Iterative observer for boundary estimation for elliptic equations
This work addresses the boundary estimation problem for elliptic PDEs, which is relevant for inverse problems in engineering and physics, but the approach is incremental as it adapts observer theory from parabolic to elliptic systems.
The paper proposes an iterative observer for estimating unknown boundary data in elliptic equations (Cauchy problem for Laplace equation), proving convergence via semigroup theory and observability concepts. Numerical simulations demonstrate the algorithm's efficiency.
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.