An algorithm is given for calculating the solution to the 3D inverse scattering problem with noisy discrete fixed energy data. The error estimates for the calculated solution are derived. The methods developed are of general nature and can be used in many applications: in nondestructive evaluation and remote sensing, in geophysical exploration, medical diagnostics and technology.

It is proved that if the scattering amplitudes for two obstacles (from a large class of obstacles) differ a little, then the obstacles differ a little, and the rate of convergence is given. An analytical formula for calculating the characteristic function of the obstacle is obtained, given the scattering amplitude at a fixed frequency. Introduction. Let D ⊂ R 3 be a bounded domain with a smooth boundary Γ, ( ∇ 2 + k 2)u = 0 in D ′: = R 3 \ D, k = const> 0; u = 0 on Γ (1) u = exp(ikα · x) + A(α ′ , α, k)r −1 exp(ikr) + o(r −1), r: = |x | → ∞, α ′: = xr −1. (2) Here α ∈ S 2 is a given unit vector, S 2 is the unit sphere in R 3, the function A(α ′ , α, k) is called the

A numerical method is given to invert surface data for the refraction coefficient.

A review of some of the author’s results in the area of inverse scattering is given. The following topics are discussed: 1) Property C and applications, 2) Stable inversion of fixedenergy 3D scattering data and its error estimate, 3) Inverse scattering with ”incomplete“ data, 4) Inverse scattering for inhomogeneous Schrödinger equation, 5) Krein’s inverse scattering method, 6) Invertibility of the steps in Gel’fand-Levitan, Marchenko, and Krein inversion methods, 7) The Newton-Sabatier and Cox-Thompson procedures are not inversion methods, 8) Resonances: existence, location, perturbation theory, 9) Born inversion as an ill-posed problem, 10) Inverse obstacle scattering with fixed-frequency data, 11) Inverse scattering with data at a fixed energy and a fixed incident direction, 12) Creating materials with a desired refraction coefficient and wave-focusing properties.

The author’s method for solving inverse scattering problem with fixed-energy data is described. Its comparison with the method based on the D-N map is given. A new inversion procedure is formulated.

In this lecture the author reviews his results on multidimensional inverse scattering. References to the works of other authors can be found in [20]. Five topics are briefly discussed: property C with constraints and new type of the uniqueness theorems

Let Y =x;, z. a, 8 ’ be a formal differential operator with constant coefficients in R”, n 3 2, j is a multi-index. We say that Y has property C iff, for any bounded domain D c R”, the set of products {urr} t(u,w~N,(Y): = {u:Zu=O in D, UEH~(D)} is complete in L’(D). Here H”‘(D) is the Sobolev space. A necessary and sutficient condition is given for 4p to have property C. 0 1991 Acadenuc Press, Inc. I.

The author’s method for solving inverse scattering problem with fixed-energy data is described. Its comparison with the method based on the D-N map is given. A new inversion procedure is formulated.

A review of the author’s results is given. Inversion formulas and stability results for the solutions to 3D inverse scattering problems with fixed energy data are obtained. Inversion of exact and noisy data is considered. The inverse potential scattering problem with fixed-energy scattering data is discussed in detail, inversion formulas for the exact and for noisy data are derived, error estimates for the inversion formulas are obtained. The inverse obstacle scattering problem is considered for non-smooth obstacles. Stability estimates are derived for inverse obstacle scattering problem in the class of smooth obstacles. Global estimates for the scttering amplitude are given when the potential grows to infinity in a bounded domain. Inverse geophysical scattering problem is discussed briefly. An algorithm for constructing the Dirichlet-to-Neumann map from the scattering amplitude and vice versa is obtained. An analytical example of non-uniqueness of the solution to a 3D inverse problem of geophysics and a uniqueness theorem for an inverse problem for parabolic equations are given.

This book presents analytic theory of random fields estimation optimal by the criterion of minimum of the variance of the error of the estimate. This theory is a generalization of the classical Wiener theory. Wiener’s theory has been developed for optimal estimation of stationary random processes, that is, random functions of one variable. Random fields are random functions of several variables. Wiener’s theory was based on the analytical solution of the basic integral equation of estimation theory. This equation for estimation of stationary random processes was Wiener-Hopf-type of equation, originally on a positive semiaxis. About 25 years later the theory of such equations has been developed for the case of finite intervals. The assumption of stationarity of the processes was vital for the theory. Analytical formulas for optimal estimates (filters) have been obtained under the assumption that the spectral density of the stationary process is a positive rational function. We generalize Wiener’s theory in several directions. First, estimation theory of random fields and not only random processes