Monochromatic Reconstruction Algorithms for Two-dimensional Multi-channel Inverse Problems

Abstract

Abstract. We consider two inverse problems for the multi-channel two-dimensional Schrödinger equation at fixed positive energy, i.e. the equation −∆ψ + V (x)ψ = Eψ at fixed positive E, where V is a matrixvalued 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 → +∞, under the assumptions that V is m-times differentiable in L 1 , for m ≥ 3, and has sufficient boundary decay.

Convexification of restricted Dirichlet-to-Neumann map

“…Some reconstruction procedures can be found in [22,24,27,28]. In the reconstruction procedure of [28], a certain infinite matrix is truncated, which is philosophically close to our truncation of that Fourier-like series. We refer to [1,9,11] for numerical studies of DN.…”

New Global Stability Estimates for Monochromatic Inverse Acoustic Scattering

“…For more information on reconstruction methods for Problems 1.1 and 1.2 see [2], [9], [16], [17], [19], [23] and references therein. Problems 1.1 and 1.2 can be also considered as examples of ill-posed problems: see [15], [5] for an introduction to this theory.…”

Application of functional-analytical Novikov algorithm for the purposes of ocean tomography

“…There are rigorous methods for solving the inverse problems -so-called functional-analytical methods [2,3,4], which were initially developed for quantum-mechanical purposes. Since the Schrödinger equation in the monochromatic (isoenergetic) case is the same as the Helmholtz equation up to notation, it gives the simple idea to apply methods which were developed in quantum mechanics for ocean acoustic applications.…”

“…In this paper we consider the possibility of application of one such a functional-analytical method -the algorithm of R.G. Novikov [2] -for the purpose of mode tomography of the ocean. This method is mathematically rigorous; it does not require both a linearity of the model and implementation of iteration procedures.…”

“…In the paper [5] the previous version of this algorithm has been numerically implemented when initial data for the solution of the inverse problem were the scattering amplitudes. A new version of the algorithm [2], discussed below, uses the fields transmitted and received at the boundary of area under investigation as input data. This situation corresponds more closely to conditions of real experiment.…”