Publications

2019

• Simulation of virtual cohorts increases predictive accuracy of cognitive decline in MCI subjects
  
  • Koval Igor
  • Allassonnière Stéphanie
  • Durrleman Stanley
  , 2019. The ability to predict the progression of biomarkers, notably in NDD, is limited by the size of the longitudinal data sets, in terms of number of patients, number of visits per patients and total follow-up time. To this end, we introduce a data augmentation technique that is able to reproduce the variability seen in a longitudinal training data set and simulate continuous biomarkers trajectories for any number of virtual patients. Thanks to this simulation framework, we propose to transform the training set into a simulated data set with more patients, more time-points per patient and longer follow-up duration. We illustrate this approach on the prediction of the MMSE of MCI subjects of the ADNI data set. We show that it allows to reach predictions with errors comparable to the noise in the data, estimated in test/retest studies, achieving a improvement of 37% of the mean absolute error compared to the same non-augmented model.
• Numerical approximations of McKean Anticipative Backward Stochastic Differential Equations arising in Initial Margin requirements
  
  • Agarwal Ankush
  • de Marco Stefano
  • Gobet Emmanuel
  • López-Salas José G
  • Noubiagain Fanny
  • Zhou Alexandre
  ESAIM: Proceedings and Surveys, EDP Sciences, 2019, 65, pp.1-26. We introduce a new class of anticipative backward stochastic differential equations with a dependence of McKean type on the law of the solution, that we name MKABSDE. We provide existence and uniqueness results in a general framework with relatively general regularity assumptions on the coefficients. We show how such stochastic equations arise within the modern paradigm of derivative pricing where a central counterparty (CCP) requires the members to deposit variation and initial margins to cover their exposure. In the case when the initial margin is proportional to the Conditional Value-at-Risk (CVaR) of the contract price, we apply our general result to define the price as a solution of a MKABSDE. We provide several linear and non-linear simpler approximations, which we solve using different numerical (deterministic and Monte-Carlo) methods. (10.1051/proc/201965001)
  DOI : 10.1051/proc/201965001
• Principal-agent problem with multiple principals
  
  • Hu Kaitong
  • Ren Zhenjie
  • Yang Junjian
  , 2019. We consider a moral hazard problem with multiple principals in a continuous-time model. The agent can only work exclusively for one principal at a given time, so faces an optimal switching problem. Using a randomized formulation, we manage to represent the agent's value function and his optimal effort by an Itô process. This representation further helps to solve the principals' problem in case we have infinite number of principals in the sense of mean field game. Finally the mean field formulation is justified by an argument of propagation of chaos.
• Automatic Detection of Interplanetary Coronal Mass Ejections from In Situ Data: A Deep Learning Approach
  
  • Nguyen Gautier
  • Aunai Nicolas
  • Fontaine Dominique
  • Le Pennec Erwan
  • van den Bossche Joris
  • Jeandet Alexis
  • Bakkali Brice
  • Vignoli Louis
  • Regaldo-Saint Blancard Bruno
  The Astrophysical Journal, American Astronomical Society, 2019, 874 (2), pp.145. Decades of studies have suggested several criteria to detect Interplanetary coronal mass ejections (ICME) in time series from in-situ spacecraft measurements. Among them the most common are an enhanced and smoothly rotating magnetic field, a low proton temperature and a low plasma beta. However, these features are not all observed for each ICME due to their strong variability. Visual detection is time-consuming and biased by the observer interpretation leading to non exhaustive, subjective and thus hardly reproducible catalogs. Using convolutional neural networks on sliding windows and peak detection, we provide a fast, automatic and multi-scale detection of ICMEs. The method has been tested on the in-situ data from WIND between 1997 and 2015 and on the 657 ICMEs that were recorded during this period. The method offers an unambiguous visual proxy of ICMEs that gives an interpretation of the data similar to what an expert observer would give. We found at a maximum 197 of the 232 ICMEs of the 2010-2015 period (recall 84 +-4.5 % including 90% of the ICMEs present in the lists of Nieves-Chinchilla et al. (2015) and Chi et al. (2016). The minimal number of False Positives was 25 out of 158 predicted ICMEs (precision 84+-2.6%). Although less accurate, the method also works with one or several missing input parameters. The method has the advantage of improving its performance by just increasing the amount of input data. The generality of the method paves the way for automatic detection of many different event signatures in spacecraft in-situ measurements. (10.3847/1538-4357/ab0d24)
  DOI : 10.3847/1538-4357/ab0d24
• From zero transmission to trapped modes in waveguides
  
