An inverse problem of reconstructing the refractive index interrain parabolic equations *

This work deals with an inverse problem of identifying the refractive index in terrain parabolic equations, which has important applications in radio systems and other applied scientific fields. Unlike other inverse coefficient problems arising in the real field, the underlying mathematical model is discussed in the complex field which may lead to many mathematical tools not available, and the corresponding theoretical analysis rather difficult. Based on the optimal control framework, the recognition problem is transformed into an optimization problem, and the existence and necessary condition of the minimizer for the cost functional are deduced. The local uniqueness and stability of the minimizer are derived from the necessary condition. Finally, a gradient descent iterative scheme is designed to solve the numerical solution of the inverse problem. The validity of the proposed algorithm is illustrated by some typical numerical examples.

Automated summary

This work focuses on solving the inverse problem of identifying the refractive index in terrain parabolic equations. The problem has important applications in various applied scientific fields. The study proposes an optimal control framework to transform the recognition problem into an optimization problem and deduces the existence and necessary condition of the minimizer for the cost functional. The paper also introduces a gradient descent iterative scheme to solve the numerical solution of the inverse problem and demonstrates its validity through numerical examples.