Semion Gutman

TWO VERSIONS OF QUASI-NEWTON METHOD FOR MULTIDIMENSIONAL INVERSE SCATTERING PROBLEM

Suppose that a medium with slowly changing spatial properties is enclosed in a bounded 3-dimensional domain and is subjected to a scattering by plane waves of a fixed frequency. Let measurements of the wave scattering field induced by this medium be available in the region outside of this domain. We study how to extract the properties of the medium from the information contained in the measurements. We are concerned with the weak scattering case of the above inverse scattering problem (ISP), that is, the unknown. spatial variations of the medium are assumed to be close to a constant. Examples can be found in the studies of the wave propagation in oceans, in the atmosphere, and in some biological media. Since the problems are nonlinear, the methods for their linearization (the Born approximation) have been developed. However, such an approach often does not produce good results. In our method, the Born approximation is just the first iteration and further iterations improve the identification by an order of magnitude. The iterative sequence is defined in the framework of a Quasi-Newton method. Using the measurements of the scattering field from a carefully chosen set of directions we are able to recover (finitely many) Fourier coefficients of the sought parameters of the model. Numerical experiments for the scattering from coaxial circular cylinders as well as for simulated data are presented.

NUMERICAL IMPLEMENTATION OF THE MRC METHOD FOR OBSTACLE SCATTERING PROBLEMS

The goal of this work is to show that the numerical solution of the obstacle scattering problem based on the modified Rayleigh conjecture (MRC) method is a competitive alternative to the boundary integral equations method, and that it has numerical advantages which may be especially important in three-dimensional scattering problems with non-smooth domains, for example, with domains whose boundaries contain corners. The MRC is formulated, the algorithm based on it is described and numerical results are presented.

Parameter identifiability for heat conduction with a boundary input

The identifiability (i.e. the unique identification) of conductivity in a heat conduction process is considered in the class of piecewise constant conductivities. The 1-D process may have nonzero boundary inputs as well as distributed inputs. Its measurements are collected at finitely many observation points. They are processed to obtain the first eigenvalue and a constant multiple of the first eigenfunction at the observation points. It is shown that the identification by the Marching Algorithm is continuous with respect to the mean convergence in the admissible set. The result is based on the continuous dependence of eigenvalues, eigenfunctions, and the solutions on the conductivities. Numerical experiments confirm the perfect identification for noiseless data. A numerical algorithm for the identification in the presence of noise is proposed and implemented.

PIECEWISE-CONSTANT POSITIVE POTENTIALS WITH PRACTICALLY THE SAME FIXED-ENERGY PHASE SHIFTS

It has recently been shown that spherically symmetric potentials of finite range are uniquely determined by the part of their phase shifts at a fixed energy level k2 > 0. However, numerical experiments show that two quite different potentials can produce almost identical phase shifts. It has been guessed by physicists that such examples are possible only for “less physical” oscillating and changing sign potentials. In this note it is shown that the above guess is incorrect: we give examples of four positive spherically symmetric compactly supported quite different potentials having practically identical phase shifts. The note also describes a hybrid stochastic deterministic method for global minimization for the construction of these potentials.

Identification of Multilayered Particles from Scattering Data by a Clustering Method

A multilayered particle is illuminated by plane acoustic or electromagnetic waves of one or several frequencies. We consider the inverse scattering problem for the identification of the layers and of the refraction coefficients of the scatterer in a non-Born region of scattering. Local deterministic and global probabilistic minimization methods are studied. A special reduction procedure is introduced to reduce the dimensionality of the minimization space. Deeps and the multilevel single-linkage methods for global minimization are used for the solution of the inverse problem. Their performance is analyzed for various multilayer configurations.

Inverse scattering by the stability index method

Abstract - 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.

Modified Rayleigh Conjecture Method for Multidimensional Obstacle Scattering Problems

ABSTRACT 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. A.G.R. 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.

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 behaviour 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 solving such a problem is examined from the theoretical as well as from the numerical points of view.

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.

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

Abstract - 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 k2 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.