We give an overview of recent techniques which use a level set representation of shapes for solving inverse scattering problems. The main focus is on electromagnetic scattering using different popular models, such as for example Maxwell's equations, TM-polarized and TE-polarized waves, impedance tomography, a transport equation or its diffusion approximation. These models are also representative of a broader class of inverse problems. Starting out from the original binary approach of Santosa for solving the corresponding shape reconstruction problem, we successively develop more recent generalizations, such as for example using colour or vector level sets. Shape sensitivity analysis and topological derivatives are discussed as well in this framework. Moreover, various techniques for incorporating regularization into the shape inverse problem using level sets are demonstrated, which also include the choice of subclasses of simple shapes, such as ellipsoids, for the inversion. Finally, we present various numerical examples in two dimensions and in three dimensions for demonstrating the performance of level set techniques in realistic applications.

https://iopscience.iop.org/article/10.1088/0266-5611/22/4/R01/pdf

Level set methods for inverse scattering

We discuss the problem of the reconstruction of the profile of an inhomogeneous object from scattered field data. Our starting point is the contrast source inversion method, where the unknown contrast sources and the unknown contrast are updated by an iterative minimization of a cost functional. We discuss the possibility of the presence of local minima of the nonlinear cost functional and under which conditions they can exist. Inspired by the successful implementation of the minimization of total variation and other edgepreserving algorithms in image restoration and inverse scattering, we have explored the use of these image-enhancement techniques as an extra regularization. The drawback of adding a regularization term to the cost functional is the presence of an artificial weighting parameter in the cost functional, which can only be determined through considerable numerical experimentation. Therefore, we first discuss the regularization as a multiplicative constraint and show that the weighting parameter is now completely prescribed by the error norm of the data equation and the object equation. Secondly, inspired by the edge-preserving algorithms, we introduce a new type of regularization, based on a weighted L2 total variation norm. The advantage is that the updating parameters in the contrast source inversion method can be determined explicitly, without the usual line minimization. In addition this new regularization shows excellent edge-preserving properties. Numerical experiments illustrate that the present multiplicative regularized inversion scheme is very robust, handling noisy as well as limited data very well, without the necessity of artificial regularization parameters. 190 Van den Berg and Abubakar

https://www.jpier.org/pier/pier34/08.0106113.B.Abubakar.pdf

Contrast Source Inversion Method: State of Art

A rigorous methodology for the analysis of initial-boundary value problems on the half-line, is applied to the nonlinear §(NLS), to the sine-Gordon (sG) in laboratory coordinates, and to the Korteweg-deVries (KdV) with dominant surface tension. Decaying initial conditions as well as a smooth subset of the boundary values are given, where n=2 for the NLS and the sG and n=3 for the KdV. For the NLS and the KdV equations, the initial condition q(x,0) = q

 0
 (x) as well as one and two boundary conditions are given respectively; for the sG equation the initial conditions q(x,0) = q

 0
 (x), q

 t
 (x,0) = q

 1
 (x), as well as one boundary condition are given. The construction of the solution q(x,t) of any of these problems involves two separate steps: (a) Given decaying initial conditions define the spectral (scattering) functions {a(k),b(k)}. Associated with the smooth functions , define the spectral functions {A(k),B(k)}. Define the function q(x,t) in terms of the solution of a matrix Riemann-Hilbert problem formulated in the complex k-plane and uniquely defined in terms of the spectral functions {a(k),b(k),A(k),B(k)}. Under the assumption that there exist functions such that the spectral functions satisfy a certain global algebraic relation, prove that the function q(x,t) is defined for all , it satisfies the given nonlinear PDE, and furthermore that . (b) Given a subset of the functions as boundary conditions, prove that the above algebraic relation characterizes the unknown part of this set. In general this involves the solution of a nonlinear Volterra integral equation which is shown to have a global solution. For a particular class of boundary conditions, called linearizable, this nonlinear equation can be bypassed and {A(k),B(k)} can be constructed using only the algebraic manipulation of the global relation. For the NLS, the sG, and the KdV, the following particular linearizable cases are solved: , respectively, where χ is a real constant.

Integrable Nonlinear Evolution Equations on the Half-Line

Solid-state chemists have been consistently successful in envisioning and making new compounds, often enlisting the tools of theoretical solid-state physics to explain some of the observed properties of the new materials. Here, a new style of collaboration between theory and experiment is discussed, whereby the desired functionality of the new material is declared first and theoretical calculations are then used to predict which stable and synthesizable compounds exhibit the required functionality. Subsequent iterative feedback cycles of prediction–synthesis–characterization result in improved predictions and promise not only to accelerate the discovery of new materials but also to enable the targeted design of materials with desired functionalities via such inverse design. The conventional theoretical approach to the study of materials typically involves explaining the properties of known materials. This approach is compared with the inverse design of materials, in which the desired properties are set as inputs and the material that exhibits them as the output.

