Iterative observer for boundary estimation for elliptic equations

TL;DR

This paper introduces an iterative observer method for boundary estimation in elliptic equations, specifically the Laplace equation, using space as a time-like variable, with proven convergence and demonstrated numerical effectiveness.

Contribution

It presents a novel iterative observer algorithm for boundary estimation in elliptic PDEs, with convergence proof and practical finite difference implementation.

Findings

Proven convergence of the iterative observer under certain conditions.

Successful numerical implementation demonstrating algorithm efficiency.

System shown to be observable using semigroup theory.

Abstract

In this paper we propose the design of an iterative observer using space as a time-like variable and prove its convergence. The iterative observer algorithm solves boundary estimation problem for a steady-state elliptic equation system namely Cauchy problem for Laplace equation. The Laplace equation is formulated as a first order state space-like system in one of the space variables and an iterative observer is developed that sweeps over the whole domain to recover the unknown data on the boundary. State operator matrix is proved to generate strongly continuous semigroup under certain conditions and the system is shown to be observable. Convergence results of proposed algorithm are established using semigroup theory and concepts of observability for distributed parameter systems. The algorithm is implemented using finite difference discretization schemes and numerical implementation is detailed. Further, the simulation results are presented towards the end to show efficiency of the algorithm.