Numerical results for Saito's uniqueness theorem in inverse scattering theory. (29th April 2020)

• Record Type: Journal Article
• Authors: Tyni, Teemu
• Is Part Of: Inverse problems. Volume 36:Number 6(2020)
• Publication Date: 2020-04-29
• Subjects: inverse scattering -- scattering theory -- Saito's formula -- Lippmann–Schwinger equation -- numerical solution -- Schrödinger operator; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/ab7d2d
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: We consider an inverse scattering problem for the Schrödinger operator in two dimensions. The aim of this work is to discuss some first numerical results on Saito's formula. Saito's formula is an explicit integral formula, which at the high-frequency limit gives a uniqueness result for the inverse scattering problem. The numeric approach is quite straight-forward: we take a large enough fixed wave number and evaluate the integrals in Saito's formula numerically. The potential function can then be recovered from the blurry measurements by using the fast Fourier transform and a high-pass filter. We also discuss in detail how the synthetic data is generated via a matrix-based approach. Several numerical examples are shown to demonstrate the results.