Optimal control approach for moving bottom detection in one-dimensional shallow waters by surface measurements
Lecaros R, Lopez-Rios J, Montecinos J, Zuazua Iriondo E (2024)
Publication Language: English
Publication Status: Accepted
Publication Type: Journal article, Original article
Future Publication Type: Journal article
Publication year: 2024
Journal
Publisher: Mathematical Methods in the Applied Sciences
Pages Range: 1-32
URI: https://dcn.nat.fau.eu/wp-content/uploads/optimalCtrlApproach.pdf
DOI: 10.1002/mma.10251
Open Access Link: https://doi.org/10.1002/mma.10251
Abstract
We consider the Boussinesq-Peregrine (BP) system as described by Lannes [Lannes, D. (2013). The water waves problem: mathematical analysis and asymptotics (Vol. 188). American Mathematical Soc.], within the shallow water regime, and study the inverse problem of determining the time and space variations of the channel bottom profile, from measurements of the wave profile and its velocity on the free surface. A well-posedness result within a Sobolev framework for (BP), considering a time dependent bottom, is presented. Then, the inverse problem is reformulated as a nonlinear PDE-constrained optimization one. An existence result of the minimum, under constraints on the admissible set of bottoms, is presented. Moreover, an implementation of the gradient descent approach, via the adjoint method, is considered. For solving numerically both, the forward (BP) and its adjoint system, we derive a universal and low-dissipation scheme, which contains non-conservative products. The scheme is based on the FORCE-α method proposed in [Toro, E. F., Saggiorato, B., Tokareva, S., and Hidalgo, A. (2020). Low-dissipation centred schemes for hyperbolic equations in conservative and non-conservative form. Journal of Computational Physics, 416, 109545]. Finally, we implement this methodology to recover three different bottom profiles; a smooth bottom, a discontinuous one, and a continuous profile with a large gradient. We compare with two classical discretizations for (BP) and the adjoint system. These results corroborate the effectiveness of the proposed methodology to recover bottom profiles.
Authors with CRIS profile
Enrique Zuazua Iriondo
Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)
Involved external institutions
Universidad Técnica Frederico Santa María
Chile (CL)
Industrial University of Santander / Universidad Industrial de Santander
Colombia (CO)
University of the Frontier / Universidad de La Frontera (UFRO)
Chile (CL)
Chile (CL)
Industrial University of Santander / Universidad Industrial de Santander
Colombia (CO)
University of the Frontier / Universidad de La Frontera (UFRO)
Chile (CL)
How to cite
APA:
Lecaros, R., Lopez-Rios, J., Montecinos, J., & Zuazua Iriondo, E. (2024). Optimal control approach for moving bottom detection in one-dimensional shallow waters by surface measurements. Mathematical Methods in the Applied Sciences, 1-32. https://doi.org/10.1002/mma.10251
MLA:
Lecaros, Rodrigo, et al. "Optimal control approach for moving bottom detection in one-dimensional shallow waters by surface measurements." Mathematical Methods in the Applied Sciences (2024): 1-32.
BibTeX: Download