Publications

2019

• Uncrowded Hypervolume Improvement: COMO-CMA-ES and the Sofomore framework
  
  • Touré Cheikh
  • Hansen Nikolaus
  • Auger Anne
  • Brockhoff Dimo
  , 2019. (10.1145/3321707.3321852)
  DOI : 10.1145/3321707.3321852
• Mixed-Integer Benchmark Problems for Single-and Bi-Objective Optimization
  
  • Tušar Tea
  • Brockhoff Dimo
  • Hansen Nikolaus
  , 2019. We introduce two suites of mixed-integer benchmark problems to be used for analyzing and comparing black-box optimization algorithms. They contain problems of diverse difficulties that are scalable in the number of decision variables. The bbob-mixint suite is designed by partially discretizing the established BBOB (Black-Box Optimization Benchmarking) problems. The bi-objective problems from the bbob-biobj-mixint suite are, on the other hand, constructed by using the bbob-mixint functions as their separate objectives. We explain the rationale behind our design decisions and show how to use the suites within the COCO (Comparing Continuous Optimizers) platform. Analyzing two chosen functions in more detail, we also provide some unexpected findings about their properties.
• Benchmarking Algorithms from the platypus Framework on the Biobjective bbob-biobj Testbed
  
  • Brockhoff Dimo
  • Tušar Tea
  , 2019, 7. One of the main goals of the COCO platform is to produce, collect , and make available benchmarking performance data sets of optimization algorithms and, more concretely, algorithm implementations. For the recently proposed biobjective bbob-biobj test suite, less than 20 algorithms have been benchmarked so far but many more are available to the public. We therefore aim in this paper to benchmark several available multiobjective optimization algorithms on the bbob-biobj test suite and discuss their performance. We focus here on algorithms implemented in the platypus framework (in Python) whose main advantage is its ease of use without the need to set up many algorithm parameters. (10.1145/3319619.3326896)
  DOI : 10.1145/3319619.3326896
• Rare Event Estimation and Robust Optimization Methods with Application to ORC Turbine Cascade
  
  • Razaaly Nassim
  , 2019. This thesis aims to formulate innovative Uncertainty Quantification (UQ) methods in both Robust Optimization (RO) and Reliability-Based Design Optimization (RBDO) problems. The targeted application is the optimization of supersonic turbines used in Organic Rankine Cycle (ORC) power systems.Typical energy sources for ORC power systems feature variable heat load and turbine inlet/outlet thermodynamic conditions. The use of organic compounds with a heavy molecular weight typically leads to supersonic turbine configurations featuring supersonic flows and shocks, which grow in relevance in the aforementioned off-design conditions; these features also depend strongly on the local blade shape, which can be influenced by the geometric tolerances of the blade manufacturing. A consensus exists about the necessity to include these uncertainties in the design process, so requiring fast UQ methods and a comprehensive tool for performing shape optimization efficiently.This work is decomposed in two main parts. The first one addresses the problem of rare events estimation, proposing two original methods for failure probability (metaAL-OIS and eAK-MCS) and one for quantile computation (QeAK-MCS). The three methods rely on surrogate-based (Kriging) adaptive strategies, aiming at refining the so-called Limit-State Surface (LSS) directly, unlike Subset Simulation (SS) derived methods. Indeed, the latter consider intermediate threshold associated with intermediate LSSs to be refined. This direct refinement property is of crucial importance since it enables the adaptability of the developed methods for RBDO algorithms. Note that the proposed algorithms are not subject to restrictive assumptions on the LSS (unlike the well-known FORM/SORM), such as the number of failure modes, however need to be formulated in the Standard Space. The eAK-MCS and QeAK-MCS methods are derived from the AK-MCS method and inherit a parallel adaptive sampling based on weighed K-Means. MetaAL-OIS features a more elaborate sequential refinement strategy based on MCMC samples drawn from a quasi-optimal ISD. It additionally proposes the construction of a Gaussian mixture ISD, permitting the accurate estimation of small failure probabilities when a large number of evaluations (several millions) is tractable, as an alternative to SS. The three methods are shown to perform very well for 2D to 8D analytical examples popular in structural reliability literature, some featuring several failure modes, all subject to very small failure probability/quantile level. Accurate estimations are performed in the cases considered using a reasonable number of calls to the performance function.The second part of this work tackles original Robust Optimization (RO) methods applied to the Shape Design of a supersonic ORC Turbine cascade. A comprehensive Uncertainty Quantification (UQ) analysis accounting for operational, fluid parameters and geometric (aleatoric) uncertainties is illustrated, permitting to provide a general overview over the impact of multiple effects and constitutes a preliminary study necessary for RO. Then, several mono-objective RO formulations under a probabilistic constraint are considered in this work, including the minimization of the mean or a high quantile of the Objective Function. A critical assessment of the (Robust) Optimal designs is finally investigated.
• Machine learning based on Hawkes processes and stochastic optimization
  
