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

Jan Bartsch; Simon Buchwald; Gabriele Ciaramella; Stefan Volkwein

doi:10.3934/mcrf.2025011

Article Contents

2025, Volume 15, Issue 4: 1262-1283. Doi: 10.3934/mcrf.2025011

This issue Previous Article Numerical analysis for a hyperbolic pde-constrained optimization problem in acoustic full waveform inversion Next Article Optimal control of quasilinear parabolic PDEs with gradient terms and pointwise constraints on the gradient of the state

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

1.
Department of Mathematics, University of Konstanz, Universitätsstr. 10, 78464 Konstanz, Germany
2.
MOX Lab, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133, Milano, Italy

^*Corresponding author: Stefan Volkwein
^*Corresponding author: Stefan Volkwein

Received: April 29, 2024

Revised: November 29, 2024

Early access: February 13, 2025

Published online: November 29, 2025

This work was financed by the Deutsche Forschungsgemeinschaft (DFG) within SFB 1432, Project-ID 425217212. GC is member of the GNCS INDAM group and acknowledges the financial support of GNCS CUP E53C23001670001. The present research is part of the activities of "Dipartimento di Eccellenza 2023-2027".

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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.

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Mathematics Subject Classification: Primary: 35K57, 35R30, 49N45; Secondary: 49M41, 93B30.

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Figure 6.1. The three different desired nonlinearities $ G^i_\star $, $ i = 1, 2, 3 $ defined in (6.1)

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Figure 6.2. Reconstruction and error plots (cf. (6.3)) for $ P = 2, 3, 5 $ and the bilinear nonlinearity with solutions curves in magenta

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Figure 6.3. Controls functions. Left: two pairs of controls obtained by Algorithm 4.1; Right: two pairs of controls with the same structure as the right-hand side of (6.4) with different $ \eta, \vartheta $

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Figure 6.4. Results of robustness test case (using random controls)

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Figure 6.5. Making the problem more convex. Left: random controls; Right: optimal controls

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Figure 6.6. Reconstruction and error plots (cf. (6.3)) for $ P = 2, 3, 5 $ and the sinusoidal nonlinearity with the solutions curves in magenta

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Figure 6.7. Reconstruction and error plots (cf. (6.3)) for $ P = 2, 3, 5 $ and the exponential nonlinearity with the solutions curves in magenta

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Figure 6.8. Difference in coefficients of the (true) Taylor expansion (cf. (6.5)) and the reconstructed nonlinearity for $ P = 2, 3, 5 $ for bilinear, sinusoidal and exponential nonlinearity

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