Approximate solution to inverse scattering problem for potentials with small support

Let q(x) be a real-valued function with compact support D subset of R(3). Given the scattering amplitude A(alpha', alpha, k) for all alpha', alpha is an element of S-2 and a fixed frequency k > 0, the moments of q(x) up to the second order are found using a computationally simple and relatively stable two-step procedure. First, one finds the zeroth moment (total intensity) and the first moment (centre of inertia) of the potential q. Second, one refines the above moments and finds the tensor of the second central moments of a. Asymptotic error estimates are given for these moments as d = diam(D) --> 0. Physically, this means that (k(2) + sup\q(x))d(2) much less than 1 and sup\q(x)\d much less than k. The found moments give an approximate position and the shape of the support of q. In particular, an ellipsoid (D) over tilde and a real constant (q) over tilde are found, such that the potential (q) over tilde(x) = (q) over tilde, x is an element of (D) over tilde, and (q) over tilde(x) = 0, x is not an element of (D) over tilde, produces the scattering data which fit best the observed scattering data and has the same zeroth, first, and second moments as the desired potential. A similar algorithm for finding the shape of D given only the modulus of the scattering amplitude A(alpha', alpha) is also developed.