Volume 21, Issue 2

Abstract. We construct and study methods for approximating the positions (localization) of discontinuities of the first kind of a one-dimensional function. Instead of the exact function, its approximation in and the perturbation level are known; smoothness conditions are imposed on the function outside the discontinuities. The number of discontinuities is countable, and all the discontinuities are divided into two sets: with the absolute value of the jump greater than some positive and discontinuities satisfying a smallness condition for the value of the jump. It is required to find the number of discontinuities in the first set and localize them using the approximately given function and the perturbation level. Since the problem is ill-posed, regularization algorithms should be used for its solution. Under additional conditions on the exact function, we construct regular methods for the localization of discontinuities and obtain estimates for the accuracy of localization and for the separability threshold, which is another important characteristic of the method. The order optimality of the constructed methods on classes of functions with discontinuities is established.

Abstract. In this paper, we consider the problem of estimating the derivative of a function from its noisy version contaminated by a stochastic white noise and argue that in certain relevant cases the reconstruction of by the derivatives of the partial sums of Fourier–Legendre series of has advantage over some standard approaches. One of the interesting observations made in the paper is that in a Hilbert scale generated by the system of Legendre polynomials the stochastic white noise does not increase, as it might be expected, the loss of accuracy compared to the deterministic noise of the same intensity. We discuss the accuracy of the considered method in the spaces L 2 and C and provide a guideline for an adaptive choice of the number of terms in differentiated partial sums (note that this number is playing the role of a regularization parameter). Moreover, we discuss the relation of the considered numerical differentiation scheme with the well-known Savitzky–Golay derivative filters, as well as possible applications in diabetes technology.

Abstract. The inversion (= reconstruction) scheme, called the dynamical probe method, is applied to active thermography to identify unknown inclusions in an heat conductor and their physical properties. We assume the physical properties of the inclusions and heat conductor are isotropic and homogeneous. The measured data for the active thermography are the so-called Neumann to Dirichlet map. By defining some indicator function via the measured data, the identification is done by looking at the behavior of the indicator function. The underlying analysis is the short time asymptotic of fundamental solution of the heat equation with discontinuous coefficients.

Abstract. Via Carleman estimates we prove uniqueness and continuous dependence results for the temporal traces of solutions to three overdetermined linear parabolic ill-posed problems with no initial condition, the first being integro-differential, the latter two being with deviating arguments. The overdetermination is prescribed in an open subset of the (space-time) lateral boundary.

Abstract. In this paper we develop a convergence analysis in an infinite dimensional setting of the Levenberg–Marquardt iteration for the solution of a hybrid conductivity imaging problem. The problem consists in determining the spatially varying conductivity σ from a series of measurements of power densities for various voltage inductions. Although this problem has been very well studied in the literature, convergence and regularizing properties of iterative algorithms in an infinite dimensional setting are still rudimentary. We provide a partial result under the assumptions that the derivative of the operator, mapping conductivities to power densities, is injective and the data is noise-free. Moreover, we implemented the Levenberg–Marquardt algorithm and tested it on simulated data.

Abstract. We prove uniqueness and continuous dependence results for a severely ill-posed linear integrodifferential boundary-value parabolic problem with no initial condition. This latter condition is replaced with an additional boundary information prescribing the temperature on an open subset of the geometric domain . The integral operators entering the equation are defined by integrals of Volterra type with respect to time. In particular, the class of integrodifferential equations dealt with in this paper include those occurring in the linear theory of heat flow in a rigid body consisting of a material with thermal memory.

Abstract. In this paper we review some recent results concerning inverse problems for thin elastic plates. The plate is assumed to be made by non-homogeneous linearly elastic material belonging to a general class of anisotropy. A first group of results concerns uniqueness and stability for the determination of unknown boundaries, including the cases of cavities and rigid inclusions. In the second group of results, we consider upper and lower estimates of the area of unknown inclusions given in terms of the work exerted by a couple field applied at the boundary of the plate. In particular, we extend previous size estimates for elastic inclusions to the case of cavities and rigid inclusions.