Volume 17, Issue 4

In this paper the author consider uniqueness of solution of determining attenuation coefficient of X-ray radiation inside absorbing and scattering medium. The known data are densities of incoming and outgoing flows of radiation in the boundary of the medium. The specific character of the problem is that the density of external sources of radiation depending on energy is disconnected in the finite number of points which corresponds to the radiation resonance. This assumption is sufficient to successful research of the problem which may be considered as the problem of X-ray tomography. The generality of the mathematical model used in research makes possible to apply the result for other problems of radiation tomography.

We consider analytical and numerical methods for solving the generalized Sommerfeld problem for time fractional diffusion equations (TFDE). This problem is to find the generalized diffusion coefficient λ of TFDE and the order α of time derivative according to an additional information about a solution of TFDE. In the case of the steady-state anomalous diffusion process, the periodic boundary value problem without initial conditions for TFDE is solved and exact analytical solutions for inverse problems are obtained. When the initial phase of the process is considered, we use the modified Monte Carlo method to obtain the inverse problem data. In this case, inverse problems are formulated as residual function minimization problems, the Levenberg–Marquardt algorithm for residual function minimization is used and numerical results are presented.

This paper is concerned with iterated soft shrinkage for minimizing Tikhonov functionals with sparsity constraints. Motivated by parameter identification problems for partial differential equations we assume, that only adaptive operator evaluations [ Ax ] h are available. Adaptive refinement strategies lead to different realizations of [ A ] h in every iteration step, hence, we cannot assume a bound on the approximation error in the operator norm. We will develop the basic convergence and regularization results for linear inverse problems with adaptive operator evaluations and penalty terms for 1 < p ≤ 2.

Determining the term structure of local volatilities with at-the-money options represents a singular situation. On the one hand, prices of options with strikes close to the current asset price are known as most sensitive to variations in the volatility. Therefore such options should be preferred for reconstruction problems. On the other hand, the analysis of corresponding inverse problems seems to be much easier, if at-the-money options are excluded. In particular, convergence rate results for regularization approaches were formulated preferably for in-the-money and out-of-the-money options. This paper is contribution to bridge the gap. By application of a generalized Tikhonov regularization approach we present convergence rate results by formulating the underling inverse problem in appropriate spaces.

In this paper, we investigate the identification of parameter with semi-linear parabolic equations. The convergence rates of the parameter and the solution of parabolic equations are discussed.

In this paper we introduce the generalization of the known technique of the gradient construction based on the conjugate problem for the case when the unknown function is complex-valued. We introduce the notion of the based frequency. The based frequency helps to estimate the possibility of simultaneous determination of dielectric permeability and conductivity.

The preconditioned iteratively regularized Gauss–Newton algorithm for the minimization of general nonlinear functionals was introduced by Smirnova, Renaut and Khan (Inverse Problems 23: 1547–1563, 2007). In this paper, we establish theoretical convergence results for an extended stabilized family of Generalized Preconditioned Iterative methods which includes ℳ-times iterated Tikhonov regularization with line search. Numerical schemes illustrating the theoretical results are also presented.