Inverse design in search of materials with target functionalities

On considere des problemes de Sturm-Liouville inverses dans lesquels les fonctions propres ont une discontinuite dans un point interieur. On montre que si le potentiel est connu sur la moitie de l'intervalle et si une condition limite est donnee, alors le potentiel et l'autre condition limite sont determines de facon unique par les valeurs propres

Discontinuous inverse eigenvalue problems

Inverse problems are those where a set of measured results is analyzed in order to get as much information as possible on a “model” which is proposed to represent a system in the real world. Exact inverse problems are related to most parts of mathematics. Applied inverse problems are the keys to other sciences. Hence the field, which is very wealthy, yields the best example of interdisciplinary research but it has nevertheless a strong individuality. The obtained results and explored directions of the 20th century are sketched in this review, with attempts to predict their evolution.

Past and future of inverse problems

The problem of finding a nonrelativistic central potential from a knowledge of all the phase shifts at one energy had been previously shown by Newton to reduce to the inversion of a given infinite matrix M. In the framework of Newton's theory, the solution is not unique but depends on one parameter. In the present work, the inverse matrix is explicitly given, together with the vectors annihilated by M. These enable one to construct all the solutions of the problem. The asymptotic behavior of the equivalent potentials is exhibited, and it is shown that one (and only one) of them decreases asymtotically faster than r−2+e, provided that the phase shifts decrease asymptotically faster than l−(3+e′) (for arbitrarily small e, e′). All the other equivalent potentials have an oscillating tail damped by a factor r−3/2. The ``transparent potentials,'' which give all phase shifts equal to zero at one energy, are also studied. In subsequent publications, analytic continuation of the potentials in the r plane and of t...

Asymptotic Properties of the Potentials in the Inverse‐Scattering Problem at Fixed Energy

It is well known that many integrable partial differential equations are consistency conditions of a “Lax pair” of linear differential equations whose variables x,t can be considered as coordinates in a plane. The standard inverse scattering method, and the standard direct linearization methods, are related to the scattering problem defined on the x-axis. In a previous paper, the scattering problem defined on the t-axis was considered. Here, the scattering on an unbounded line of the plane, an “elbow,” is defined and it is shown how it can yield new reconstruction methods for the solutions of the consistency condition. The results are neither as complete or as much interesting as the standard IST, but their derivation shows that defining the “wanted” class of solutions is crucial, and it helps understanding why the standard IST is so “simple.”

Elbow scattering and inverse scattering applications to LKdV and KdV

In many problems of potential scattering, and particularly in the inverse‐scattering problem, two equations prove to be of essential interest: the Gel'fand‐Levitan equation and the Regge‐Newton equation. These and their generalizations apply, respectively, to energy‐independent and to λ‐independent potentials. In this paper, a method is devised for obtaining equations applying to more general cases. The requirement is the existence of analytic properties of the wavefunctions corresponding to special classes of potentials, enabling one to construct interpolation formulas of the Lagrange form. From these formulas, it is possible to derive integral equations which may then be generalizable to much larger classes of potentials. This method is fully developed in the case of potentials depending linearly on λ. Interpolation formulas and analytic properties of the wavefunctions in the λ plane are exhibited. Integral equations are given and proved to apply to very large classes of potentials. Existence, uniqueness, and analytic properties of their solutions are thoroughly studied. An example is given for which all the wavefunctions are calculated exactly. Application of the method to the inverse scattering problem in the presence of a spin‐orbit potential will be the object of a forthcoming paper.

Approach to Scattering Problems through Interpolation Formulas and Application to Spin‐Orbit Potentials

A method for studying the inverse scattering problem at fixed energy is given; it enables one to get at a fairly large class of potentials C. It is shown that the problem has in C an infinity of solutions, depending on an infinity of parameters. If the study is restricted to the potentials of C which can be continued as even analytic functions in a circle centered at the origin, the problem has only one solution. A linear approximation is given and is shown to yield inverse formulas of the Born approximation. For even potentials of C, the relevance of this approximation is studied as follows: Comparison is made between a given static potential and the potential obtained from the scattering amplitude through the inverse formula. The standard deviation between these two potentials is shown to go to zero as the energy goes to ∞. Miscellaneous properties of potentials, wavefunctions, and Jost functions are given in the framework of this method.

Pierre C. Sabatier

Papers

Past and future of inverse problems

Asymptotic Properties of the Potentials in the Inverse‐Scattering Problem at Fixed Energy

Elbow scattering and inverse scattering applications to LKdV and KdV

Approach to Scattering Problems through Interpolation Formulas and Application to Spin‐Orbit Potentials

General method for the inverse scattering problem at fixed energy.