Approximate Solution to Inverse Scattering Problem for Potentials With Small Support

Let q ( x ) be a real‐valued function with compact support D ⊂ℝ 3 . Given the scattering amplitude A (α′, α, k ) for all α′, α∈ 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 q . Asymptotic error estimates are given for these moments as d = diam( D )→0. Physically, this means that ( k 2 +sup∣ q ( x )) d 2 <1 and sup∣ q ( x )∣ d < k . The found moments give an approximate position and the shape of the support of q . In particular, an ellipsoid D̃ and a real constant q̃ are found, such that the potential q̃ ( x ) = q̃, x ∈D̃, and q̃ ( x ) = 0, x ∉ D̃, 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 (α′,α) is also developed.