The Schur Algorithm Applied to the One-Dimensional Continuous Inverse Scattering Problem

Abstract

The one-dimensional continuous inverse scattering problem can be solved by the Schur algorithm in the discrete-time domain using sampled scattering data. The sampling rate of the scattering data should be increased to reduce the discretization error, but the complexity of the Schur algorithm is proportional to the square of the sampling rate. To improve this tradeoff between the complexity and the accuracy, we propose a Schur algorithm with the Richardson extrapolation (SARE). The asymptotic expansion of the Schur algorithm, necessary for the Richardson extrapolation, is derived in powers of the discretization step, which shows that the accuracy order (with respect to the discretization step) of the Schur algorithm is 1. The accuracy order of the SARE with the -step Richardson extrapolation is increased to with comparable complexity to the Schur algorithm. Therefore, the discretization error of the Schur algorithm can be decreased in a computationally efficient manner by the SARE.

Fast Nonlinear Fourier Transform Algorithms Using Higher Order Exponential Integrators

“…Note that the range of |Re(λ)| that can be resolved is determined by the larger of the two step-sizes h (see Section IV-A). We also remark that Richardson extrapolation can also be applied to the slow algorithms in Section III-C. 5 It was used to improve an inverse NFT algorithm for the defocusing case in [42].…”

Fast Nonlinear Fourier Transform Algorithms Using Higher Order Exponential Integrators

“…5 It was used to improve an inverse NFT algorithm for the defocusing case in [42]. We apply this idea to FCF [2] 1 and FCF [4] 2 to obtain the algorithms FCF_RE [2] 1 and FCF_RE [4] 2 respectively.…”