave field in an observation domain. The inverse problem has applications to the reconstruction of the refractive index of an inhomogeneous medium, as well as to acoustic sounding, medical imaging, etc. In the article, a new linear three-dimensional Fredholm integral equation of the first kind is introduced from which it is possible to find the unknown coefficient. We present and substantiate a numerical algorithm for solving this integral equation. The algorithm does not require significant computational resources and a long solution time. It is based on the use of fast Fourier transform under some a priori assumptions about unknown coefficient and observation region of the wave field. Typical results of solving this three-dimensional inverse problem on a personal computer for simulated data demonstrate high capabilities of the proposed algorithm.

This article presents the solution of a special inverse elastography problem: knowing vertical displacements of compressed biological tissue to find a piecewise constant distribution of Young’s modulus in an investigated specimen. Our goal is to detect homogeneous inclusions in the tissue, which can be interpreted as oncological. To this end, we consider the specimen as two-dimensional elastic solid, displacements of which satisfy the differential equations of the linear static theory of elasticity in the plain strain statement. The inclusions to be found are specified by parametric functions with unknown geometric parameters and unknown Young’s modulus. Reducing this inverse problem to the search for all unknown parameters, we solve it applying the modified method of extending compacts by V. K. Ivanov and I. N. Dombrovskaya. A posteriori error estimate is carried out for the obtained approximate solutions.

This paper is concerned with the numerical solution to a three-dimensional coefficient inverse problem for buried objects with multi-frequency experimental data. The measured data, which are associated with a single direction of an incident plane wave, are backscatter data for targets buried in a sandbox. These raw scattering data were collected using a microwave scattering facility at the University of North Carolina at Charlotte. We develop a data preprocessing procedure and exploit a newly developed globally convergent inversion method for solving the inverse problem with these preprocessed data. It is shown that dielectric constants of the buried targets as well as their locations are reconstructed with a very good accuracy. We also prove a new analytical result which rigorously justifies an important step of the so-called “data propagation” procedure.

We consider the inverse problem of source determination in nonuniformly parabolic equation under the additional condition of integral observation. We investigate the questions of existence and uniqueness of solution. Two types of sufficient conditions for the unique solvability of the inverse problem are obtained. Examples of inverse problems are given for which the conditions of the proved theorems are fulfilled.

The problem of correlating the error estimates at a point and on a correctness class is of interest to many mathematicians. Since the desired solution to a real ill-posed problem is unique, the error estimate obtained on the class becomes crude. In this paper, by assuming that an exact solution is a piecewise smooth function, we prove, for a special class of incorrect problems, that an error estimate at a point is an infinitely small quantity compared with an exact estimate on a correctness set.

An applied inverse problem in the form of a nonlinear operator equation on a pair of two Hilbert spaces is considered. In the lack of stability, a version of Broyden’s method regularized by truncated singular value decomposition (TSVD) is proposed in order to suppress the potential noise propagation and to reduce the cost associated with storage and inversion of the Fréchet derivative operator. An attractive feature of this study is that it does not rely on any restrictions on the nonlinearity of the operator and/or on the spectrum of the Fréchet derivative. For the numerical analysis of the proposed algorithm, a parameter identification problem in epidemiology is investigated.