Reconstructing initial data using observers : error analysis of the semi-discrete and fully discrete approximations

TL;DR

This paper analyzes the convergence of a new iterative observer-based algorithm for reconstructing initial data in PDEs, providing detailed error analysis for semi-discrete and fully discrete numerical schemes using finite elements and finite differences.

Contribution

It offers a comprehensive convergence analysis of an observer-based inverse problem algorithm for Schrödinger and wave systems, including error estimates for discretized approximations.

Findings

Convergence of the algorithm is established for semi-discrete and fully discrete schemes.

Error bounds are derived for finite element and finite difference discretizations.

The analysis applies to Schrödinger and wave systems with bounded local observations.

Abstract

A new iterative algorithm for solving initial data inverse problems from partial observations has been recently proposed in Ramdani, Tucsnak and Weiss [15]. Based on the concept of observers (also called Luenberger observers), this algorithm covers a large class of abstract evolution PDE's. In this paper, we are concerned with the convergence analysis of this algorithm. More precisely, we provide a complete numerical analysis for semi-discrete (in space) and fully discrete approximations derived using finite elements in space and finite differences in time. The analysis is carried out for abstract Schr\"odinger and wave conservative systems with bounded observation (locally distributed).