Approximate Solution to Inverse Scattering Problem for Potentials With Small Support

✍ Scribed by A. I. Katsevich; A. G. Ramm

• Publisher: John Wiley and Sons
• Year: 1996
• Tongue: English
• Weight: 549 KB
• Volume: 19
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/(sici)1099-1476(19960925)19:14<1121::aid-mma805>3.0.co;2-0

✦ Synopsis

Let q ( x ) be. a real-valued function with compact support D c R3. Given the scattering amplitude A(a', a, k) for all a', a E 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) +O. Physically, this means that (k2 + sup)q(x))d* < 1 and suplq(x)ld 4 k. The found moments give an approximate position and the shape of the support of q. In particular, an ellipsoid d and a real constant 4 are found, such that the potential Q(x) = 4, XED, and ij(x) = 0, xed, 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(a', a) is also developed.