Topological Sensitivity Analysis for the Anisotropic Laplace Problem

This paper is concerned with the reconstruction of objects immersed in anisotropic media from boundary measurements. The aim of this paper is to propose an alternative approach based on the Kohn-Vogelius formulation and the topological sensitivity analysis method. The idea is to formulate the reconstruction problem as a topology optimization one minimizing an energy-like function. We derive a topological asymptotic expansion for the anisotropic Laplace operator. The unknown object is reconstructed using level-set curve of the topological gradient. We make finally some numerical examples proving the efficiency and accuracy of the proposed algorithm.

A noniterative reconstruction method for solving a time-fractional inverse source problem from partial boundary measurements

In this paper, a noniterative method for solving an inverse source problem governed by the two-dimensional time-fractional diffusion equation is proposed. The basic idea consists in reconstructing the geometrical support of the unknown source from partial boundary measurements of the associated potential. A Kohn-Vogelius type shape functional is considered together with a regularization term penalizing the relative perimeter of the unknown set of anomalies. Identifiability result is derived and uniqueness of a minimizer is ensured. The shape functional measuring the misfit between the solutions of two auxiliary problems containing information about the boundary measurements is minimized with respect to a finite number of ball-shaped trial anomalies by using the topological derivative method. In particular, the second-order topological gradient is exploited to devise an efficient and fast noniterative reconstruction algorithm. Finally, some numerical experiments are presented, showing different features of the proposed approach in reconstructing multiple anomalies of varying shapes and sizes by taking noisy data into account .

Sound speed uncertainty in Acousto-Electric Tomography

The goal in Acousto-Electric Tomography (AET) is to reconstruct an image of the unknown electric conductivity inside an object from boundary measurements of electrostatic currents and voltages collected while the object is penetrated by propagating ultrasound waves. This problem is a coupled-physics inverse problem. Accurate knowledge of the propagating ultrasound wave is usually assumed and required, but in practice tracking the propagating wave is hard due to inexact knowledge of the interior acoustic properties of the object. In this work, we model uncertainty in the sound speed of the acoustic wave, and formulate a suitable reconstruction method for the interior power density and conductivity. We also establish theoretical error bounds, and show that the suggested approach can be understood as a regularization strategy for the inverse problem. Finally, we numerically simulate the sound speed variations from a numerical breast tissue model, and computationally explore the effect of using an inaccurate sound speed on the error in reconstructions. Our results show that with reasonable uncertainty in the sound speed reliable reconstruction is still possible.

Edge channels of broken-symmetry quantum Hall states in graphene visualized by atomic force microscopy

The quantum Hall (QH) effect, a topologically non-trivial quantum phase, expanded the concept of topological order in physics bringing into focus the intimate relation between the “bulk” topology and the edge states. The QH effect in graphene is distinguished by its four-fold degenerate zero energy Landau level (zLL), where the symmetry is broken by electron interactions on top of lattice-scale potentials. However, the broken-symmetry edge states have eluded spatial measurements. In this article, we spatially map the quantum Hall broken-symmetry edge states comprising the graphene zLL at integer filling factors of $${{\nu }}={{0}},\pm {{1}}$$ ν = 0 , ± 1 across the quantum Hall edge boundary using high-resolution atomic force microscopy (AFM) and show a gapped ground state proceeding from the bulk through to the QH edge boundary. Measurements of the chemical potential resolve the energies of the four-fold degenerate zLL as a function of magnetic field and show the interplay of the moiré superlattice potential of the graphene/boron nitride system and spin/valley symmetry-breaking effects in large magnetic fields.

Detection of velocity and attenuation inclusions in the medical ultrasound tomography

The paper is devoted to numerical research of the medical ultrasound tomography problem. This problem consists in finding small inclusions in the breast tissue by boundary measurements of the acoustic waves generated by sources located on the boundary. For medical diagnostic, it is important to recover the image of the acoustical medium and to determine the values of velocity, attenuation and density. In the paper, we describe a numerical experiment of visualization of several inclusions using the energy version of the reverse time migration (RTM). Certainly, the RTM image does not separate velocity and attenuation inclusions. However, kinematic and amplitude analysis gives a possibility to estimate values of the velocity and attenuation. As a result, we split the RTM image into two ones. Note that in this paper we consider the scalar value of sound velocity.

Recovery of transversal metric tensor in the Schrödinger equation from the Dirichlet-to-Neumann map

<p style='text-indent:20px;'>In this paper, we deal with the inverse problem of determining simple metrics on a compact Riemannian manifold from boundary measurements. We take this information in the dynamical Dirichlet-to-Neumann map associated to the Schrödinger equation. We prove in dimension <inline-formula><tex-math id="M1">\begin{document}$ n\geq 2 $\end{document}</tex-math></inline-formula> that the knowledge of the Dirichlet-to-Neumann map for the Schrödinger equation uniquely determines the simple metric (up to an admissible set). We also prove a Hölder-type stability estimate by the construction of geometrical optics solutions of the Schrödinger equation and the direct use of the invertibility of the geodesical X-ray transform.</p>

Refined stability estimates in electrical impedance tomography with multi-layer structure

<p style='text-indent:20px;'>In this paper we study the inverse problem of determining an electrical inclusion in a multi-layer composite from boundary measurements in 2D. We assume the conductivities in different layers are different and derive a stability estimate for the linearized map with explicit formulae on the conductivity and the thickness of each layer. Intuitively, if an inclusion is surrounded by a highly conductive layer, then, in view of "the principle of the least work", the current will take a path in the highly conductive layer and disregard the existence of the inclusion. Consequently, a worse stability of identifying the hidden inclusion is expected in this case. Our estimates indeed show that the ill-posedness of the problem increases as long as the conductivity of some layer becomes large. This work is an extension of the previous result by Nagayasu-Uhlmann-Wang[<xref ref-type="bibr" rid="b15">15</xref>], where a depth-dependent estimate is derived when an inclusion is deeply hidden in a conductor. Estimates in this work also show the influence of the depth of the inclusion.</p>