In this work, identification of the shear force in the Atomic Force Microscopy cantilever tip-sample interaction is considered. This interaction is governed by the following dynamic Euler-Bernoulli beam equation. $$ \left\{ \begin{array}{ll} \rho_A(x) u_{tt}+\mu(x)u_{t}+ (r(x)u_{xx}+\kappa(x)u_{xxt})_{xx} =0,\, (x,t)\in \Omega_{T},\\ [1pt] u(x,0)=u_{t}(x,0)=0, ~x \in (0,\ell ), \\ [1pt] u(0,t)=u_x(0,t)=0,~ \left (r(x)u_{xx}+\kappa(x)u_{xxt}\right)_{x=\ell}=M(t),\\ \qquad \qquad \qquad \left (-(r(x)u_{xx}+\kappa(x)u_{xxt})_x \right )_{x=\ell}=g(t),~t \in [0,T], \end{array} \right. $$ where the momentum $M(t)$ is correlated with the transverse shear force $g(t)$ by a certain formula. For the identification of $g(t)$ the deflection on the right hand tip is used as an measured data to minimize the corresponding objective functional by an explicit gradient formula. As a next step, the conjugate gradient algorithm (CGA) is designed for the reconstruction process to have numerical solution of the considered inverse problem. This algorithm is based on the weak solution theory, adjoint problem approach and method of lines combined with Hermite finite elements. Computational results, obtained for noisy output data, are illustrated to show an efficiency and accuracy of the proposed approach, for typical classes of shear force functions with realistic problem parameters.

The talk is focused on recent developments of reconstruction algorithms that can be used to approximate admittivity distributions in Electrical Impedance Tomography. The algorithms are non-iterative and are based on linearized integral equation formulations [1,2] which have been extended to reconstruct the conductivity and/or permittivity distributions of two and three-dimensional domains from boundary measurements of both low and high-frequency alternating input currents and induced potentials [3]. The linearized approaches rely on the solutions to the Laplace equation on a disk and a hemispherical domain subject to appropriate idealized Neumann boundary conditions corresponding to applied spatial varying trigonometric current patterns. Reconstructions from noisy simulated data are obtained from single-time, time-difference and multiple-times data. Moreover, a proposed design of a prototype for a novel integrated circuit based electrical impedance mammographic system embedded in a brassiere will be presented.

[1] C. Hähnlein, K. Schilcher, C. Sebu, H. Spiesberger. Conductivity imaging with interior potential measurements, Inverse Problems in Science and Engineering 19(5): 729-750, 2011.

[2] K-H. Georgi, C. Hähnlein, K. Schilcher, C. Sebu, H. Spiesberger. Conductivity Reconstructions Using Real Data from a New Planar Electrical Impedance Device, Inverse Problems in Science and Engineering, Inverse Problems in Science and Engineering 21(5): 801-822, 2013.

[3] C. Sebu, A. Amaira, J. Curmi. A linearized integral equation reconstruction method of admittivity distributions using Electrical Impedance Tomography, Engineering Analysis with Boundary Elements 150: 103-110, 2023.

Let $\mathcal{H}$ be a separable Hilbert space and let $\mathcal{L}$ be operator with a discrete spectrum on $\mathcal{H}.$ For $$\begin{cases} &\mathcal{D}_t^\alpha u(t)+a(t)\mathcal{L}u(t)=f(t) \,\, \hbox{in} \,\, \mathcal{H},\,0<t \leq T,\\ &u (0) =h \; \text{in}\; \mathcal{H}, \end{cases}\;\;\;\;(1) $$ we study

Coefficient inverse problem: Given $f(t), h$ and $E(t), $ find a pair of functions $\{a(t),u(t)\}$ satisfying the problem (1) and the additional condition $$ F[u(t)]=E(t),\; t\in[0,T], $$ where $F$ is the linear bounded functional.

As for this kind of inverse problem for parabolic equation, see [1]. Under suitable restrictions on the given data, we prove the existence, uniqueness and continuous dependence of the solution on the data.

[1] Z. Zhang. An undetermined time-dependent coefficient in a fractional diffusion equation, Inverse Problems and Imaging. 11: 875–900, 2017.

The main purpose of this talk is to explain how quantum currents induced by either the classical or the time-fractional Schrödinger equation, associated with an Iwatsuka Hamiltonian, uniquely determine the magnetic potential.