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We discuss a time-harmonic inverse scattering problem for the Helmholtz equation with compactly supported penetrable and possibly inhomogeneous scattering objects in an unbounded homogeneous background medium, and we develop a monotonicity relation for the far field operator that maps superpositions of incident plane waves to the far field patterns of the corresponding scattered waves. We utilize this monotonicity relation to establish novel characterizations of the support of the scattering objects in terms of the far field operator. These are related to and extend corresponding results known from factorization and linear sampling methods to determine the support of unknown scattering objects from far field observations of scattered fields. An attraction of the new characterizations is that they only require the refractive index of the scattering objects to be above or below the refractive index of the background medium locally and near the boundary of the scatterers. An important tool to prove these results are so-called localized wave functions that have arbitrarily large norm in some prescribed region while at the same time having arbitrarily small norm in some other prescribed region. We present numerical examples to illustrate our theoretical findings. Mathematics subject classifications (MSC2010): 35R30, (65N21)

We discuss a simple non-iterative method to reconstruct the support of a collection of obstacles from the measurements of far-field patterns of acoustic or electromagnetic waves corresponding to plane-wave incident fields with one or few incident directions at several frequencies. The method is a variant of the orthogonality sampling algorithm recently studied by Potthast (2010 Inverse Problems 26 074015). Our theoretical analysis of the algorithm relies on an asymptotic expansion of the far-field pattern of the scattered field as the size of the scatterers tends to zero with respect to the wavelength of the incident field that holds not only at a single frequency, but also across appropriate frequency bands. This expansion suggests some modifications to the original orthogonality sampling algorithm and yields a theoretical motivation for its multi-frequency version. We illustrate the performance of the reconstruction method by numerical examples.

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Abstract. We consider the boundary value problem of calculating the electrostatic potential for a homogeneous conductor containing finitely many small insulating inclusions. We give a new proof of the asymptotic expansion of the electrostatic potential in terms of the background potential, the location of the inhomogeneities and their geometry, as the size of the inhomogeneities tends to zero. Such asymptotic expansions have already been used to design direct (i.e. noniterative) reconstruction algorithms for the determination of the location of the small inclusions from electrostatic measurements on the boundary, e.g. MUSIC-type methods. Our derivation of the asymptotic formulas is based on integral equation methods. It demonstrates the strong relation between factorization methods and MUSIC-type methods for the solution of this inverse problem.