  • Chesnel Lucas
  • Pagneux Vincent
  Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2019. We consider the time-harmonic scattering wave problem in a 2D waveguide at wavenumber k such that one mode is propagating in the far field. In a first step, for a given k, playing with one scattering branch of finite length, we demonstrate how to construct geometries with zero transmission. The main novelty in this result is that the symmetry of the geometry is not needed: the proof relies on the unitary structure of the scattering matrix. Then, in a second step, from a waveguide with zero transmission, we show how to build geometries supporting trapped modes associated with eigenvalues embedded in the continuous spectrum. For this second construction, using the augmented scattering matrix and its unitarity, we play both with the geometry and the wavenumber. Finally, the mathematical analysis is supplemented by numerical illustrations of the results.
• COCO: The Large Scale Black-Box Optimization Benchmarking (bbob-largescale) Test Suite
  
  • Elhara Ouassim Ait
  • Varelas Konstantinos
  • Nguyen Duc Hung
  • Tušar Tea
  • Brockhoff Dimo
  • Hansen Nikolaus
  • Auger Anne
  , 2019. The bbob-largescale test suite, containing 24 single-objective functions in continuous domain, extends the well-known single-objective noiseless bbob test suite, which has been used since 2009 in the BBOB workshop series, to large dimension. The core idea is to make the rotational transformations R, Q in search space that appear in the bbob test suite computationally cheaper while retaining some desired properties. This documentation presents an approach that replaces a full rotational transformation with a combination of a block-diagonal matrix and two permutation matrices in order to construct test functions whose computational and memory costs scale linearly in the dimension of the problem.
• Numerical simulation of a H5 chondrite radiative field: comparison with the experiments performed at the VKI plasmatron facility
  
  • Dias Bruno
  • Scoggins James B
  • Soucasse Laurent
  • Rivière Philippe
  • Soufiani Anouar
  • Magin Thierry E.
  , 2019, pp.68. Meteor entry is characterized by complex shock layer physics such as radiation, evaporation of the meteoroid surface and the resulting chemistry process with the air constituents. Moreover, several meteor observations include spectral measurements from which their composition can be inferred. Recently a ground facility experiment of an H5 chondrite was performed at the VKI plasmatron facility which includes the measurement of time resolved optical emission spectroscopy data of ablated species. In this work we present a model able to reproduce the ablation of meteors and the comparison of the numerical radiative field with the one observed in the experiments. It is observed that the numerical results agree generally well with the experimental data when nitridation and oxidation gas-surface reactions are included due to the presence of a cork sample holder.
• A power plant valuation under an asymmetric risk criterion taking into account maintenance costs
  
  • Alasseur Clémence
  • Gobet Emmanuel
  • Pimentel Isaque
  • Warin Xavier
  , 2019. Power producers are interested in valuing their power plant production. By trading into forward contracts, we propose to reduce the contingency of the associated income considering the fixed costs and using an asymmetric risk criterion. In an asymptotic framework, we provide an optimal hedging strategy through a solution of a nonlinear partial differential equation. As a numerical experiment, we analyze the impact of the fixed costs structure on the hedging policy and the value of the assets.
• Different Measure Approximations for Efficient Constrained Multi-Objective Optimization under Uncertainty
  
  • Rivier Mickael
  • Congedo Pietro Marco
  , 2019.
• Surrogate-based inversion for first-arrival seismic tomography
  
