Volume 24, Issue 3

In this paper, the inverse scattering problem for the nonstationary Dirac system on the half-plane is considered. The uniqueness criterion for the inverse scattering problem in terms of boundary condition is described and the restoration of the potential from the scattering operator is proved.

Photoacoustic tomography is the most well-known example of a hybrid imaging method. In this article, we define a Radon-type transform arising in a version of photoacoustic tomography that uses integrating circular detectors and describe how the Radon transform integrating over all circles with a fixed radius is determined from this Radon-type transform. We consider three situations in which the centers of the circular detectors are located on a cylinder, on a plane, and on a sphere. This transform is similar to a toroidal Radon transform, which maps a given function to its integrals over a set of tori. We also study this object.

We consider a pencil of matrix Sturm–Liouville operators on a finite interval. We study the properties of its spectral characteristics and inverse problems that consist in the recovering of the pencil by the spectral data, that is, eigenvalues and so-called weight matrices. This inverse problem is reduced to a linear equation in a Banach space by the method of spectral mappings. A constructive algorithm for the solution of the inverse problem is provided.

Resonance frequency is examined for its effect on the linear sampling method (LSM). The aim of the study is to present an evaluation of occurrence of anomaly at some special frequencies in the LSM procedure. A comprehensive survey is done to analyze the destructive effect of resonance for LSM analysis. Analytical resonance calculations are used to verify the claim. Finally, a frequency selection method is proposed in order to overcome this phenomenon.

We consider a parabolic problem with degeneracy in the interior of the spatial domain and we focus on the well-posedness of the problem and on inverse source problems. The novelties of the present paper are two. First, the degeneracy point is in the interior of the spatial domain. Second, we consider Neumann boundary conditions so that no previous result can be adapted to this situation.

We are interested on an inverse problem of distributed parameter estimation in a partial differential equation (PDE) from measures of the PDE's solution. The considered parameter is supposed to be a piecewise constant function. Identifying the parameterization consists on identifying both values of the parameter and shapes of zones where the parameter is constant. We develop a posteriori error estimators for the considered inverse problem and we present a new algorithm, result of the combination between adaptive parameterization technique , leading to overcome the underdetermination problem and, an adaptive mesh technique, guided by a posteriori error estimators , providing more precise results.

Size-structured population models provide a popular means to mathematically describe phenomena such as bacterial aggregation, schooling fish, and planetesimal evolution. For parameter estimation, a generalized sensitivity function (GSF) provides a tool that quantifies the impact of data from specific regions of the experimental domain. This function helps to identify the most relevant data subdomains, which enhances the optimization of experimental design. To our knowledge, GSFs have not been used in the partial differential equation (PDE) realm, so we provide a novel PDE extension of the discrete and continuous ordinary differential equation (ODE) concepts of Thomaseth and Cobelli and Banks et al. respectively. We analyze a GSF in the context of size-structured population models, and specifically analyze the Smoluchowski coagulation equation to determine the most relevant time and volume domains for three, distinct aggregation kernels. Finally, we provide evidence that parameter estimation for the Smoluchowski coagulation equation does not require post-gelation data.

We study the standard gradient projection method in a Hilbert space, as applied to minimization of the residual functional for nonlinear operator equations with differentiable operators. The functional is minimized over a closed, convex and bounded set, which contains a solution to the equation. It is assumed that the inverse problem associated with the operator equation is conditionally well-posed with a Hölder-type modulus of relative continuity. We prove that the iterative process is asymptotically stable with respect to errors in the right part of the operator equation. Moreover, the process delivers in the limit an order optimal approximation to the desired solution.

We consider a model time-harmonic wave equation of acoustic tomography of moving fluid in an open bounded domain in ℝ d , d ≥ 2, with variable sound speed c , density ρ, fluid velocity v and absorption coefficient α. We give global uniqueness results for related inverse boundary value problem for the cases of boundary measurements given for two and for three fixed frequencies. Besides, we also give a non-uniqueness result for this inverse problem for the case of boundary measurements given for all frequencies.

Tikhonov regularization in Banach spaces with convex penalty and convex fidelity term for linear ill-posed operator equations is studied. As a main result, convergence rates in terms of the Bregman distance of the regularized solution to the exact solution is proven by imposing a generalization of the established variational inequality conditions on the exact solution. This condition only involves a decay rate of the difference of the penalty functionals in terms of the residual.

We prove the global uniqueness in determination of the conductivity, the permeability and the permittivity of the two-dimensional Maxwell equations by the partial Dirichlet-to-Neumann map limited to an arbitrary subboundary.