Uniqueness in inverse transmission scattering problems for multilayered obstacles
Johannes Elschner Guanghui Hu
Inverse Problems & Imaging 2011, 5(4): 793-813 doi: 10.3934/ipi.2011.5.793
Assume a time-harmonic electromagnetic wave is scattered by an infinitely long cylindrical conductor surrounded by an unknown piecewise homogenous medium remaining invariant along the cylinder axis. We prove that, in TM mode, the far field patterns for all incident and observation directions at a fixed frequency uniquely determine the unknown surrounding medium as well as the shape of the cylindrical conductor. A similar uniqueness result is obtained for the scattering by multilayered penetrable periodic structures in a piecewise homogeneous medium. The periodic interfaces and refractive indices can be uniquely identified from the near field data measured only above (or below) the structure for all quasi-periodic incident waves with a fixed phase-shift. The proofs are based on the singularity of the Green function to a two dimensional elliptic equation with piecewise constant leading coefficients.
keywords: Inverse electromagnetic scattering uniqueness piecewise homogeneous medium periodic structure. TM mode
Dietmar Hömberg Guanghui Hu
Discrete & Continuous Dynamical Systems - S 2015, 8(3): i-i doi: 10.3934/dcdss.2015.8.3i
The workshop Electromagnetics-Modelling, Simulation, Control and Industrial Applications was held at Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin during May 13-17, 2013. Organizers of this workshop were Dietmar Hömberg (WIAS), Ronald H. W. Hoppe (University of Augsburg/University of Houston), Olaf Klein (WIAS), Jürgen Sprekels (WIAS) and Fredi Tröltzsch (Technical University of Berlin). More than sixty researchers from mathematical, physical, engineering and industrial communities participated in this scientific meeting. This special issue of DCDS-S, which contains eleven research-level articles, is based on the talks presented during the workshop. Electromagnetism plays an important role in many modern high-technological applications. Our workshop brought together prominent worldwide experts from academia and industry to discuss recent achievements and future trends of modelling, computations and analysis in electromagnetics. The contributions to this volume cover the following topics: finite and boundary element discretization methods for the electromagnetic field equations in frequency and time domain, optimal control and model reduction for multi-physics problems involving electromagnetics, mathematical analysis of Maxwell's equations as well as direct and inverse scattering problems.
    We particularly emphasize that most articles are devoted to mathematically challenging issues in applied sciences and industrial applications. The optimal control and model reduction presented by S. Nicaise et al. arise from electromagnetic flow measurement in the real world. The contribution by G. Beck et al. focuses on a generalized telegrapher's model which describes the propagation of electromagnetic waves in non-homogeneous conductor cables with multi-wires. The numerical analysis of boundary integral formulations carried out by K. Schmidt et al. is motivated by asymptotic models for thin conducting sheets. The integral equation system established by B. Bugert et al. and the analysis and experiments performed by H. Gross et al. make new contributions to direct and inverse electromagnetic scattering from diffraction gratings, respectively. The locating and inversion schemes for detecting unknown configurations proposed by H. Ammari, G. Bao, J. Li and X. Liu et al. could be important and useful in radar and medical imaging, non-destructive testing and geophysical exploration. Last but not least, one can also find important mathematical applications regarding the estimate of the second Maxwell eigenvalues obtained by D. Pauly and the regularity of solutions to Maxwell's system at low frequencies due to P-E. Druet.
    We hope the presented papers will find a large audience and they may stimulate novel studies on electromagnetism. Finally we would like to express our gratitude to the Research Center MATHEON and the Weierstrass Institute, whose financial support made the workshop possible.
Determination of singular time-dependent coefficients for wave equations from full and partial data
Guanghui Hu Yavar Kian
Inverse Problems & Imaging 2018, 12(3): 745-772 doi: 10.3934/ipi.2018032

We study the problem of determining uniquely a time-dependent singular potential $q$, appearing in the wave equation $\partial_t^2u-Δ_x u+q(t,x)u = 0$ in $Q = (0,T)×Ω$ with $T>0$ and $Ω$ a $ \mathcal C^2$ bounded domain of $\mathbb{R}^n$, $n≥2$. We start by considering the unique determination of some general singular time-dependent coefficients. Then, by weakening the singularities of the set of admissible coefficients, we manage to reduce the set of data that still guaranties unique recovery of such a coefficient. To our best knowledge, this paper is the first claiming unique determination of unbounded time-dependent coefficients, which is motivated by the problem of determining general nonlinear terms appearing in nonlinear wave equations.

keywords: Inverse problems wave equation time dependent coefficient singular coefficients Carleman estimate
Inverse source problems in electrodynamics
Guanghui Hu Peijun Li Xiaodong Liu Yue Zhao
Inverse Problems & Imaging 2018, 12(6): 1411-1428 doi: 10.3934/ipi.2018059

This paper concerns inverse source problems for the time-dependent Maxwell equations. The electric current density is assumed to be the product of a spatial function and a temporal function. We prove uniqueness and stability in determining the spatial or temporal function from the electric field, which is measured on a sphere or at a point over a finite time interval.

keywords: Stability uniqueness inverse source problem Maxwell's equations
Uniqueness in inverse elastic scattering from unbounded rigid surfaces of rectangular type
Johannes Elschner Guanghui Hu Masahiro Yamamoto
Inverse Problems & Imaging 2015, 9(1): 127-141 doi: 10.3934/ipi.2015.9.127
Consider the two-dimensional inverse elastic scattering problem of recovering a piecewise linear rigid rough or periodic surface of rectangular type for which the neighboring line segments are always perpendicular. We prove the global uniqueness with at most two incident elastic plane waves by using near-field data. If the Lamé constants satisfy a certain condition, then the data of a single plane wave is sufficient to imply the uniqueness. Our proof is based on a transcendental equation for the Navier equation, which is derived from the expansion of analytic solutions to the Helmholtz equation. The uniqueness results apply also to an inverse scattering problem for non-convex bounded rigid bodies of rectangular type.
keywords: Inverse scattering rough surface Navier equation uniqueness diffraction grating. linear elasticity Dirichlet boundary condition
Factorization method in inverse interaction problems with bi-periodic interfaces between acoustic and elastic waves
Guanghui Hu Andreas Kirsch Tao Yin
Inverse Problems & Imaging 2016, 10(1): 103-129 doi: 10.3934/ipi.2016.10.103
Consider a time-harmonic acoustic wave incident onto a doubly periodic (biperiodic) interface from a homogeneous compressible inviscid fluid. The region below the interface is supposed to be an isotropic linearly elastic solid. This paper is concerned with the inverse fluid-solid interaction (FSI) problem of recovering the unbounded periodic interface separating the fluid and solid. We provide a theoretical justification of the factorization method for precisely characterizing the region occupied by the elastic solid by utilizing the scattered acoustic waves measured in the fluid. A computational criterion and a uniqueness result are presented with infinitely many incident acoustic waves having common quasiperiodicity parameters. Numerical examples in 2D are demonstrated to show the validity and accuracy of the inversion algorithm.
keywords: linear elasticity. biperiodic surface Factorization method Helmholtz equation Navier equation inverse scattering fluid-solid interaction

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