Inverse scattering by the stability index method

Abstract

A novel numerical method for solving inverse scattering problem with fixed-energy data is proposed. The method contains a new important concept: the stability index of the inversion problem. This is a number, computed from the data, which shows how stable the inversion is. If this index is small, then the inversion provides a set of potentials which differ so little, that practically one can represent this set by one potential. If this index is larger than some threshold, then practically one concludes that with the given data the inversion is unstable and the potential cannot be identified uniquely from the data. Inversion of the fixed-energy phase shifts for several model potentials is considered. The results show practical efficiency of the proposed method. The method is of general nature and is applicable to a very wide variety of the inverse problems.

Modified Rayleigh conjecture method and its applications

“…Since most objective functions contain many local minima, this is a highly nontrivial task. In several papers, starting with [14], the authors developed and tested the Stability Index Method (SIM) for global minimization. In our presentation here we follow [11], which also contains a convergence analysis and additional numerical results.…”

“…Originally, we have applied the Stability Index minimization method to inverse scattering problems arising in quantum mechanical scattering, [14]. Such potential scattering problems are important in quantum mechanics, where they appear in the context of scattering of particles bombarding an atom nucleus.…”

“…Assuming a particular structure of the potential, the scattering results can be computed and compared to the given scattering data. Thus the inverse scattering problem is reduced to the minimization of the discrepancy (best fit to data), see [14,31] for details.…”