  • Bompaire Martin
  , 2019. The common thread of this thesis is the study of Hawkes processes. These point processes decrypt the cross-causality that occurs across several event series. Namely, they retrieve the influence that the events of one series have on the future events of all series. For example, in the context of social networks, they describe how likely an action of a certain user (such as a Tweet) will trigger reactions from the others.The first chapter consists in a general introduction on point processes followed by a focus on Hawkes processes and more specifically on the properties of the widely used exponential kernels parametrization. In the following chapter, we introduce an adaptive penalization technique to model, with Hawkes processes, the information propagation on social networks. This penalization is able to take into account the prior knowledge on the social network characteristics, such as the sparse interactions between users or the community structure, to reflect them on the estimated model. Our technique uses data-driven weighted penalties induced by a careful analysis of the generalization error.Next, we focus on convex optimization and recall the recent progresses made with stochastic first order methods using variance reduction techniques. The fourth chapter is dedicated to an adaptation of these techniques to optimize the most commonly used goodness-of-fit of Hawkes processes. Indeed, this goodness-of-fit does not meet the gradient-Lipschitz assumption that is required by the latest first order methods. Thus, we work under another smoothness assumption, and obtain a linear convergence rate for a shifted version of Stochastic Dual Coordinate Ascent that improves the current state-of-the-art. Besides, such objectives include many linear constraints that are easily violated by classic first order algorithms, but in the Fenchel-dual problem these constraints are easier to deal with. Hence, our algorithm's robustness is comparable to second order methods that are very expensive in high dimensions.Finally, the last chapter introduces a new statistical learning library for Python 3 with a particular emphasis on time-dependent models, tools for generalized linear models and survival analysis. Called tick, this library relies on a C++ implementation and state-of-the-art optimization algorithms to provide very fast computations in a single node multi-core setting. Open-sourced and published on Github, this library has been used all along this thesis to perform benchmarks and experiments.
• Backtracking strategies for accelerated descent methods with smooth composite objectives
  
  • Calatroni Luca
  • Chambolle Antonin
  SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2019, 29 (3), pp.1772--1798. We present and analyse a backtracking strategy for a general Fast Iterative Shrink-age/Thresholding Algorithm which has been recently proposed in [11] for strongly convex objective functions. Differently from classical Armijo-type line searching, our backtracking rule allows for local increase and decrease of the Lipschitz constant estimate along the iterations. For such strategy accelerated convergence rates are proved and numerical results are shown for some exemplar imaging problems.
• Uncertainty Quantification For Stochastic Approximation limits and applications to risk/performance metrics in finance
  
  • Crépey Stéphane
  • Fort Gersende
  • Gobet Emmanuel
  • Stazhynski Uladzislau
  , 2019.
• Population dynamics and epidemic processes on a trade network
  
