3,021 research outputs found
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 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
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
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 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
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
is ill-posed, and the reconstruction is unstable. In this paper we introduce a
quantitative measure of this instability. The diameters of minimizing
sets
are used to study the change in the stability with the change of ,
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
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 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
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
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
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