Applied Mathematics

Picture : morphing based on Riemannian optimization concepts. Click here for details.

The Applied Mathematics group gathers 8 professors and about twenty researchers who are working on several subfields.

Principal Investigators :

Pierre-Antoine AbsilVincent Blondel, Jean-Charles Delvenne, François Glineur, Geovani Grapiglia, Laurent Jacques, Raphaël Jungers, Estelle Massart

Research Lab :

INMA (Mathematical Engineering research division)

Research Areas :

Research in the algebra team focuses on various structures whose automorphism groups are linear algebraic groups, notably quadratic forms and algebras over arbitrary fields. These structures are studied using methods from number theory and algebraic geometry, such as valuation theory and Galois cohomology. The current projects aim at developing new cohomological invariants and a noncommutative valuation theory for central simple algebras with involution. This activity is run in cooperation with the group theory team of the IRMP.

Balance laws are hyperbolic partial differential equations that are commonly used to express the fundamental dynamics of open conservative systems. Many physical systems having an engineering interest are described by systems of one-dimensional hyperbolic balance laws. Typical examples are for instance the telegrapher equations for electrical lines, the shallow water (Saint-Venant) equations for open channels, the Euler equations for gas flow in pipelines or the Aw-Rascle equations for road traffic. In this research, our concern is to analyse the exponential stability (in the sense of Lyapunov) of the steady-states of such systems.

This research relies on the use of non-negative convex algebra for solving underdetermined linear systems of equations under positive constraints. Such problems arise in various domains of Systems Biology. We are particularly concerned with the decomposition of complex metabolic networks into elementary pathways and with the metabolic flux analysis which aims at computing the entire intracellular flux distribution from a limited number of flux measurements.

The group works on numerical methods for rational approximation, linear algebra and optimization with applications in systems and control, economy, biology and medicine. In approximation theory we look at approximation problems in the complex plane (orthogonal polynomials, quadrature formulas) and at the solution of functional equations, with applications in science, technology and economy. In linear algebra we study the model reduction problem via interpolation and projection of state-space models. We also look at optimal Hankel-norm approximations and their formulation via convex optimization techniques.  In optimization, we are looking for general schemes with provable global complexity estimates. This extends onto the methods for solving systems of nonlinear equations and optimization on nonlinear manifolds. These techniques are applied to problems in signals and systems.

We study several types of matrix factorization techniques, in particular variants where nonnegative factors are required. We focus on both algorithmic (mehods and computational complexity) and applicative (machine learning, graph problems, polyhedral combinatorics) points of view.

The complex rheological behaviour of non-Newtonian liquids is dictated by the flow induced evolution of their internal microstructure. For example, in homogeneous polymeric fluids, the relevant microstructure is the conformation of the macromolecules. Each macroscopic fluid element contains a large number of polymers with a statistical distribution of conformations. During flow, the polymer conformations evolve along the fluid trajectories. Also, the macroscopic stress carried by each fluid element is itself governed by the distribution of conformations within that element. One thus faces a highly non-linear coupling between rheological behaviour, flow-induced evolution of the microstructure, and flow conditions. The fundamental scientific challenges in rheology and non-Newtonian fluid mechanics are indeed to fully comprehend the nature of this non-linear coupling and to predict its consequences in flow problems of interest. We currently focus on the development of molecular models of kinetic theory and methods of computational rheology.

Most recent publications

Below are listed the 10 most recent journal articles and conference papers produced in this research area. You also can access all publications by following this link : see all applied mathematics publications.

Journal Articles

1. Wang, Zheming; Jungers, Raphaël M.; Petreczky, Mihaly; Chen, Bo; Yu, Li. Learning stability of partially observed switched linear systems. (Soumis).

2. Banse, Adrien; Wang, Zheming; Jungers, Raphaël M. Learning stability guarantees for constrained switching linear systems from noisy observations. In: Nonlinear Analysis: Hybrid Systems, (2023). (Soumis).

3. Wang, Zheming; Jungers, Raphaël M.; Ong, Chong Jin. Computation of invariant sets via immersion for discrete-time nonlinear systems. In: Automatica, Vol. 147, p. 110686 (2023). doi:10.1016/j.automatica.2022.110686.

