28 research outputs found
Polarization tensors of planar domains as functions of the admittivity contrast
(Electric) polarization tensors describe part of the leading order term of
asymptotic voltage perturbations caused by low volume fraction inhomogeneities
of the electrical properties of a medium. They depend on the geometry of the
support of the inhomogeneities and on their admittivity contrast. Corresponding
asymptotic formulas are of particular interest in the design of reconstruction
algorithms for determining the locations and the material properties of
inhomogeneities inside a body from measurements of current flows and associated
voltage potentials on the body's surface. In this work we consider the
two-dimensional case only and provide an analytic representation of the
polarization tensor in terms of spectral properties of the double layer
integral operator associated with the support of simply connected conductivity
inhomogeneities. Furthermore, we establish that an (infinitesimal) simply
connected inhomogeneity has the shape of an ellipse, if and only if the
polarization tensor is a rational function of the admittivity contrast with at
most two poles whose residues satisfy a certain algebraic constraint. We also
use the analytic representation to provide a proof of the so-called
Hashin-Shtrikman bounds for polarization tensors; a similar approach has been
taken previously by Golden and Papanicolaou and Kohn and Milton in the context
of anisotropic composite materials
Monotonicity in inverse obstacle scattering on unbounded domains
We consider an inverse obstacle scattering problem for the Helmholtz equation with obstacles that carry mixed Dirichlet and Neumann boundary conditions. We discuss far field operators that map superpositions of plane wave incident fields to far field patterns of scattered waves, and we derive monotonicity relations for the eigenvalues of suitable modifications of these operators. These monotonicity relations are then used to establish a novel characterization of the support of mixed obstacles in terms of the corresponding far field operators. We apply this characterization in reconstruction schemes for shape detection and object classification, and we present numerical results to illustrate our theoretical findings
Monotonicity in inverse scattering for Maxwell’s equations
We consider the inverse scattering problem to recover the support of penetrable scattering objects in three-dimensional free space from far field observations of scattered time-harmonic electromagnetic waves. The observed far field data are described by far field operators that map superpositions of plane wave incident fields to the far field patterns of the corresponding scattered waves. We discuss monotonicity relations for the eigenvalues of linear combinations of these operators with suitable probing operators. These monotonicity relations yield criteria and algorithms for reconstructing the support of scattering objects
from the corresponding far field operators. To establish these results we combine the monotonicity relations with certain localized vector wave functions that have arbitrarily large energy in some prescribed region while at the same time having arbitrarily small energy on some other prescribed region. Throughout we suppose that the relative magnetic permeability of the scattering objects is one, while their real-valued relative electric permittivity maybe inhomogeneous and the permittivity contrast may even change sign. Numerical examples
illustrate our theoretical findings
The factorization method and Capon’s method for random source identification in experimental aeroacoustics
Experimental aeroacoustics is concerned with the estimation of acoustic source power distributions, which are for instance caused by fluid structure interactions on scaled aircraft models inside a wind tunnel, from microphone array measurements of associated sound pressure fluctuations. In the frequency domain aeroacoustic sound propagation can be modeled as a random source problem for a convected Helmholtz equation. This article is concerned with the inverse random source problem to recover the support of an uncorrelated aeroacoustic source from correlations of observed pressure signals. We show that a variant of the factorization method from inverse scattering theory can be used for this purpose. We also discuss a surprising relation between the factorization method and a commonly used beamforming algorithm from experimental aeroacoustics, which is known as Capon\u27s method or as the minimum variance method. Numerical examples illustrate our theoretical findings
The factorization method and Capon’s method for random source identification in experimental aeroacoustics
Experimental aeroacoustics is concerned with the estimation of acoustic source power distributions, which are for instance caused by fluid structure interactions on scaled aircraft models inside a wind tunnel, from microphone array measurements of associated sound pressure fluctuations. In the frequency domain aeroacoustic sound propagation can be modelled as a random source problem for a convected Helmholtz equation. This article is concerned with the inverse random source problem to recover the support of an uncorrelated aeroacoustic source from correlations of observed pressure signals. We show a variant of the factorization method from inverse scattering theory can be used for this purpose. We also discuss a surprising relation between the factorization method and a commonly used beam-forming algorithm from experimental aeroacoustics, which is known as Capon’s method or as the minimum variance method. Numerical examples illustrate our theoretical findings
Tangential cone condition for the full waveform forward operator in the viscoelastic regime: the non-local case
We discuss mapping properties of the parameter-to-state map of full waveform inversion and generalize the results of [M. Eller and A. Rieder, Inverse Problems 37 (2021) 085011] from the acoustic to the viscoelastic wave equation. In particular we establish injectivity of the Fréchet derivative of the parameter-to-state map for a semi-discrete seismic inverse problem in the viscoelastic regime. Here, the finite dimensional parameter space is restricted to functions having global support in the propagation medium (the non-local case) and that are locally linearly independent. As a consequence we deduce local conditional wellposedness of this nonlinear inverse problem. Furthermore, we show that the tangential cone condition holds, which is an essential prerequisite in the convergence analysis of a variety of inversion algorithms for nonlinear illposed problems