The goal of the review is to provide a state-of-the-art survey on sampling and probe methods for the solution of inverse problems. Further, a configuration approach to some of the problems will be presented. We study the concepts and analytical results for several recent sampling and probe methods. We will give an introduction to the basic idea behind each method using a simple model problem and then provide some general formulation in terms of particular configurations to study the range of the arguments which are used to set up the method. This provides a novel way to present the algorithms and the analytic arguments for their investigation in a variety of different settings. In detail we investigate the probe method (Ikehata), linear sampling method (Colton–Kirsch) and the factorization method (Kirsch), singular sources method (Potthast), no response test (Luke–Potthast), range test (Kusiak, Potthast and Sylvester) and the enclosure method (Ikehata) for the solution of inverse acoustic and electromagnetic scattering problems. The main ideas, approaches and convergence results of the methods are presented. For each method, we provide a historical survey about applications to different situations.

https://iopscience.iop.org/article/10.1088/0266-5611/22/2/R01/pdf

A survey on sampling and probe methods for inverse problems

The theory and applications of integral equations is an very important subject within applied mathematics. Integral equations are used as mathematical models for many and varied physical situations but the integral equations also occur as reformulations of other mathematical problems.

https://link.springer.com/content/pdf/10.1007%2F978-94-011-5338-6_4.pdf

Numerical Treatment of the Integral Equations

The Danish Eulerian Air Pollution Model (DEM model), developed by the Danish National Environmental Research Institute, simulates the transport and distribution of air pollutants in the atmosphere. This paper describes the deployment of the DEM model as a grid service by using Globus Toolkit version 3.2 (GT3). The project starts by porting the DEM model onto the IBM pSeries machine at the University of Reading and investigates a coarse-grain grid implementation. Components including resource management, grid service development and client application development are designed and implemented within a working grid infrastructure.

Grid enablement of the danish eulerian air pollution model

We consider a near-field inverse scattering problem for the wave equation: find the shape of a Dirichlet scattering object from time domain measurements of scattered waves. For this time-domain inverse problem, we propose a linear sampling method, a well-known technique for corresponding frequency domain inverse scattering problems. The problem setting and the algorithm incorporate two basic features. First, the data for the method consist of measurements of causal waves, that is, of waves that vanish before some moment in time. Second, the inversion algorithm directly works on the time-domain data without using a Fourier transformation. The first point is related to the applications we have in mind, which include for instance ground-penetrating radar imaging. The second feature allows us to naturally incorporate multiple (in fact, a continuum of) frequencies in the inversion algorithm. Consequently, it offers the potential of improving the quality of the reconstruction compared to frequency domain methods working with a single frequency. We demonstrate this potential by several numerical examples.

https://iopscience.iop.org/article/10.1088/0266-5611/26/8/085001/pdf

A sampling method for inverse scattering in the time domain

This paper is concerned with an inverse rough surface scattering problem in near-field optical imaging, which is to reconstruct the scattering surface with a resolution beyond the diffraction limit. The surface is assumed to be a small and smooth deformation of a plane surface. Based on a transformed field expansion, the boundary value problem with complex scattering surface is converted into a successive sequence of two-point boundary value problems in the frequency domain, where an analytic solution for the direct scattering problem is derived from the method of integrated solution. By neglecting the high order terms in the power series expansion, the nonlinear inverse problem is linearized and an explicit inversion formula is obtained. A spectral cut-off regularization is adopted to suppress the exponential growth of the noise in the evanescent wave components, which carry high spatial frequency of the scattering surface and contribute to the superresolution in the near-field regime. The method works f...

/pdf/near-field-imaging-of-infinite-rough-surfaces-1zooyjew1p.pdf

Near-Field Imaging of Infinite Rough Surfaces

In this paper we propose a new method to determine the location and shape of an unbounded rough surface from measurements of scattered electromagnetic waves. The proposed method is based on the point source method of Potthast (IMA J. Appl. Math. 61, 119–140, 1998) for inverse scattering by bounded obstacles. We propose a version for inverse rough surface scattering which can reconstruct the total field when the incident field is not necessarily time harmonic. We present numerical results for the case of a perfectly conducting surface in TE polarization, in which case a homogeneous Dirichlet condition applies on the boundary. The results show great accuracy of reconstruction of the total field and of the prediction of the surface location.

/pdf/a-time-domain-point-source-method-for-inverse-scattering-by-283nlyji69.pdf

C. D. Lines

Papers

A Time Domain Point Source Method for Inverse Scattering by Rough Surfaces