Band 26, Heft 5

This work is devoted to the identification of a mathematical model of bacteria population under the antibiotic influence, based on the solution of the corresponding inverse problems. These problems are solved by the gradient method, genetic algorithm and Nelder–Mead method. Calculations are made using model and real data.

Inverse problems for differential pencils with nonlocal conditions are considered. Uniqueness theorems of inverse problems from the Weyl-type function and spectra are proved, which are generalizations of the well-known Weyl function and Borg’s inverse problem for the classical Sturm–Liouville operators.

This paper deals with some inverse problems for the linear elasticity system with origin in elastography: we try to identify the material coefficients from some extra information on (a part of) the boundary. In our main result, we assume that the total variation of the coefficient matrix is a priori bounded. We reformulate the problem as the minimization of a function in an appropriate constraint set. We prove that this extremal problem possesses at least one solution with the help of some regularity results. Two crucial ingredients are a Meyers-like theorem that holds in the context of linear elasticity and a nonlinear interpolation result by Luc Tartar. We also perform some numerical experiments that provide satisfactory results. To this end, we apply the Augmented Lagrangian algorithm, completed with a limited-memory BFGS subalgorithm. Finally, on the basis of these experiments, we illustrate the influence of the starting guess and the errors in the data on the behavior of the iterates.

We consider the inverse scattering problem in a layer-homogeneous elastic structure filling a half-plane. Scattering data is the seismic wave field registered on the boundary with a surface point source. We prove that velocities of longitudinal and transverse waves, and a density of the medium are recovered by a unique way. The algorithm of the inverse problem solution is based on the τ- p Radon transform. Also, we present some results of numerical modeling for waves propagation and solving of the inverse problem.

This paper considers the problem of estimating the permeability in a nonlinear convection-diffusion equation. To overcome the large calculation burden of conventional methods, we apply an adaptive multigrid conjugate gradient method to solve this inverse problem. This new method combines the multigrid multiscale idea with the conjugate gradient method, and adopts the necessary condition that the optimum solution should be the fixed point of the multigrid inversion method. Some numerical results verify that the proposed method both dramatically reduces the required computations and improves the inversion quality.

In this paper, we give Ambarzumyan-type theorems for a Sturm–Liouville dynamic equation with Robin boundary conditions on a time scale. Under certain conditions, we prove that the potential can be specified from only the first eigenvalue.

We consider Tikhonov-type variational regularization of ill-posed linear operator equations in Banach spaces with general convex penalty functionals. Upper bounds for certain error measures expressing the distance between exact and regularized solutions, especially for Bregman distances, can be obtained from variational source conditions. We prove that such bounds are optimal in case of twisted Bregman distances, a specific a priori parameter choice, and low regularity of the exact solution, that is, the rate function is also an asymptotic lower bound for the error measure. This result extends existing converse results from Hilbert space settings to Banach spaces without adhering to spectral theory.

In this article, we consider a fractional diffusion equation of half order in time. We study inverse problems of determining the space-dependent factor in the source term from additional data at a fixed time and interior or boundary data over an appropriate time interval. We establish the global Lipschitz stability estimates in the inverse source problems. Our methods are based on Carleman estimates. Here we prove and use the Carleman estimates for a fractional diffusion equation of half order in time.

We consider inverse dynamic and spectral problems for the one-dimensional Dirac system on a finite tree. Our aim will be to recover the topology of a tree (lengths and connectivity of edges) as well as the matrix potentials on each edge. As inverse data we use the Weyl–Titchmarsh matrix function or the dynamic response operator.

Two phaseless inverse problems for elliptic equations are considered. We assume that information related to modulus of full fields, which consist of sums of incident and scattering fields, is given. These full fields are the interference fields generated by point sources. We introduce a set of auxiliary point sources for solving the inverse problems and demonstrate that the corresponding data allow us to solve the inverse problems in a way similar to the case of measurements of scattering waves.

We consider the method of quasi-solutions (also referred to as Ivanov regularization) for the regularization of linear ill-posed problems in non-reflexive Banach spaces. Using the equivalence to a metric projection onto the image of the forward operator, it is possible to show regularization properties and to characterize parameter choice rules that lead to a convergent regularization method, which includes the Morozov discrepancy principle. Convergence rates in a suitably chosen Bregman distance can be obtained as well. We also address the numerical computation of quasi-solutions to inverse source problems for partial differential equations in L∞⁢(Ω){L^{\infty}(\Omega)} using a semi-smooth Newton method and a backtracking line search for the parameter choice according to the discrepancy principle. Numerical examples illustrate the behavior of quasi-solutions in this setting.