Description du sujet de thèse

Problem and context: Tsunamis are among the most devastating natural disasters, causing significant human and material losses. The implementation of early warning systems is crucial to provide valuable time for the evacuation of coastal populations. Currently, the sensor networks used in tsunami monitoring include buoys anchored to the seabed, geolocated and capable of transmitting measurements of water column pressure (DART buoys) or water surface variations (LIDAR measurements). The literature on the optimal placement of sensors for tsunami detection remains very limited. For example, Ferrolino et al. (2020) proposed an approach to minimize tsunami detection time based on 2D Saint-Venant equations.
Objective and description of the thesis: The main objective of this thesis is to study the problem of optimal sensor placement for Saint-Venant partial differential equations governing free surface flows, with a specific application to tsunami detection. More specifically, the thesis will be based on a model based on Boussinesq equations, which extend Saint-Venant equations by including dispersive terms, thus allowing for more accurate modeling of tsunami propagation, in particular the effects of dispersion and nonlinearity.
This thesis will consist in:
1. Completing and analyzing the literature review: the optimal placement of sensors based on dynamic models currently relies mainly on finite dimension approaches, with the exception of a few notable references, such as Privat et al. (2015) or Demetriou (2010), Burns and Rautenberg (2014) in the case of mobile sensors. We are interested in approaches that maximize an observability criterion, often based on the observability Gramian (trace, determinant, minimum eigenvalue, or Gramian conditioning) [Georges 1995, Herzog et al. 2017], defined in the case of linear systems (in finite dimension and extended to infinite dimension in the context of the theory of generators of C0-semi-groups [Curtain and Zwart 1995]); this maximization of observability is related to the numerical conditioning of the inverse problem. There is a direct link between the notion of Gramian of observability/identifiability and sensitivity analysis.
2. Developing an approach to optimize the resolution of inverse problems defined from Boussinesq PDEs (source, parameter, and state estimation) based on a sensitivity analysis approach (such as Fisher information matrix, for example). To do this, we will develop numerical methods such as Physics-Informed Machine Learning (PIML), in which solutions are approximated using neural networks (PINNs [Raissi et al. 2019]) or PIML radial basis functions [Tominec and Breznik 2020, Lopez-Ferber et al. 2024]. We will mathematically and numerically characterize a sensitivity/performance criterion for the inverse problem using PIML in order to deduce a sensor placement optimization problem and study its solution. The theoretical link between the PIML approach and adjoint formulations for solving inverse problems will be studied. A comparison between these two approaches will be carried out numerically.
3. Applying the methodology to a concrete case study: study of the optimal deployment of a DART/LIDAR sensor network to effectively detect a tsunami in a high-risk area, the Aegean Sea (including the Cyclads and the Dodecanese), a region prone to numerous faults capable of generating tsunami-generating earthquakes. For example, the Amorgos Island fault was the scene of a major tsunami in 1956, causing an earthquake with an estimated magnitude of between 7.2 and 7.8, a displacement of the seabed of 9 to 16 meters, and waves reaching up to 20 meters on some Greek coasts, causing considerable damage [Leclerc et al. 2024]. The aim will be to optimize the resolution of the inverse problem of determining the source (location and intensity) of a tsunami to ensure effective prediction of surface wave propagation.Références bibliographiques
References
[Bouchard et al. 2024] R. Bouchard, N. Younes, O. Millet, A. Wautier (2024): Parameter optimization of phase-field-based LBM model for calculating capillary forces, Computers and Geotechnics, Volume 172, August 2024, 106391. https://doi.org/10.1016/j.compgeo.2024.106391 .
[Bourgeois et al. 2025] L. Bourgeois, J.-F. Mercier, R.Terrine. Identification of bottom deformations of the ocean from surface measurements. Inverse Problems and Imaging, American Institute of Mathematical Sciences, 2025. https://doi.org/10.3934/ipi.2025008 .
[Burns et Rautenberg 2014] Burns, J. A., & Rautenberg, C. N. (2014). The Infinite-Dimensional Optimal Filtering Problem with Mobile and Stationary Sensor Networks. Numerical Functional Analysis and Optimization, 36(2), 181–224. https://doi.org/10.1080/01630563.2014.970647 .
[Curtain et Zwart 1995] An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, Springer Verlag, https://doi.org/10.1007/978-1-4612-4224-6
[Demetriou 2010] M. A. Demetriou, "Guidance of Mobile Actuator-Plus-Sensor Networks for Improved Control and Estimation of Distributed Parameter Systems," in IEEE Transactions on Automatic Control, vol. 55, no. 7, pp. 1570-1584, July 2010. https://doi.org/10.1109/TAC.2010.2042229 .
[Ferrolinio et al. 2020] AR Ferrolino, Lope JEC, Mendoza RG. Optimal Location of Sensors for Early Detection of Tsunami Waves. Computational Science – ICCS 2020. https://doi.org/10.1007/978-3-030-50417-5_42 .
[Georges 1995] Didier Georges. The use of observability and controllability gramians or functions for optimal sensor and actuator location in finite-dimensional systems, proceedings of 34th IEEE Conference on Decision and Control, https://doi.org/10.1109/CDC.1995.478999 .
[Herzog et al. 2017] Herzog, R., Riedel, I. & Uciński, D. Optimal sensor placement for joint parameter and state estimation problems in large-scale dynamical systems with applications to thermo-mechanics. Optim. Eng. 19, 591–627 (2018). https://doi.org/10.1007/s11081-018-9391-8 .
[Leclerc et al. 2024] Leclerc, F., Palagonia, S., Feuillet, N. et al. Large seafloor rupture caused by the 1956 Amorgos tsunamigenic earthquake, Greece. Commun Earth Environ 5, 663 (2024). https://doi.org/10.1038/s43247-024-01839-0
[Oliver et al. 2022] M. Oliver, D. Georges, C. Prieur. Spatialized Epidemiological Forecasting applied to Covid-19 Pandemic at Departmental Scale in France. Systems and Control Letters, 2022, 164 (June), pp.105240. https://doi.org/10.1016/j.sysconle.2022.105240 .
[Privat et al. 2015] Y. Privat, Trélat, E., and Zuazua, E. (2015). Optimal Shape and Location of Sensors for Parabolic Equations with Random Initial Data. Arch. Rational Mech. Anal. 216 (2015) 921981. https://doi.org/10.1007/s00205-014-0823-0 .
[Raissi et al. 2019] M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational physics, 378 :686–707, 2019. https://doi.org/10.1016/j.jcp.2018.10.045.
[Tominec et Breznik 2020] I. Tominec, E. Breznik. An unfitted RBF-FD method in a least-squares setting for elliptic PDEs on complex geometries, arXiv, maths, https://doi.org/10.1016/j.jcp.2021.110283 .

