Works on COVID-19

I was not born an epidemiologist, but given the gravity of the current Covid-19 pandemic I am joining forces with colleagues from PSL, Inria and Sorbonne University to help include mobility data in the modeling of the epidemy.

• Communications:
• Dec 21, 2020: Report 2 on the effects of mobility and population density on the Covid-19 spread.
• May 26, 2020: Report 1 on the effects of confinement and deconfinement on mobility.
• Media coverage: Europe 1 (June 1st, 2020: Summary, Interview)
• Face au Virus: Research initiative at PSL where I am actively working
• Seminars organized by Gabriel Turinici.

State Estimation using reduced models and measurement data

The general topic here is to build a coherent mathematical framework to reconstruct the state of physical systems by using two types information from different nature. The first are mathematical models, often based on differential or integral equations, and the second are large and, possibly, noisy data sets which are now routinely available in many applications. This types of problems fall into the general scope of inverse problems, which have classically been addressed under a bayesian perspective due to their ill-posedness and high-dimensionality. In our case, we develop a more deterministic point of view where reduced modeling of parametrized PDEs play a prominent role to address the high-dimensionality. In this framework, we work on the following aspects:

• Optimal reconstruction algorithms for state estimation with reduced models
• Optimal sensor placement strategies
• Bringing the methodology to applications.
  • Hemodynamics (and the risk of arterial stenosis)
  • Real time monitoring of air pollution in the city
  • Real time monitoring of the population of neutrons in a reactor core

Numerical solution of linear Boltzmann models

Kinetic problems accumulate a number of obstructions that make their numerical solution challenging:

• The solutions depend on a relatively high number of variables (time, space and velocity).
• The solutions have usually low regularity
• The collision operator induces a global coupling which gives rise to large, densely populated matrices using standard discretizations.
• The coefficients of the equation, usually called cross-sections, are highly oscilatory as a function of the kinetic energy of the particles.

As a result of these difficulties, one often tacitly assumes that the computed numerical solution represents the corresponding continuous solution reasonably well, without being, however, able to actually quantify the quality in any rigorous sense. I am interested in developing a robust mathematical framework to overcome this issue. My main contributions are the following:

• In a recent contribution, we have developed a new algorithm to solve the neutron transport equation (a linear Boltzmann model also known as radiative transfer equation) where the deviation of the numerical result from the exact continuous solution is certifiably quantified and set to meet a given target accuracy with respect to a relevant norm.
• Due to the high oscillations in energy, the cross-sections that are used in the neutron transport equation are the result of an involved preprocessing step based on physical arguments and whose mathematical foundations are hard to pin down. In a recent work, we have considered the homogenization of the energy variable without this preprocessing to understand the nature of the limiting equation in presence of high energy oscillations of the coefficients. The homogenized equation presents a very involved memory kernel.
• During my PhD, I have also taken part in the implementation of a production neutron transport code in CEA called Minaret.

Parallelization by Decomposition of the Time domain

• An Adaptive Parareal Algorithm
• Application of the algorithm to the time-dependent neutron transport equation