  • Montagnon Pierre
  , 2019. This thesis discusses the mathematical modelling of population dynamics on cattle trade networks coupled with epidemic processes.We first consider metapopulation models taking into account local demographic dynamics (immigration, births, deaths and animal movements due to trade between the nodes of the network). Recurrence and ergodicity criteria are stated for Markovian models with deterministic local dynamics and stochastic inter-nodal transferts, for a multitype branching process with immigration and for a jump process with logistic rates on a finite state space. In these last two cases, we study scaling limits of processes over finite time intervals and their stability over time scales that are exponentials of the scaling parameter.In a second part, we define a coupling of the jump population models considered with an SIR (Susceptible --- Infected --- Removed) epidemic dynamics. The resulting process accounts for local infectious contacts and pathogen propagation on the network due to movements of infective animals. We approximate the epidemic process by a multitype branching process on finite time intervals, then provide an iterative method to compute the probability of a emph{major epidemic outbreak}, defined as the event of survival of the approximating branching process. We then show that conditionally on a major epidemic outbreak and under a stability condition for an endemic equilibrium of the associated dynamical system, the extinction time and final size of the epidemic grow at least exponentially with respect to the scaling parameter of the model.We finally perform a numerical application of the theoretical results obtained on the SIR model coupled with logistic population dynamics. Calibrating the demographical model parameters on the 2015 Finistère cattle trade network, we compute indicators of the epidemic vulnerability of the network induced by individual holdings. We detail a protocol to assess the relative efficiency of three types of control strategies (screening at importation, isolation and vaccination) targeting the holdings identified as critical for the computed indicators.
• Microwave Tomographic Imaging of Cerebrovascular Accidents by Using High-Performance Computing
  
  • Tournier Pierre-Henri
  • Aliferis Ioannis
  • Bonazzoli Marcella
  • de Buhan Maya
  • Darbas Marion
  • Dolean Victorita
  • Hecht Frédéric
  • Jolivet Pierre
  • El Kanfoud Ibtissam
  • Migliaccio Claire
  • Nataf Frédéric
  • Pichot Christian
  • Semenov Serguei
  Parallel Computing, Elsevier, 2019, 85, pp.88-97. The motivation of this work is the detection of cerebrovascular accidents by microwave tomographic imaging. This requires the solution of an inverse problem relying on a minimization algorithm (for example, gradient-based), where successive iterations consist in repeated solutions of a direct problem. The reconstruction algorithm is extremely computationally intensive and makes use of efficient parallel algorithms and high-performance computing. The feasibility of this type of imaging is conditioned on one hand by an accurate reconstruction of the material properties of the propagation medium and on the other hand by a considerable reduction in simulation time. Fulfilling these two requirements will enable a very rapid and accurate diagnosis. From the mathematical and numerical point of view, this means solving Maxwell's equations in time-harmonic regime by appropriate domain decomposition methods, which are naturally adapted to parallel architectures. (10.1016/j.parco.2019.02.004)
  DOI : 10.1016/j.parco.2019.02.004
• The nonlinear Schrödinger equation driven by jump processes
  
  • de Bouard Anne
  • Hausenblas Erika
  Australian Journal of Mathematical Analysis and Applications, Austral Internet Publishing, 2019, 475 (1), pp.215-252. (10.1016/j.jmaa.2019.02.036)
  DOI : 10.1016/j.jmaa.2019.02.036
• Schauder Estimates for a Class of Potential Mean Field Games of Controls
  
  • Bonnans Joseph Frédéric
  • Hadikhanloo Saeed
  • Pfeiffer Laurent
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2019, pp.34. An existence result for a class of mean field games of controls is provided. In the considered model, the cost functional to be minimized by each agent involves a price depending at a given time on the controls of all agents and a congestion term. The existence of a classical solution is demonstrated with the Leray-Schauder theorem; the proof relies in particular on a priori bounds for the solution, which are obtained with the help of a potential formulation of the problem.
• Eventual linear convergence rate of an exchange algorithm for superresolution
  
  • de Gournay Frédéric
  • Weiss Pierre
  • Flinth Axel
  , 2019. We study an iterative discretization algorithm for solving optimization problems regularized by the total variation norm. Its design relies on ideas from the Frank-Wolfe algorithm, as well as from semi-infinite programming. For smooth regularity terms, we prove an eventual linear convergence rate guarantee.
• A stochastic SIR model on a graph with epidemiological and population dynamics occurring over the same time scale
  
