Consider an operator equation $F(u)=0$ in a real Hilbert space. The problem of solving this equation is ill-posed if the operator $F'(u)$ is not boundedly invertible, and well-posed otherwise. A general method, dynamical systems method (DSM) for solving linear and nonlinear ill-posed problems in a Hilbert space is presented. This method consists of the construction of a nonlinear dynamical system, that is, a Cauchy problem, which has the following properties: 1) it has a global solution, 2) this solution tends to a limit as time tends to infinity, 3) the limit solves the original linear or non-linear problem. New convergence and discretization theorems are obtained. Examples of the applications of this approach are given. The method works for a wide range of well-posed problems as well.Comment: 21p

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An overview of the author's results is given. The inverse problems for obstacle, geophysical and potential scattering are considered. The basic method for proving uniqueness theorems in one-and multi-dimensional inverse problems is discussed and illustrated by numerous examples.The method is based on property C for pairs of differential operators. Property C stands for completeness of the sets of products of solutions to homogeneous differential equations. To prove a uniqueness theorem in inverse scattering problem one assumes that there are two operators which generate the same scattering data. This

A rigorous reduction of the many-body wave scattering problem to solving a linear algebraic system is given bypassing solving the usual system of integral equation. The limiting case of infinitely many small particles embedded into a medium is considered and the limiting equation for the field in the medium is derived. The impedance boundary conditions are imposed on the boundaries of small bodies. The case of Neumann boundary conditions (acoustically hard particles) is also considered. Applications to creating materials with a desired refraction coefficient are given. It is proved that by embedding suitable number of small particles per unit volume of the original material with suitable boundary impedances one can create a new material with any desired refraction coefficient. The governing equation is a scalar Helmholtz equation, which one obtains by Fourier transforming the wave equation.

ABSTRACT. Based on a regularized Volterra equation, two different approaches for numerical differentiation are considered. The first approach consists of solving a regularized Volterra equation while the second approach is based on solving a disretized version of the regularized Volterra equation. Numerical experiments show that these methods are efficient and compete favorably with the variational regularization method for stable calculating the derivatives of noisy functions.

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PrefaceWave scattering by bodies, small in comparison with the wavelength, is of interest in many applications: light scattering by cosmic and other dust, light scattering in colloidal solutions, scattering of waves in media with small inhomogeneities, such as holes in metal, for example, ultrasound mammography, ocean acoustics, etc. In 1871 Rayleigh started his classical work on wave scattering by small bodies. He understood that the main input in the far-zone field, scattered by a dielectric body, small in comparison with the wavelenghth A, is made by the dipole radiation. However, he did not give methods for calculating this radiation for bodies of arbitrary shapes. The body is small if ka < 0.1, where k = ^ is the wavenumber, and a is the characteristic dimension of the body. Practically in some cases one may consider the body small if ka < 0.2. Thomson (1893) understood that the main part of the far-zone field, scattered by a small perfectly conducting body, consists not only of the electric dipole radiation, but also of the magnetic dipole radiation which is of the same order of magnitude. Many papers and books dealing with the wave scattering from small bodies and its applications have been published since then.However only in the author's works ([85], [112], [113]) have analytic formulas for calculation with arbitrary accuracy of the electric and magnetic polarizability tensors for bodies of arbitrary shapes been derived. These formulas allow one to calculate with the desired accuracy the dipole radiation from bodies of arbitrary shapes, and the electric and magnetic polarizability tensors for these bodies in terms of their geometries and material properties (dielectric permeability e, magnetic permittivity fj,, and conductivity a) of the bodies. Using these formulas the author has derived analytic formulas for the 5-matrix for acoustic and electromagnetic wave scattering by small bodies of arbitrary shapes. viii Preface for various functionate of practical interest in scattering theory, such as electrical capacitances of the conductors of arbitrary shapes and elements of the polarizability tensors of dielectric bodies of arbitrary shapes. These results allow the author to solve the inverse radiation problem. Iterative methods for calculating static fields play an important role in the theory developed in this monograph. These methods are presented for interior and exterior boundary-value problems and for various boundary conditions. Boundary-value problems are reduced to boundary integral equations, and these equations are solved by means of iterative processes. There is a common feature of the static problems we study. Namely, these problems are reduced to solving Predholm integral equations at the largest eigenvalue (smallest characteristic value, which is reciprocal to the eigenvalue) of the corresponding compact integral operator. The right-hand side of the equation is such that this equation is solvable. The largest eigenvalue is semisimple, that is, it is a simple pole of the resolvent of the cor...

Theory of wave scattering by many small bodies is developed under various assumptions concerning the ratio a d , where a is the characteristic dimension of a small body and d is the distance between neighboring bodies d = O(a κ 1 ), 0 < κ 1 < 1. On the boundary S m of every small body an impedance-type condition is assumed