Modified Rayleigh Conjecture method for multidimensional obstacle scattering problems

The Rayleigh conjecture on the representation of the scattered field in the exterior of an obstacle $D$ is widely used in applications. However this conjecture is false for some obstacles. AGR introduced the Modified Rayleigh Conjecture (MRC), and in this paper we present successful numerical algorithms based on the MRC for various 2D and 3D obstacle scattering problems. The 3D obstacles include a cube and an ellipsoid. The MRC method is easy to implement for both simple and complex geometries. It is shown to be a viable alternative for other obstacle scattering methods.Comment: 10p

Modified Rayleigh Conjecture Method and Its Applications

The Rayleigh conjecture about convergence up to the boundary of the series representing the scattered field in the exterior of an obstacle $D$ is widely used by engineers in applications. However this conjecture is false for some obstacles. AGR introduced the Modified Rayleigh Conjecture (MRC), which is an exact mathematical result. In this paper we review the theoretical basis for the MRC method for 2D and 3D obstacle scattering problems, for static problems, and for scattering by periodic structures. We also present successful numerical algorithms based on the MRC for various scattering problems. The MRC method is easy to implement for both simple and complex geometries. It is shown to be a viable alternative for other obstacle scattering methods. Various direct and inverse scattering problems require finding global minima of functions of several variables. The Stability Index Method (SIM) combines stochastic and deterministic method to accomplish such a minimization

Application of the hybrid stochastic-deterministic minimization method to a surface data inverse scattering problem

A method for the identification of small inhomogeneities from a surface data is presented in the framework of an inverse scattering problem for the Helmholtz equation. Using the assumptions of smallness of the scatterers one reduces this inverse problem to an identification of the positions of the small scatterers. These positions are found by a global minimization search. Such a search is implemented by a novel Hybrid Stochastic-Deterministic Minimization method. The method combines random tries and a deterministic minimization. The effectiveness of this approach is illustrated by numerical experiments. In the modeling part our method is valid when the Born approximation fails. In the numerical part, an algorithm for the estimate of the number of the small scatterers is proposed

Stable identification of piecewise-constant potentials from fixed-energy phase shifts

An identification of a spherically symmetric potential by its phase shifts is an important physical problem. Recent theoretical results assure that such a potential is uniquely defined by a sufficiently large subset of its phase shifts at any one fixed energy level. However, two different potentials can produce almost identical phase shifts. That is, the inverse problem of the identification of a potential from its phase shifts at one energy level $k^2$ is ill-posed, and the reconstruction is unstable. In this paper we introduce a quantitative measure $D(k)$ of this instability. The diameters of minimizing sets $D(k)$ are used to study the change in the stability with the change of $k$, and the influence of noise on the identification. They are also used in the stopping criterion for the nonlinear minimization method IRRS (Iterative Random Reduced Search). IRRS combines probabilistic global and deterministic local search methods and it is used for the numerical recovery of the potential by the set of its phase shifts. The results of the identification for noiseless as well as noise corrupted data are presented

Analysis of a method for identification of obstacles

Some difficulties are pointed out in the methods for identification of obstacles based on the numerical verification of the inclusion of a function in the range of an operator. Numerical examples are given to illustrate theoretical conclusions. Alternative methods of identification of obstacles are mentioned: the Support Function Method (SFM) and the Modified Rayleigh Conjecture (MRC) method.Comment: 9 pages, 2 figure

Computational method for acoustic wave focusing

Scattering properties of a material are changed when the material is injected with small acoustically soft particles. It is shown that its new scattering behavior can be understood as a solution of a potential scattering problem with the potential $q$ explicitly related to the density of the small particles. In this paper we examine the inverse problem of designing a material with the desired focusing properties. An algorithm for such a problem is examined from the theoretical as well as from the numerical perspective.Comment: 13 pages, 4 figure

Numerical Solution of Obstacle Scattering Problems

Some novel numerical approaches to solving direct and inverse obstacle scattering problems (IOSP) are presented. Scattering by finite obstacles and by periodic structures is considered. The emphasis for solving direct scattering problem is on the Modified Rayleigh Conjecture (MRC) method, recently introduced and tested by the authors. This method is used numerically in scattering by finite obstacles and by periodic structures. Numerical results it produces are very encouraging. The support function method (SFM) for solving the IOSP is described and tested in some examples. Analysis of the various versions of linear sampling methods for solving IOSP is given and the limitations of these methods are described.Comment: 25 page

Optimization methods in direct and inverse scattering

In many Direct and Inverse Scattering problems one has to use a parameter-fitting procedure, because analytical inversion procedures are often not available. In this paper a variety of such methods is presented with a discussion of theoretical and computational issues. The problem of finding small subsurface inclusions from surface scattering data is solved by the Hybrid Stochastic-Deterministic minimization algorithm. A similar approach is used to determine layers in a particle from the scattering data. The Inverse potential scattering problem for spherically symmetric potentials and fixed-energy phase shifts as the scattering data is described. It is solved by the Stability Index Method. This general approach estimates the size of the minimizing sets, and gives a practically useful stopping criterion for global minimization algorithms. The 3D inverse scattering problem with fixed-energy data and its solution by the Ramm's method are discussed. An Obstacle Direct Scattering problem is treated by a Modified Rayleigh Conjecture (MRC) method. A special minimization procedure allows one to inexpensively compute scattered fields for 2D and 3D obstacles having smooth as well as nonsmooth surfaces. A new Support Function Method (SFM) is used for Inverse Obstacle Scattering problems. Another method for Inverse scattering problems, the Linear Sampling Method (LSM), is analyzed.Comment: 52 pages, 8 figure

Modified Rayleigh Conjecture for scattering by periodic structures

This paper contains a self-contained brief presentation of the scattering theory for periodic structures. Its main result is a theorem (the Modified Rayleigh Conjecture, or MRC), which gives a rigorous foundation for a numerical method for solving the direct scattering problem for periodic structures. A numerical example illustrating the procedure is presented

Inverse Scattering by the Stability Index Method

A novel numerical method for solving inverse scattering problem with fixed-energy data is proposed. The method contains a new important concept: the stability index of the inversion problem. This is a number, computed from the data, which shows how stable the inversion is. If this index is small, then the inversion provides a set of potentials which differ so little, that practically one can represent this set by one potential. If this index is larger than some threshold, then practically one concludes that with the given data the inversion is unstable and the potential cannot be identified uniquely from the data. Inversion of the fixed-energy phase shifts for several model potentials is considered. The results show practical efficiency of the proposed method. The method is of general nature and is applicable to a very wide variety of the inverse problems