4. Della Rossa, Matteo; Jungers, Raphaël M. Interpretability of Path-Complete Techniques and Memory-based Lyapunov functions. In: IEEE Control Systems Letters, Vol. 7, p. 781-786 (2023). doi:10.1109/LCSYS.2022.3226627.

5. Wang, Zheming; Berger, Guillaume; Jungers, Raphaël M. Data-driven control of switched linear systems with probabilistic stability guarantees. doi:10.48550/arxiv.2103.10823 (Soumis).

6. legat, Benoît; Jungers, Raphaël M. Geometric control of hybrid systems. In: Nonlinear Analysis: Hybrid Systems, Vol. 47, p. 101289 (2023). doi:10.1016/j.nahs.2022.101289.

7. Ren, Wei; Dimarogonas, Dimos V. Event-Triggered Tracking Control of Networked Multiagent Systems. In: IEEE Transactions on Automatic Control, Vol. 67, no.10, p. 5332-5347 (2022). doi:10.1109/tac.2022.3180789.

8. Ren, Wei; Jungers, Raphaël M.; Dimarogonas, Dimos V. Razumikhin and Krasovskii approaches for safe stabilization. In: Automatica, Vol. 146, p. 110563 (2022). doi:10.1016/j.automatica.2022.110563.

9. Doikov, Nikita; Nesterov, Yurii. Affine-invariant contracting-point methods for Convex Optimization. In: Mathematical Programming, (2022). doi:10.1007/s10107-021-01761-9.

10. Rodomanov, Anton; Nesterov, Yurii. Subgradient ellipsoid method for nonsmooth convex problems. In: Mathematical Programming, (2022). doi:10.1007/s10107-022-01833-4.

Conference Papers

1. Debauche, Virginie; Jungers, Raphaël M.. Formal Synthesis of Path-Complete Lyapunov Functions on Neural Templates. 2023 xxx.

2. Debauche, Virginie; Della Rossa, Matteo; Jungers, Raphaël M.. Characterization of the ordering of path-complete stability certificates with addition-closed templates. 2023 xxx.

3. Berger, Guillaume O.; Jungers, Raphaël M.; Wang, Zheming. Data-driven invariant subspace identification for black-box switched linear systems. 2022 xxx. doi:10.1109/CDC51059.2022.9993022.

4. Banse, Adrien; Romao, Licio; Abate, Alessandro; Jungers, Raphaël M.. Data-driven memory-dependent abstractions of dynamical systems. 2022 xxx.

5. Wang, Zheming; Jungers, Raphaël M.. Probabilistic guarantees on the objective value for the scenario approach via sensitivity analysis. In: 2022 IEEE 61st Conference on Decision and Control (CDC), I E E E, 2022, 978-1-6654-6762-9 xxx. doi:10.1109/cdc51059.2022.9993351.

6. Wang, Zheming; Jungers, Raphaël M.. Immersion-based model predictive control of constrained nonlinear systems: Polyflow approximation. In: 2021 European Control Conference (ECC). p. 1099-1104 (2021). IEEE Xplore, 2022 xxx. doi:10.23919/ecc54610.2021.9655233.

7. Della Rossa, Matteo; Jungers, Raphaël M.. Almost sure Stability of Stochastic Switched Systems: Graph lifts-based Approach. In: 2022 IEEE 61st Conference on Decision and Control (CDC). Vol. 62. I E E E, 2022 xxx. doi:10.1109/cdc51059.2022.9993079.

8. Gao, Bin; Vary, Simon; Ablin, Pierre; Absil, Pierre-Antoine. Optimization flows landing on the Stiefel manifold. In: IFAC-PapersOnLine. Vol. 55, no.30, p. 25-30 (2022). Elsevier BV, 2022 xxx. doi:10.1016/j.ifacol.2022.11.023.

9. Banse, Adrien; Wang, Zheming; Jungers, Raphaël M.. Black-box stability analysis of hybrid systems with sample-based multiple Lyapunov functions. 2022 xxx.

10. Banse, Adrien; Wang, Zheming; Jungers, Raphaël M.. Learning stability guarantees for data-driven constrained switching linear systems. 2022 xxx.