  • Montagnon Pierre
  Journal of Mathematical Biology, Springer, 2019, 79 (1), pp.31-62. We define and study an open stochastic SIR (Susceptible-Infected-Removed) model on a graph in order to describe the spread of an epidemic on a cattle trade network with epidemiological and demographic dynamics occurring over the same time scale. Population transition intensities are assumed to be density-dependent with a constant component, the amplitude of which determines the overall scale of the population process. Standard branching approximation results for the epidemic process are first given, along with a numerical computation method for the probability of a major epidemic outbreak. This procedure is illustrated using real data on trade-related cattle movements from a densely populated livestock farming region in western France (Finistere) and epidemiological parameters corresponding to an infectious epizootic disease. Then we exhibit an exponential lower bound for the extinction time and the total size of the epidemic in the stable endemic case as a scaling parameter goes to infinity using results inspired by the Freidlin-Wentzell theory of large deviations from a dynamical system. (10.1007/s00285-019-01349-0)
  DOI : 10.1007/s00285-019-01349-0
• The time-dependent diffusivity in the abdominal ganglion of Aplysia californica: experiments and simulations
  
  • Nguyen Khieu Van
  • Le Bihan Denis
  • Ciobanu Luisa
  • Li Jing-Rebecca
  Biomedical Physics & Engineering Express, IOP Publishing, 2019, 5 (4), pp.045036. The nerve cells of theAplysiaare much larger than mammalian neurons. Using theAplysiaganglia tostudy the relationship between the cellular structure and the diffusion MRI signal can potentially shedlight on this relationship for more complex organisms. We measured the dMRI signal of chemically-fixed abdominal ganglia of theAplysiaat several diffusion times. At the diffusion times measured andobserved at low b-values, the dMRI signal is mono-exponential and can be accurately represented bythe parameter ADC(Apparent Diffusion Coefficient). We performed numerical simulations of waterdiffusion for the large cell neurons in the abdominal ganglia after creating geometrical configurationsby segmenting high resolution T2-weighted(T2w)images to obtain the cell outline and thenincorporating a manually generated nucleus. The results of the numerical simulations validate theclaim that water diffusion in the large cell neurons is in the short diffusion time regime at ourexperimental diffusion times. Then, using the analytical short time approximation(STA)formula forthe ADC, we showed that in order to explain the experimentally observed behavior, it is necessary toconsider the nucleus and the cytoplasm as two separate diffusion compartments. By using a twocompartment STA model, we were able to illustrate the effect of the highly irregular shape of the cell nucleus on the ADC. (10.1088/2057-1976/ab301e)
  DOI : 10.1088/2057-1976/ab301e
• Travelling wave mathematical analysis and efficient numerical resolution for a realistic model of solid propellant combustion
  
  • François Laurent
  • Dupays Joel
  • Davidenko Dmitry
  • Massot Marc
  , 2019. We investigate a model of solid propellant combustion involving surface pyrolysis coupled to finite activation energy gas phase combustion at unit Lewis number. Existence and uniqueness of a travelling wave solution are established by extending dynamical system tools classically used for premixed flames, dealing with the additional difficulty arising from the surface regression and pyrolysis. An efficient shooting method allows to solve the problem in phase space without resorting to space dis-cretization nor fixed-point Newton iterations. The results are compared to solutions from a CFD code developed at ONERA, assessing the efficiency and potential of the method. (10.13009/EUCASS2019-624)
  DOI : 10.13009/EUCASS2019-624
• Computational study of Al droplet combustion in a burning solid propellant environment
  
  • Mathieu Muller
  • Dmitry Davidenko
  • Vincent Giovangigli
  , 2019. Combustion of a single aluminum particle in a solid propellant flame is studied with complex chemistry and detailed molecular transport. The reacting flow around the droplet is treated as spherically symmetric. In a new droplet model, an average effect of the alumina cap is taken into account. The combustion model is validated with respect to experimental results and an axisymmetric simulation. The combustion of two particle classes is simulated with different surface reaction mechanisms. The effect of surface mechanism on the temporal evolution of particle parameters is investigated. The burning time and residue size are evaluated for the simulated cases. (10.13009/EUCASS2019-202)
  DOI : 10.13009/EUCASS2019-202
• Sub-Riemannian Ricci curvatures and universal diameter bounds for 3-Sasakian manifolds
  