  • Sochala Pierre
  • Gesret Alexandrine
  • Le Maitre Olivier
  , 2019.
• Uncertainty propagation in multiphysics systems of solvers: application to robust space object reentry predictions
  
  • Sanson Francois
  • Le Maitre Olivier
  • Congedo Pietro Marco
  , 2019.
• Density estimation for RWRE
  
  • Havet Antoine
  • Lerasle Matthieu
  • Moulines Éric
  Mathematical Methods of Statistics, Springer, 2019. We consider the problem of non-parametric density estimation of a random environment from the observation of a single trajectory of a random walk in this environment. We first construct a density estimator using the beta-moments. We then show that the Goldenshluger-Lepski method can be used to select the beta-moment. We prove non-asymptotic bounds for the supremum norm of these estimators for both the recurrent and the transient to the right cases. A simulation study supports our theoretical findings. (10.3103/S1066530719010022)
  DOI : 10.3103/S1066530719010022
• On Bi-Objective convex-quadratic problems
  
  • Touré Cheikh
  • Auger Anne
  • Brockhoff Dimo
  • Hansen Nikolaus
  , 2019. In this paper we analyze theoretical properties of bi-objective convex-quadratic problems. We give a complete description of their Pareto set and prove the convexity of their Pareto front. We show that the Pareto set is a line segment when both Hessian matrices are proportional. We then propose a novel set of convex-quadratic test problems, describe their theoretical properties and the algorithm abilities required by those test problems. This includes in particular testing the sensitivity with respect to separability, ill-conditioned problems, rotational invariance, and whether the Pareto set is aligned with the coordinate axis.
• Recursive computation of the invariant distributions of Feller processes: Revisited examples and new applications
  
  • Pagès Gilles
  • Rey Clément
  Monte Carlo Methods and Applications, De Gruyter, 2019, 25 (1), pp.1-36. Abstract In this paper, we show that the abstract framework developed in [G. Pagès and C. Rey, Recursive computation of the invariant distribution of Markov and Feller processes, preprint 2017, https://arxiv.org/abs/1703.04557 ] and inspired by [D. Lamberton and G. Pagès, Recursive computation of the invariant distribution of a diffusion, Bernoulli 8 2002, 3, 367–405] can be used to build invariant distributions for Brownian diffusion processes using the Milstein scheme and for diffusion processes with censored jump using the Euler scheme. Both studies rely on a weakly mean-reverting setting for both cases. For the Milstein scheme we prove the convergence for test functions with polynomial (Wasserstein convergence) and exponential growth. For the Euler scheme of diffusion processes with censored jump we prove the convergence for test functions with polynomial growth. (10.1515/mcma-2018-2027)
  DOI : 10.1515/mcma-2018-2027
• Kriging-sparse Polynomial Dimensional Decomposition surrogate model with adaptive refinement
  
  • Cortesi Andrea Francesco
  • Jannoun Ghina
  • Congedo Pietro Marco
  Journal of Computational Physics, Elsevier, 2019, 380, pp.212-242. Uncertainty Quantification and Sensitivity Analysis problems are made more difficult in the case of applications involving expensive computer simulations. This is because a limited amount of simulations is available to build a sufficiently accurate metamodel of the quantities of interest. In this work, an algorithm for the construction of a low-cost and accurate metamodel is proposed, having in mind computationally expensive applications. It has two main features. First, Universal Kriging is coupled with sparse Polynomial Dimensional Decomposition (PDD) to build a metamodel with improved accuracy. The polynomials selected by the adaptive PDD representation are used as a sparse basis to build a Universal Kriging surrogate model. Secondly, a numerical method, derived from anisotropic mesh adaptation, is formulated in order to adaptively insert a fixed number of new training points to an existing Design of Experiments. The convergence of the proposed algorithm is analyzed and assessed on different test functions with an increasing size of the input space. Finally, the algorithm is used to propagate uncertainties in two high-dimensional real problems related to the atmospheric reentry. (10.1016/j.jcp.2018.10.051)
  DOI : 10.1016/j.jcp.2018.10.051
• Structural optimization under internal porosity constraints using topological derivatives
  
