Identification and stability of small-sized dislocations using a direct algorithm

<p style='text-indent:20px;'>This paper considers the problem of identifying dislocation lines of curvilinear form in three-dimensional materials from boundary measurements, when the areas surrounded by the dislocation lines are assumed to be small-sized. The objective of this inverse problem is to reconstruct the number, the initial position and certain characteristics of these dislocations and establish, using certain test functions, a Hölder stability of the centers. This paper can be considered as a generalization of [<xref ref-type="bibr" rid="b9">9</xref>], where instead of reconstructing point-wise dislocations, as done in the latter paper, our aim is to recover the parameters of line dislocations by employing a direct algebraic algorithm.</p>

On variational regularization: Finite dimension and Hölder stability

In this paper, we analyze the convergence rates for finite-dimensional variational regularization in Banach spaces by taking into account the noisy data and operator approximations. In particular, we determine the convergence rates by incorporating the smoothness concepts of Hölder stability estimates and the variational inequalities. Additionally, we discuss two ill-posed inverse problems to complement the abstract theory presented in our main results.

Estimation of the Time-Dependent Body Force Needed to Exert on a Membrane to Reach a Desired State at the Final Time

We are concerned with the wave propagation in a homogeneous 2D or 3D membrane Ω of finite size. We assume that either the membrane is initially at rest or we know its initial shape (but not necessarily both) and its boundary is subject to a known boundary force. We address the question of estimating the needed time-dependent body force to exert on the membrane to reach a desired state at a given final time T. As an additional information, we ask for the displacement on the boundary. We consider the displacement either at a single point of the boundary or on the whole boundary. First, we show the uniqueness of solution of these inverse problems under natural conditions on the final time T. If, in addition, the displacement on the whole boundary is only time dependent (which means that the boundary moves with a constant speed), this condition on T is removed if Ω satisfies Schiffer’s property. Second, we derive a conditional Hölder stability inequality for estimating such a time-dependent force. Third, we propose a numerical procedure based on the application of the satisfier function along with the standard Fourier expansion of the solution to the problems. Numerical tests are given to illustrate the applicability of the proposed procedure.