  • Rizzi Luca
  • Silveira Pavel
  Journal of the Institute of Mathematics of Jussieu, Cambridge University Press, 2019, 18 (4), pp.783-827. For a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev-Zelenko-Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet-Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian. As an application, we consider the sub-Riemannian structure of 3-Sasakian manifolds, for which we provide explicit curvature formulas. We prove that any complete 3-Sasakian structure of dimension 4d + 3, with d > 1, has sub-Riemannian diameter bounded by π. When d = 1, a similar statement holds under additional Ricci bounds. These results are sharp for the natural sub-Riemannian structure of the quaternionic Hopf fibrations on the 4d+3 dimensional sphere, whose exact sub-Riemannian diameter is π, for all d ≥ 1. (10.1017/S1474748017000226)
  DOI : 10.1017/S1474748017000226
• Assessment of Deterministic Shape Optimizations Within a Stochastic Framework for Supersonic Organic Rankine Cycle Nozzle Cascades
  
  • Romei Alessandro
  • Congedo Pietro Marco
  • Persico Giacomo
  Journal of Engineering for Gas Turbines and Power, American Society of Mechanical Engineers, 2019, 141 (7). The design of converging–diverging blades for organic Rankine cycle (ORC) applications widely relies on automated shape-optimization processes. As a result, the optimization produces an adapted-nozzle cascade at the design conditions. However, only few works account for the uncertainties in those conditions and their consequences on cascade performance. The proposed solution, i.e., including uncertainties within the optimization routine, demands an overall huge computational cost to estimate the target output statistic at each iteration of the optimization algorithm. With the aim of understanding if this computational cost is avoidable, we study the impact of uncertainties in the design conditions on the robustness of deterministically optimized profiles. Several optimized blades, obtained with different objective functions, constraints, and design variables, are considered in the present numerical analysis, which features a turbulent compressible flow solver and a state-of-the-art uncertainty-quantification (UQ) method. By including measured field variations in the formulation of the UQ problem, we show that a deterministic shape optimization already improves the robustness of the profile with respect to the baseline configuration. Guidelines about objective functions and blade parametrizations for deterministic optimizations are also provided. Finally, a novel methodology to estimate the mass-flow-rate probability density function (PDF) for choked supersonic turbines is proposed, along with a robust reformulation of the constraint problem without increasing the computational cost. (10.1115/1.4042925)
  DOI : 10.1115/1.4042925
• Acoustic and geoacoustic inverse problems in randomly perturbed shallow-water environments
  
  • Dumaz Laure
  • Garnier Josselin
  • Lepoultier Guilhem
  Journal of the Acoustical Society of America, Acoustical Society of America, 2019, 146 (1), pp.458-469. (10.1121/1.5116569)
  DOI : 10.1121/1.5116569
• Central limit theorem for discretization errors based on general stopping time sampling
  
  • Gobet Emmanuel
  • Landon Nicolas
  • Stazhynski Uladzislau
  , 2019.
• Total roto-translational variation
  
  • Chambolle Antonin
  • Pock Thomas
  Numerische Mathematik, Springer Verlag, 2019, 142 (3), pp.611-666. We consider curvature depending variational models for image regularization, such as Euler's elastica. These models are known to provide strong priors for the continuity of edges and hence have important applications in shape-and image processing. We consider a lifted convex representation of these models in the roto-translation space: In this space, curvature depending variational energies are represented by means of a convex functional defined on divergence free vector fields. The line energies are then easily extended to any scalar function. It yields a natural generalization of the total variation to the roto-translation space. As our main result, we show that the proposed convex representation is tight for characteristic functions of smooth shapes. We also discuss cases where this representation fails. For numerical solution, we propose a staggered grid discretization based on an averaged Raviart-Thomas finite elements approximation. This discretization is consistent, up to minor details, with the underlying continuous model. The resulting non-smooth convex optimization problem is solved using a first-order primal-dual algorithm. We illustrate the results of our numerical algorithm on various problems from shape-and image processing. (10.1007/s00211-019-01026-w)
  DOI : 10.1007/s00211-019-01026-w
• A cortical-inspired model for orientation-dependent contrast perception: a link with Wilson-Cowan equations
  