  • Martínez-Frutos Jesus
  • Allaire Grégoire
  • Dapogny Charles
  • Periago Francisco
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2019, 345, pp.1-25. Porosity is a well-known phenomenon occurring during various manufacturing processes (casting, welding, additive manufacturing) of solid structures, which undermines their reliability and mechanical performance. The main purpose of this article is to introduce a new constraint functional of the domain which controls the negative impact of porosity on elastic structures in the framework of shape and topology optimization. The main ingredient of our modelling is the notion of topological derivative, which is used in a slightly unusual way: instead of being an indicator of where to nucleate holes in the course of the optimization process, it is a component of a new constraint functional which assesses the influence of pores on the mechanical performance of structures. The shape derivative of this constraint is calculated and incorporated into a level set based shape optimization algorithm. Our approach is illustrated by several two-and three-dimensional numerical experiments of topology optimization problems constrained by a control on the porosity effect. (10.1016/j.cma.2018.10.036)
  DOI : 10.1016/j.cma.2018.10.036
• Statistical description of turbulent particle-laden flows in the very dilute regime using the Anisotropic Gaussian Moment Method
  
  • Sabat Macole
  • Vié Aymeric
  • Larat Adam
  • Massot Marc
  International Journal of Multiphase Flow, Elsevier, 2019, 112, pp.243-257. The present work aims at investigating the ability of a Kinetic-Based Moment Method (KBMM) to reproduce the statistics of turbulent particle-laden flows using the Anisotropic Gaussian (AG) closure. This method is the simplest KBMM member that can account for Particle Trajectory Crossing (PTC) properly with a well-posed mathematical structure [1]. In order to validate this model further, we investigate here 3D turbulent flows that are more representative of the mixing processes, which occurs in realistic applications. The chosen configuration is a 3D statistically-stationary Homogeneous Isotropic Turbulence (HIT) loaded with particles in a very dilute regime. The analysis focuses on the description of the first three lowest order moments of the particulate flow: the number density, the Eulerian velocity and the internal energy. A thorough numerical study on a large range of particle inertia allows us to show that the AG closure extends the ability of the Eulerian models to correctly reproduce the particle dynamics up to a Stokes number based on the Eulerian turbulence macro-scale equal to one, but also highlights the necessity of high-order numerical schemes to reach mesh convergence, especially for the number density field. (10.1016/j.ijmultiphaseflow.2018.10.004)
  DOI : 10.1016/j.ijmultiphaseflow.2018.10.004
• The tropical analogue of the Helton–Nie conjecture is true
  
  • Allamigeon Xavier
  • Gaubert Stéphane
  • Skomra Mateusz
  Journal of Symbolic Computation, Elsevier, 2019, 91, pp.129-148. Helton and Nie conjectured that every convex semialgebraic set over the field of real numbers can be written as the projection of a spectrahedron. Recently, Scheiderer disproved this conjecture. We show, however, that the following result, which may be thought of as a tropical analogue of this conjecture, is true: over a real closed nonarchimedean field of Puiseux series, the convex semialgebraic sets and the projections of spectrahedra have precisely the same images by the nonarchimedean valuation. The proof relies on game theory methods. (10.1016/j.jsc.2018.06.017)
  DOI : 10.1016/j.jsc.2018.06.017
• Dramatic Acceleration of Wave Condensation Mediated by Disorder in Multimode Fibers
  
