The Factorization Method for Inverse Problems

Abstract

Ever since Christian Hülsmeyer showed in 1904 that one could use radio waves to detect metallic objects at a distance (the range of the first apparatus was 3000 meters), the race has been on to tease out ever more information from scattered waves. Within months of his first detection demonstration, Hülsmeyer devised a way to determine the distance to the object. At that rate of improvement one might have extrapolated to unimaginable twenty-first-century capabilities. Unfortunately, what has proved to be unimaginable is the difficulty of doing much more than the original device had already accomplished. It would seem that some forms of bionic vision have gone the way of rocket backpacks -that is, until recently.The newest book by Andreas Kirsch with coauthor Natalia Grinberg, The Factorization Method for Inverse Problems, collects over a decade of work by Kirsch and collaborators on a simple method for shape identification in inverse scattering. This book belongs to the next generation of monographs on inverse scattering following the now standard works of Colton and Kress [2] (Inverse Acoustic and Electromagnetic Scattering Theory (1998)) and Isakov [7] (Inverse Problems for Partial Differential Equations).Kirsch's factorization method arose from experimentation with noniterative inverse scattering methods that avoid the computational expense of calculating the solution to the forward problem at each iteration. Noniterative methods attack head-on the inverse problem of determining the scatterer from measured scattered fields by attempting, in principle, simply to invert the scattering operator. In most situations of interest, however, the scattering operator is nonlinear and the inverse problem is ill-posed. Early ideas focused on operator splitting techniques that decompose the scattering operator into a well-posed nonlinear part and an ill-posed linear part, each of which can be inverted stably. Another class of noniterative methods use indicator functions to detect the inconsistency or unsolvability of an easily computed auxiliary problem parameterized by points in space. The shape and location of the object is then determined by those points where the auxiliary problem is solvable. This latter generation of techniques, of which the factorization method is one, is both stable and computationally fast. What separates the factorization method from most of the other noniterative techniques is that it is mathematically complete: the computable criterion for determining the shape and location of the scatterers is both sufficient and necessary while most other techniques rely on only sufficient criteria.The book consists of seven chapters treating the application of the factorization method for, respectively: simplest cases (namely where the far field operator is normal), refinements for more complicated settings (namely, where the far field operator is not normal), so-called mixed boundary value scattering problems, scattering from inhomogeneous media, Maxwell's equations, impedance tomography, and finally a short ...

Factorization Method for Electromagnetic Inverse Scattering from Biperiodic Structures

“…Our analysis extends approaches in [4,20,22] to Maxwell's equations in a biperiodic setting. We adapt the special plane incident fields introduced in [4] for the periodic scalar case to the vectorial problem, which allows us to suitably factorize the near field operator.…”

“…For the convenience of the reader, we give a rather complete proof, see also in [20,22]. First, we introduce real and imaginary part of a bounded linear operator.…”

“…[1,8,9,17,32]. In the general context of acoustic and electromagnetic inverse scattering, qualitative methods has been received much considerable attentions, see Chapter 5 of [20]. Among those methods, the most developed is the linear sampling method which was first introduced in [13] for the scalar case of obstacle inverse scattering.…”

“…The latter method has rigorous justification, keeps the previous advantages and of course is an interesting tool for reconstruction problems in inverse scattering. However, there is only a restricted class of scattering problems to which the Factorization method can be applied, see [20]. Recently this method has been extended to periodic inverse scattering problems.…”

“…Since this generalization is of some importance for the problem under consideration, we give a complete proof. Finally the necessary properties of the middle operator are obtained by the approach in [20] for obstacle inverse scattering of electromagnetic waves.…”