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
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
An inverse problem for fluid-solid interaction
Johannes Elschner George C. Hsiao Andreas Rathsfeld
Inverse Problems & Imaging 2008, 2(1): 83-120 doi: 10.3934/ipi.2008.2.83
Any acoustic plane wave incident to an elastic obstacle results in a scattered field with a corresponding far field pattern. Mathematically, the scattered field is the solution of a transmission problem coupling the reduced elastodynamic equations over the obstacle with the Helmholtz equation in the exterior. The inverse problem is to reconstruct the elastic body represented by a parametrization of its boundary.
    We define an objective functional depending on a non-negative regularization parameter such that, for any positive regularization parameter, there exists a regularized solution minimizing the functional. Moreover, for the regularization parameter tending to zero, these regularized solutions converge to the solution of the inverse problem provided the latter is uniquely determined by the given far field patterns. The whole approach is based on the variational form of the partial differential operators involved. Hence, numerical approximations can be found applying finite element discretization. Note that, though the transmission problem may have non-unique solutions for domains with so-called Jones frequencies, the scattered field and its far field pattern is unique and depend continuously on the shape of the obstacle.
keywords: Gauß-Newton method. inverse scattering gradients acoustic and elastic waves

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