Applied Mathematics

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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. De Klerk, Etienne; Glineur, François; Taylor, Adrien B. Worst-Case Convergence Analysis of Inexact Gradient and Newton Methods Through Semidefinite Programming Performance Estimation. In: SIAM Journal on Optimization, Vol. 30, no.3, p. 2053-2082 (2020). doi:10.1137/19m1281368.

2. Haile, Darrell; Rowen, Louis; Tignol, Jean-Pierre. On Quaternion Algebras Split by a Given Extension, Clifford Algebras and Hyperelliptic Curves. In: Algebras and Representation Theory, Vol. 23, no.4, p. 1807-1826 (2020). doi:10.1007/s10468-019-09914-3.

3. Gutiérrez-Gómez, Leonardo; Vohryzek, Jakub; Chiêm, Benjamin; Baumann, Philipp S.; Conus, Philippe; Cuenod, Kim Do; Hagmann, Patric; Delvenne, Jean-Charles. Stable biomarker identification for predicting schizophrenia in the human connectome. In: NeuroImage: Clinical, Vol. 27, p. 102316 (2020). doi:10.1016/j.nicl.2020.102316.

4. Legat, Benoît; Tabuada, Paulo; Jungers, Raphaël M. Sum-of-Squares methods for controlled invariant sets with applications to model-predictive control. In: Nonlinear Analysis: Hybrid Systems, Vol. 36, p. 100858 (2020). doi:10.1016/j.nahs.2020.100858.

5. Gomez, Marco A.; Jungers, Raphaël M.; Michiels, Wim. On the m-dimensional Cayley–Hamilton theorem and its application to an algebraic decision problem inferred from the H2 norm analysis of delay systems. In: Automatica, Vol. 113, p. 108761 (2020). doi:10.1016/j.automatica.2019.108761.

6. Dechouniotis, Dimitrios; Athanasopoulos, Nikolaos; Leivadeas, Aris; Mitton, Nathalie; Jungers, Raphaël M.; Papavassiliou, Symeon. Edge Computing Resource Allocation for Dynamic Networks: The DRUID-NET Vision and Perspective. In: Sensors, Vol. 20, no.8, p. 2191 (2020). doi:10.3390/s20082191.

7. Berger, Guillaume; Absil, Pierre-Antoine; Jungers, Raphaël M.; Nesterov, Yurii. On the Quality of First-Order Approximation of Functions with Hölder Continuous Gradient. In: Journal of Optimization Theory and Applications, Vol. 185, p. 17-33 (2020). doi:10.1007/s10957-020-01632-x.

8. Quéguiner-Mathieu, Anne; Tignol, Jean-Pierre. Decomposability of orthogonal involutions in degree 12. In: Pacific Journal of Mathematics, Vol. 304, no.1, p. 169-180 (2020). doi:10.2140/pjm.2020.304.169.

9. Scheuer, A.; Grégoire, G.; Abisset-Chavanne, E.; Chinesta, F.; Keunings, Roland. Modelling the effect of particle inertia on the orientation kinematics of fibres and spheroids immersed in a simple shear flow. In: Computers & Mathematics with Applications, Vol. 79, no.3, p. 539-554 (2020). doi:10.1016/j.camwa.2018.12.039.

10. Berger, Guillaume; Jungers, Raphaël M. Formal Methods for Computing Hyperbolic Invariant Sets for Nonlinear Systems. In: IEEE Control Systems Letters, Vol. 4, no.1, p. 235-240 (2020). doi:10.1109/lcsys.2019.2923923.

Conference Papers

1. Dewez, Julien; Glineur, François. Lower bounds on the nonnegative rank using a nested polytopes formulation.

2. Farjadnia, Mahsa; Wang, Zheming; Jungers, Raphaël M.. Stability Analysis of Data Driven Complex Systems.

3. Jungers, Raphaël M.; Tabuada, Paulo. Non-local Linearization of Nonlinear Differential Equations via Polyflows.

4. Azfar, Umer; Catalano, Costanza; Charlier, Ludovic; Jungers, Raphaël M.. A Linear Bound on the K-Rendezvous Time for Primitive Sets of NZ Matrices. In: Developments in Language Theory : Lecture Notes in Computer Science, 2019, 9783030248857, p. 59-73. doi:10.1007/978-3-030-24886-4_4.

5. Athanasopoulos, Nikolaos; Jungers, Raphaël M.. Polyhedral Path-Complete Lyapunov Functions. In: 2019 IEEE 58th Conference on Decision and Control (CDC), IEEE, 2019, 9781728113982. doi:10.1109/cdc40024.2019.9029905.

6. Philippe, Matthew; Jungers, Raphaël M.. A complete characterization of the ordering of path-complete methods. In: Proceedings of the 22nd ACM International Conference on Hybrid Systems Computation and Control - HSCC '19, ACM Press, 2019, 9781450362825, p. 138-146. doi:10.1145/3302504.3311803.

7. Hamaide, Valentin; Glineur, François. Predictive maintenance of a rotating condenser inside a synchrocyclotron. In: CEUR Workshop Proceedings. Vol. 2491, p. 1-12 (2019).

8. Hautecoeur, Cécile; Glineur, François. Accelerating Nonnegative Matrix Factorization Over Polynomial Signals With Faster Projections. In: 2019 IEEE 29th International Workshop on Machine Learning for Signal Processing (MLSP), IEEE, 2019, 9781728108247, p. 1-6. doi:10.1109/mlsp.2019.8918844.

9. Wang, Zheming; Jungers, Raphaël M.; Ong, Chong-Jin. Computation of the maximal invariant set of linear systems with quasi-smooth nonlinear constraints. In: 2019 18th European Control Conference (ECC), IEEE, 2019, 9783907144008, p. 3803-3808. doi:10.23919/ecc.2019.8796145.

10. Wang, Zheming; Jungers, Raphaël M.; Flandroy, Quentin; Herregods, Baptiste; Hernalsteens, Cedric. Finite-horizon covariance control of state-affine nonlinear systems with application to proton beamline calibration. In: 2019 18th European Control Conference (ECC), IEEE, 2019, 9783907144008, p. 3740-3745. doi:10.23919/ecc.2019.8796121.