Contexte de travail

The Gipsa-lab is a joint research laboratory of the CNRS, Grenoble-INP -UGA and the University of Grenoble Alpes. It is under agreement with Inria and the Observatory of Sciences of the Universe of Grenoble. He conducts theoretical and applied research on AUTOMATICS, SIGNAL, IMAGES, SPEECH, COGNITION, ROBOTICS and LEARNING.
Multidisciplinary and at the interface between the human, the physical and digital worlds, our research is confronted with measurements, data, observations from physical, physiological and cognitive systems. They focus on the design of methodologies and algorithms for processing and extracting information, decisions, actions and communications that are viable, efficient and compatible with physical and human reality. Our work is based on mathematical and computer theories for the development of models and algorithms, validated by hardware and software implementations.
By relying on its platforms and partnerships, Gipsa-lab maintains a constant link with applications in a wide variety of fields: health, environment, energy, geophysics, embedded systems, mechatronics, processes and industrial systems, telecommunications, networks, transport and vehicles, operational safety and security, human-computer interaction, linguistic engineering, physiology and biomechanics, etc.
Due to the nature of its research, Gipsa-lab is in direct and constant contact with the economic environment and society.
Its potential as teacher-researchers and researchers is invested in training at the level of universities and engineering schools on the Grenoble site (Grenoble Alpes University).
Gipsa-lab develops its research through 16 teams or themes organized into 4 divisions:
• Automatic and Diagnosis (PAD)
• Data Science (PSD)
• Speech and Cognition (PPC)
• Geometries, Learning, Information and Algorithms (GAIA).
The staff supporting research (38 engineers and technicians) is distributed in the common services distributed within 2 divisions:
• The Administrative and Financial Pole
• The Technical Pole
Gipsa-lab has around 150 permanent staff, including 70 teacher-researchers and 41 researchers. It also welcomes guest researchers and post-docs.
Gipsa-lab supervises nearly 150 theses, including around 50 new ones each year. All the theses carried out in the laboratory are financed and supervised by teacher-researchers and researchers, including 50 holders of an HDR.
Finally, around sixty Master's trainees come each spring to swell the ranks of the laboratory.
The doctoral student will be attached to the INFINITY team of the Control and Diagnostic Division at GIPSA-lab.

Le poste se situe dans un secteur relevant de la protection du potentiel scientifique et technique (PPST), et nécessite donc, conformément à la réglementation, que votre arrivée soit autorisée par l'autorité compétente du MESR.

Contraintes et risques

Required profile: Research Master's degree in Mathematics or Automatic Control.Occasional travel is planned during the thesis within the laboratories of the supervisory team (LJK, LASIE), no risks associated with the proposed research.

Informations complémentaires

Beginners accepted