  • Bertalmío Marcelo
  • Calatroni Luca
  • Franceschi Valentina
  • Franceschiello Benedetta
  • Prandi Dario
  , 2019. We consider a differential model describing neuro-physiological contrast perception phenomena induced by surrounding orientations. The mathematical formulation relies on a cortical-inspired modelling [10] largely used over the last years to describe neuron interactions in the primary visual cortex (V1) and applied to several image processing problems [12,19,13]. Our model connects to Wilson-Cowan-type equations [23] and it is analogous to the one used in [3,2,14] to describe assimilation and contrast phenomena, the main novelty being its explicit dependence on local image orientation. To confirm the validity of the model, we report some numerical tests showing its ability to explain orientation-dependent phenomena (such as grating induction) and geometric-optical illusions [21,16] classically explained only by filtering-based techniques [6,18]. (10.1007/978-3-030-22368-7_37)
  DOI : 10.1007/978-3-030-22368-7_37
• Optimal control of an artificial microbial differentiation system for protein bioproductioń
  
  • Weill Elise
  • Andreani Virgile
  • Aditya Chetan
  • Martinon Pierre
  • Ruess Jakob
  • Batt Grégory
  • Bonnans Frédéric
  , 2019. The production of recombinant proteins is a problem of significant interest in bioengineering. Because of the existing trade-off between cellular growth and protein production, these two processes are separated in time in most commonly-employed strategies: a growth phase is followed by a production phase. Here, we investigate the potential of an alternative strategy using artificial cell specialization and differentiation systems in which cells either grow ("growers") or produce proteins ("producers") and growers can irreversibly "differentiate" into producers. Inspired by an existing two-population system implemented in yeast, we propose a model of a "yeast synthetic stem cell system" and define an optimal control problem to maximize bioproduction. Analytically, we first establish the well-posedness of the problem. Then, we prove the existence of an optimal control and derive non trivial optimality conditions. We finally use these results to find numerical optimal solutions. We conclude by a discussion of extensions of this work to models that capture the heterogeneity of the cell response to differentiation signals.
• Main effects and interactions in mixed and incomplete data frames
  
  • Robin Geneviève
  • Klopp Olga
  • Josse Julie
  • Moulines Éric
  • Tibshirani Robert
  Journal of the American Statistical Association, Taylor & Francis, 2019. A mixed data frame (MDF) is a table collecting categorical, numerical and count observations. The use of MDF is widespread in statistics and the applications are numerous from abundance data in ecology to recommender systems. In many cases, an MDF exhibits simultaneously main effects, such as row, column or group effects and interactions, for which a low-rank model has often been suggested. Although the literature on low-rank approximations is very substantial, with few exceptions, existing methods do not allow to incorporate main effects and interactions while providing statistical guarantees. The present work fills this gap. * This work has been funded by the DataScience Inititiative (Ecole Polytechnique) and the Russian Academic Excellence Project '5-100 (10.1080/01621459.2019.1623041)
  DOI : 10.1080/01621459.2019.1623041
• Non-asymptotic Analysis of Biased Stochastic Approximation Scheme
  
  • Karimi Belhal
  • Miasojedow Blazej
  • Moulines Éric
  • Wai Hoi-To
  , 2019, pp.1 - 33. Stochastic approximation (SA) is a key method used in statistical learning. Recently, its non-asymptotic convergence analysis has been considered in many papers. However, most of the prior analyses are made under restrictive assumptions such as unbiased gradient estimates and convex objective function, which significantly limit their applications to sophisticated tasks such as online and reinforcement learning. These restrictions are all essentially relaxed in this work. In particular, we analyze a general SA scheme to minimize a non-convex, smooth objective function. We consider update procedure whose drift term depends on a state-dependent Markov chain and the mean field is not necessarily of gradient type, covering approximate second-order method and allowing asymptotic bias for the one-step updates. We illustrate these settings with the online EM algorithm and the policy-gradient method for average reward maximization in reinforcement learning.