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, Raphaël Jungers, Roland Keunings, Yurii Nesterov, Vincent Wertz

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. Bergmann, Ronny; Gousenbourger, Pierre-Yves. A Variational Model for Data Fitting on Manifolds by Minimizing the Acceleration of a Bézier Curve. In: Frontiers in Applied Mathematics and Statistics, Vol. 4, no.59, p. 1--16 (2018). doi:10.3389/fams.2018.00059.

2. Quéguiner-Mathieu, Anne; Tignol, Jean-Pierre. Orthogonal involutions on central simple algebras and function fields of Severi-Brauer varieties. In: Advances in mathematics, Vol. 336, p. 455-476 (October 2018). doi:10.1016/j.aim.2018.07.030.

3. Becher, Karim Johannes; Grenier-Boley, Nicolas; Tignol, Jean-Pierre. Transfer of quadratic forms and of quaternion algebras over quadratic field extensions. In: Archiv der Mathematik, Vol. 111, no.2, p. 135-143 (2018). doi:10.1007/s00013-018-1198-5.

4. Scheuer, Adrien; Abisset-Chavanne, Emmanuelle; Chinesta, Francisco; Keunings, Roland. Microscopic modelling of orientation kinematics of non-spherical particles suspended in confined flows using unilateral mechanics. In: Comptes Rendus Mécanique, Vol. 346, no.1, p. 48-56 (2018). doi:10.1016/j.crme.2017.11.003.

5. Quéguiner-Mathieu, Anne; Tignol, Jean-Pierre. Outer automorphisms of classical algebraic groups. In: Annales scientifiques de l’Ecole normale supérieure, Vol. 51, no.1, p. 113-141 (janvier-février 2018). doi:10.24033/asens.2352.

6. Becher, Karim Johannes; Grenier-Boley, Nicolas; Tignol, Jean-Pierre. Involutions and stable subalgebras. In: Journal of Algebra, Vol. 493, no.1, p. 381-409 (2018). doi:10.1016/j.jalgebra.2017.09.026.

7. Ammar, Amine; Abisset-Chavanne, Emmanuelle; Chinesta, Francisco; Keunings, Roland. Flow modelling of quasi-Newtonian fluids in two-scale fibrous fabrics : Advanced simulations. In: International Journal of Material Forming, Vol. 10, no.4, p. 547-556 (2016). doi:10.1007/s12289-016-1300-0.

8. Nadal, Enrique; Aguado, Jose Vicente; Abisset-Chavanne, Emmanuelle; Chinesta, Francisco; Keunings, Roland; Cueto, Elias. A physically-based fractional diffusion model for semi-dilute suspensions of rods in a Newtonian fluid. In: Applied Mathematical Modelling, Vol. 51, p. 58-67 (2017). doi:10.1016/j.apm.2017.06.009.

9. Perez, M.; Guevelou, S.; Abisset-Chavanne, E.; Chinesta, F.; Keunings, Roland. From dilute to entangled fibre suspensions involved in the flow of reinforced polymers: A unified framework. In: Journal of Non-Newtonian Fluid Mechanics, Vol. 250, p. 8-17 (2017). doi:10.1016/j.jnnfm.2017.10.003.

10. Kiselev, Vladimir Yu; Kirschner, Kristina; Schaub, Michael T; Andrews, Tallulah; Yiu, Andrew; Chandra, Tamir; Natarajan, Kedar N; Reik, Wolf; Barahona, Mauricio; Green, Anthony R; Hemberg, Martin. SC3: consensus clustering of single-cell RNA-seq data. In: Nature Methods, Vol. 14, p. 483-486 (2017). doi:10.1038/nmeth.4236.

Conference Papers

1. Gousenbourger, Pierre-Yves; Massart, Estelle; Absil, Pierre-Antoine. Blended smoothing splines on Riemannian manifolds.

2. Martin, Benoît; Glineur, François; De Rua, Philippe; De Jaeger, Emmanuel. Loss reduction in a windfarm participating in primary voltage control using an extension of the Convex DistFlow OPF. In: Proceedings of the 20th Power Systems Computation Conference, IEEE, 2018, 978-1-910963-10-4. doi:10.23919/PSCC.2018.8442758.

3. Bhowmick, Ayan Kumar; GUEUNING, Martin; Delvenne, Jean-Charles; Lambiotte, Renaud; Mitra, Bivas. Temporal Pattern of (Re)tweets Reveal Cascade Migration. In: Proceedings of the 2017 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining 2017 - ASONAM '17, ACM Press, 2017, p. 483-488. doi:10.1145/3110025.3110084.

4. Gonze, François; Gusev, Vladimir; Gerencser, Balazs; Jungers, Raphaël M.; Volkov, Mikhail V.. On the Interplay Between Babai and Černý’s Conjectures. In: Developments in Language Theory : Lecture Notes in Computer Science, Springer International Publishing, 2017, 9783319628080, p. 185-197. doi:10.1007/978-3-319-62809-7_13.

5. Gonze, François; Simonetto, Andrea; Huens, Etienne; Boucquey, Jean; Jungers, Raphaël M.. Probabilistic Occupancy Counts and Flight Criticality Measures for ATM.

6. Angeli, David; Athanasopoulos, Nikolaos; Jungers, Raphaël M.; Philippe, Matthew. Path-Complete Graphs and Common Lyapunov Functions. In: Proceedings of the 20th International Conference on Hybrid Systems: Computation and Control - HSCC '17, ACM Press, 2017. doi:10.1145/3049797.3049817.

7. Boucquey, Jean; Hately, andrew; Irvine, Richard; Steurs, Stefan; Gonze, François; Huens, Etienne; Jungers, Raphaël M.. Probabilistic Traffic Models for Occupancy Counting.

8. Angeli, David; Athanasopoulos, Nikolaos; Jungers, Raphaël M.; Philippe, Matthew. A linear program to compare path-complete Lyapunov functions. In: 2017 IEEE 56th Annual Conference on Decision and Control (CDC), IEEE, 2017, 978-1-5090-2873-3, p. 5888-5893. doi:10.1109/cdc.2017.8264550.

9. Taylor, Adrien; Hendrickx, Julien; Glineur, François. Performance Estimation Toolbox (PESTO): automated worst-case analysis of first-order optimization methods. In: Proceedings of the 56th IEEE Conference on Decision and Control (CDC 2017), 2017, p. 1278-1283. doi:10.1109/CDC.2017.8263832.

10. Gousenbourger, Pierre-Yves; Jacques, Laurent; Absil, Pierre-Antoine. Fast Method to Fit a C1 Piecewise-Bézier Function to Manifold-Valued Data Points: How Suboptimal is the Curve Obtained on the Sphere S2?. In: Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science, Springer International, 2017, 978-3-319-68445-1. doi:10.1007/978-3-319-68445-1_69.