Reconstruction of unknown monotone nonlinear operators in semilinear elliptic models using optimal inputs

TL;DR

This paper introduces a greedy algorithm within an offline-online framework to effectively reconstruct unknown monotone nonlinear operators in semilinear elliptic models by designing optimal controls for system excitation.

Contribution

It presents a novel greedy reconstruction algorithm that optimally designs controls for system excitation, enabling accurate identification of unknown nonlinear relations in elliptic models.

Findings

Numerical simulations confirm the effectiveness of the proposed approach.

The method is adaptable to various types of equations beyond the tested model.

Optimal controls improve the accuracy of reconstructing unknown operators.

Abstract

Physical models often contain unknown functions and relations. The goal of our work is to answer the question of how one should excite or control a system under consideration in an appropriate way to be able to reconstruct an unknown nonlinear relation. To answer this question, we propose a greedy reconstruction algorithm within an offline-online strategy. We apply this strategy to a two-dimensional semilinear elliptic model. Our identification is based on the application of several space-dependent excitations (also called controls). These specific controls are designed by the algorithm in order to obtain a deeper insight into the underlying physical problem and a more precise reconstruction of the unknown relation. We perform numerical simulations that demonstrate the effectiveness of our approach which is not limited to the current type of equation. Since our algorithm provides not only a way to determine unknown operators by existing data but also protocols for new experiments, it is a holistic concept to tackle the problem of improving physical models.