Monochromatic reconstruction algorithms for two-dimensional multi-channel inverse problems

We consider two inverse problems for the multi-channel two-dimensional Schrodinger equation at fixed positive energy, i.e., the equation -Delta psi + V(x)psi = E psi at fixed positive E, where V is a matrix-valued potential. The first is the Gel'fand inverse problem on a bounded domain D at fixed energy and the second is the inverse fixed-energy scattering problem on the whole plane R-2. We present in this paper two algorithms which give efficient approximate solutions to these problems: in particular, in both cases, we show that the potential V is reconstructed with Lipschitz stability by these algorithms up to O(E-(m-2)/2) in the uniform norm as E -> +infinity, under the assumptions that V is m-times differentiable in L-1, for m >= 3, and has sufficient boundary decay.