Unified approach to classical equations of inverse problem theory

TL;DR

This paper demonstrates that classical inverse problem integral equations can be derived and solved within the boundary control method, unifying their treatment through control theory and inverse data analysis.

Contribution

It provides a unified framework linking classical integral equations of inverse problems to the boundary control method, enhancing understanding and solution strategies.

Findings

Classical inverse equations are derived via the BC-method.

Solving these equations corresponds to boundary control problems.

Solutions are explicitly determined by inverse data.

Abstract

The boundary control (BC-) method is an approach to inverse problems based upon their deep relations to control and system theory. We show that the classical integral equations of inverse problem theory (Gelfand-Levitan, Krein and Marchenko equations) can be derived in the framework of the BC-method in a unified way. Namely, to solve each of these equations is in fact to solve a relevant boundary control problem, whereas its solution is determined by the inverse data.