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

TL;DR

The paper addresses identifying unknown nonlinear operators in semilinear elliptic PDEs by formulating the problem as recovering a finite-dimensional coefficient vector in a monotone operator, using an offline-online greedy reconstruction strategy to design informative controls.It establishes theoretical guarantees for the forward and inverse mappings, notably Lipschitz continuity of the control-to-state map and the inverse parameter-to-state map, under monotonicity and boundedness assumptions on the basis functions.The proposed Optimized Nonlinear Greedy Reconstruction (ONGR) algorithm iteratively selects controls and basis elements to maximize local convexity of the identification objective, enabling robust, data-efficient operator reconstruction in 2D.Numerical experiments with bilinear, sinusoidal, and exponential nonlinearities demonstrate the method’s effectiveness and reveal how the required polynomial degree and control design influence reconstruction accuracy and Taylor-coefficient estimation.Overall, the offline-online framework with active data acquisition provides a principled approach to improve physical models by reliably uncovering unknown nonlinearities in semilinear elliptic systems.

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.

Figures (8)

• Figure 6.1: The three different desired nonlinearities $G^i_\star$, $i=1,2,3$ defined in \ref{['eq:Nonlinearities']}.
• Figure 6.2: Reconstruction and error plots (cf. \ref{['eq:definition_error']}) for $P=2,3,5$ and the bilinear nonlinearity with solutions curves in magenta.
• Figure 6.3: Controls functions. Left: two pairs of controls obtained by \ref{['algo:ONGR']}; Right: two pairs of controls with the same structure as the right-hand side of \ref{['eq:constructed_controls']} with different $\eta,\vartheta$.
• Figure 6.4: Results of robustness test case (using random controls).
• Figure 6.5: Making the problem more convex. Left: random controls; Right: optimal controls.

Theorems & Definitions (19)

• Corollary 3.3
• proof
• Remark 3.4
• Definition 3.5
• Theorem 3.6
• proof
• Remark 3.7
• Theorem 3.8
• Proposition 3.9
• Lemma 3.10