Monotonicity in Inverse Medium Scattering on Unbounded Domains

Abstract

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)

Monotonicity-Based Inversion of the Fractional Schrödinger Equation II. General Potentials and Stability

“…the list of references in the introduction. Extensions of monotonicity relations to subspaces of finite codimensions have first been studied in [45,33], and we follow the general approach from there. A sharper bound on the dimension of the excluded subspaces has recently been obtained for the standard Helmholtz equation in [44].…”

On Localizing and Concentrating Electromagnetic Fields

“…Moreover, the combination of localized fields and monotonicity relations have led to the development of monotonicity-based methods for obstacle/inclusion detection, cf. [22,39] for the origins and mathematical justification of this approach, [5, 7, 9-11, 16-19, 21, 23, 32, 38, 40, 41, 44] for further recent contributions, and the recent works [13,20] for the Helmholtz equation. Theoretical uniqueness results for inverse coefficient problems have also been obtained by this approach in [4,14,15,21,24]).…”

Dimension Bounds in Monotonicity Methods for the Helmholtz Equation

“…It turns out that the basic monotonicity property may fail in this case, but monotonicity still holds up to a finite dimensional space and [HPS] shows that shape detection methods and local uniqueness results can be developed also in this situation. [GH18] extends this idea to farfield inverse scattering and shows numerical reconstructions.…”

“…Let us give some more references to earlier and related work, and comment on the relevance of our results. Monotonicity estimates and localized potentials techniques have been used in different ways for the study of inverse problems [Har09, HS10, Har12, AH13, HU13, BHHM17, HU17, BHKS18, GH18 GH18,HL19] cover the case of positive frequency imaging where the monotonicity only holds up to a finite dimensional space. For extending monotonicity-based theoretical uniqueness and stability results, as well as monotonicity-based numerical reconstruction methods, it seems to be of utmost importance to have a good bound on the number of eigenvalues that have to be disregarded.…”