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Job Record #15722
TitlePhD in the Fluminance Inria group
CategoryJob in Academia
LocationFrance, Britany, Rennes
InternationalYes, international applications are welcome
Closure DateFriday, May 31, 2019

We seek a candidate for a PhD position within the Fluminance team, (INRIA Rennes, France). This study at 
the cross section between fluid mechanics, physical geophysics, applied mathematics and statistical 
learning theory. We will investigate the use of semi-group theory techniques together with reproducible 
kernel Hilbert space theory for the characterization of dynamical systems from observations (see section 
« Detailed subject » for more information).



The candidate will work in the Fluminance team located in Rennes. The team is part of INRIA (, 
which is one of the leading research center in computer sciences in France. Fluminance is as well affiliated 
to the mathematics research institute of the Rennes University (IRMAR). This PhD thesis will be led within 
a strong collaboration with Imperial College and Ifremer (french oceanic research institute). The main 
research activities of Fluminance focus on the study of turbulent flows from image data sequences, which 
encompasses many issues for the analysis of experimental and geophysical flows. We refer the candidates 
to the team’s website for more information:


Duration: 36 month.


Detailed subject            

The precise characterization of geophysical phenomena is becoming a crucial need in many aspects of our 
everyday life as it may strongly impact many environmental and economical fields. We may think, among 
others, to applications related to climate studies, oceanographic analysis or weather forecasting which are 
of paramount importance for the study of global warming, the tracking of polluting sheets or the 
prediction of catastrophic events. Unfortunately, the laws ruling such geophysical processes depend on 
state variables evolving in huge dimensional spaces with a strong scale coupling in space and time. The 
range of these interactions is so large that only large-scale representations of the system of interest can 
be simulated. In the other hand, one may have access nowadays to series of finely resolved data 
sequences depicting the footprint of the small-scale flow action. 


The goal of this PhD will be to explore the constitution of large-scale representations of flow dynamics 
from a spectral representation of the infinitesimal generator of Frobenious-Perron and Koopman operators 
driving the dynamical system’s measure and observables respectively [7]. To devise affordable techniques 
for such a spectral representation of infinite dimensional operators, we will embed these operators onto a 
family of reproducible kernel Hilbert spaces [1,2,4] driven by the dynamics. This embedding onto a 
manifold of smooth functions should allow us to characterize several meaningful mathematical properties 
of the resulting operator semi-groups authorizing to apply classical finite dimensional numerical methods 
for their spectral analysis and estimation. Those spectral representations will then be used to infer and 
study fast data assimilation techniques to couple the dynamical system to high resolution data. They also 
open routes to estimate efficiently the leading Lyapunov exponents associated to the underlying dynamics 
or to infer meaningful ergodic properties. The characterization of the spectrum of these operators has 
recently raised a huge interest in the fluid mechanics community [10,12,13]. However, the techniques 
proposed so far use implicitly strong questionable finite dimensional assumptions. They furthermore rely 
only on sequence of data. In this work, will use the data as well, but also a (possibly imperfect) 
representation of the dynamics. 


In a second time, we will study the extension of this deterministic setting to a stochastic representation of 
the dynamics as recently proposed in [5] and [11]. As a matter of fact, in order to incorporate inherent 
uncertainties or errors and to better represent the effect of the neglected scales, there is a growing 
interest to set up random representations for those flows [5,9,11]. The modelling and the handling along 
time of uncertainties are crucial for instance for ensemble forecasting and data assimilation issues.  In this 
study, we propose to stick to a recent derivation [11] that naturally emerges from a decomposition of the 
flow velocity field into a differentiable drift component and a time uncorrelated uncertainty random term. 
This framework has shown to enable the derivation of meaningful dynamical random oceanic models with 
results greatly improved when compared to deterministic simulations [3].


The resulting models open very exciting mathematical questions on those modified equations for which 
little is known. The characterization of the spectrum associated to random infinitesimal generator of the 
corresponding Frobenius-Perron and Koopman operators as well as ergodic properties should allow us to 
extend the data assimilation context tackled in the first part of this thesis beyond the Gaussian coupling 
strategies, which constitute indeed the Achile’s heel of the usual data assimilation procedures based 
either on optimal control strategies [8] or on ensemble filtering techniques [5]. The mathematical study of 
the corresponding stochastic systems is also of major interest in order to exhibit their asymptotic 
properties or to derive stable discrete schemes for their numerical simulation.


N. Aronszajn. Theory of reproducing kernels. Trans. of the American Math. Society, 68(337-404), 1950. 
A. Berlinet and C. Thomas-Agnan. Reproducible kernel Hilbert spaces in Probability and Statistics. Kluwer 
Academic Publishers, 2001.
B. Chapron, P. Dérian, E. Mémin, V. Resseguier
"Large scale flows under location uncertainty: a consistent stochastic framework",
Quarterly Journal of the Royal Meteorological Society, 144 (710): 251-260.
F. Cucker and S. Smale. On the mathematical foundation of learning. Bul. of the Amer. Math. Soc., 39(1):1–
49, 2001. 
G. Evensen. Sequential data assimilation with a non linear quasi-geostrophic model using Monte Carlo 
methods to forecast error statistics. J. Geophys. Res., 99 (C5)(10):143–162, 1994.
D. Holm, (2015), Variational principles for stochastic fluid dynamics., Proc Math Phys Eng Sci, Vol: 471, 
ISSN: 1364-5021.
A. Lasota and M. Mackey. Chaos, fractals, and noise, stochastic aspect of dynamics,, volume 97 of Applied 
Mathematical Sciences. Spinger-Verlag New York, second edition edition, 1994. 
J. L. Lions. Optimal control of systems governed by PDEs. Springer-Verlag, New-York, 1971.
Majda A (2012) Challenges in climate science and contemporary applied mathematics. Commun Pure Appl 
Math 65(7):920–
A. Mauroy and I. Mezic. Global stability analysis using the eigenfunctions of the koopman operator. IEEE 
Transactions on Automatic Control, 61(3):3356–3369, 2016. 
E. Mémin. (2014). Fluid flow dynamics under location uncertainty. Geophysical & Astrophysical Fluid 
Dynamics , 108(2): 119-146.
I. Mezic. Spectral properties of dynamical systems, model reduction and decomposition. Nonlinear 
Dynamics, 41:309–325, 2005.
C. Rowley, I. Mezic, S. Bagheri, P. Schlatter and D. Henningson. Spectral analysis of nonlinear flows, J. 
Fluid Mech., 641: 115-127, 2009. 


Skills and profile

The candidate should have a solid background in fluid mechanics/oceanography and/or in applied 
mathematics. She/he must have a good knowledge of Matlab or Python, and Fortran or C/C++. He/She 
must have  a master degree or engineer school degree related to fluid mechanics, computational physics, 
geophysics  or applied mathematics.

Instruction to apply

Application should be done from the web site:
Please submit online : your resume, cover letter and letters of recommendation eventually

For more information, please contact

Benefits package
Subsidized meals
Professional equipment available (videoconferencing, loan of computer equipment, etc.)
Social, cultural and sports events and activities
Access to vocational training
Social security coverage
monthly gross salary amounting to 1982 euros for the first and second years and 2085 euros for the third 
Contact Information:
Please mention the CFD Jobs Database, record #15722 when responding to this ad.
NameMemin Etienne
Email ApplicationYes
Record Data:
Last Modified17:26:23, Tuesday, March 12, 2019

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