Mathematical Aspects and Numerical Computations of an Inverse Boundary Value Identification Using the Adjoint Method

TL;DR

This paper explores the mathematical foundations and numerical implementation of the adjoint method for inverse boundary value problems, providing convergence analysis and practical numerical results.

Contribution

It offers new theoretical convergence conditions and explicit step size criteria for the adjoint method in inverse boundary problems.

Findings

Theoretical convergence conditions are established.

Explicit step size criteria are derived.

Numerical experiments confirm the effectiveness of the proposed theories.

Abstract

The purpose of this study is to show some mathematical aspects of the adjoint method that is a numerical method for the Cauchy problem, an inverse boundary value problem. The adjoint method is an iterative method based on the variational formulation, and the steepest descent method minimizes an objective functional derived from our original problem. The conventional adjoint method is time-consuming in numerical computations because of the Armijo criterion, which is used to numerically determine the step size of the steepest descent method. It is important to find explicit conditions for the convergence and the optimal step size. Some theoretical results about the convergence for the numerical method are obtained. Through numerical experiments, it is concluded that our theories are effective.