  • Fusaro Adrien
  • Garnier Josselin
  • Krupa Katarzyna
  • Millot Guy
  • Picozzi Antonio
  Physical Review Letters, American Physical Society, 2019, 122 (12). Classical nonlinear waves exhibit a phenomenon of condensation that results from the natural irreversible process of thermalization, in analogy with the quantum Bose-Einstein condensation. Wave condensation originates in the divergence of the thermodynamic equilibrium Rayleigh-Jeans distribution, which is responsible for the macroscopic population of the fundamental mode of the system. However, achieving complete thermalization and condensation of incoherent waves through nonlinear optical propagation is known to require prohibitive large interaction lengths. Here, we derive a discrete kinetic equation describing the nonequilibrium evolution of the random wave in the presence of a structural disorder of the medium. Our theory reveals that a weak disorder accelerates the rate of thermalization and condensation by several order of magnitudes. Such a counterintu-itive dramatic acceleration of condensation can provide an explanation for the recently discovered phenomenon of optical beam self-cleaning. We report experiments in multimode optical fibers that highlight the transition from an incoherent thermal distribution to wave condensation, with a con-densate fraction of up to 60% in the fundamental mode of the waveguide trapping potential. (10.1103/PhysRevLett.122.123902)
  DOI : 10.1103/PhysRevLett.122.123902
• Different measure approximations for efficient constrained multi-objective optimization under uncertainty
  
  • Rivier Mickael
  • Congedo Pietro Marco
  , 2019.
• Robust supervised classification and feature selection using a primal-dual method
  
  • Barlaud Michel
  • Chambolle Antonin
  • Caillau Jean-Baptiste
  , 2019. This paper deals with supervised classification and feature selection in high dimensional space. A classical approach is to project data on a low dimensional space and classify by minimizing an appropriate quadratic cost. A strict control on sparsity is moreover obtained by adding an 1 constraint, here on the matrix of weights used for projecting the data. Tuning the sparsity bound results in selecting the relevant features for supervised classification. However, an issue is that using a quadratic cost (a squared 2 norm, in practice) for the data term is not robust to outliers. In this paper, we cope with this problem by using an 1 norm both for the constraint and for the loss function. In this case, the criterion is convex but not gradient Lipschitz anymore. Another second issue is that we optimize simultaneously the projection matrix and the centers used for classification. To do so, notwithstanding the lack of regularity, we provide a novel tailored constrained primal-dual method to compute jointly both variables with convergence proofs. We demonstrate the effectiveness of our method on three datasets (one synthetic, two from biological data), and provide a comparison between 1 and 2 costs.
• Local structure of multi-dimensional martingale optimal transport
  
  • de March Hadrien
  , 2019. This paper analyzes the support of the conditional distribution of optimal martingale transport plans in higher dimension. In the context of a distance coupling in dimension larger than 2, previous results established by Ghoussoub, Kim & Lim show that this conditional transport is concentrated on its own Choquet boundary. Moreover, when the target measure is atomic, they prove that the support is concentrated on d+1 points, and conjecture that this result is valid for arbitrary target measure. We provide a structure result of the support of the conditional optimal transport for general Lipschitz couplings. Using tools from algebraic geometry, we provide sufficient conditions for finiteness of this conditional support, together with (optimal) lower bounds on the maximal cardinality for a given coupling function. More results are obtained for specific examples of coupling functions based on distance functions. In particular, we show that the above conjecture of Ghoussoub, Kim & Lim is not valid beyond the context of atomic target distributions.
• Irreducible convex paving for decomposition of multi-dimensional martingale transport plans
  
  • de March Hadrien
  • Touzi Nizar
  , 2019. Martingale transport plans on the line are known from Beiglbock & Juillet to have an irreducible decomposition on a (at most) countable union of intervals. We provide an extension of this decomposition for martingale transport plans in R^d, d larger than one. Our decomposition is a partition of R^d consisting of a possibly uncountable family of relatively open convex components, with the required measurability so that the disintegration is well-defined. We justify the relevance of our decomposition by proving the existence of a martingale transport plan filling these components. We also deduce from this decomposition a characterization of the structure of polar sets with respect to all martingale transport plans.
• Quasi-sure duality for multi-dimensional martingale optimal transport
  
  • de March Hadrien
  , 2019. Based on the multidimensional irreducible paving of De March & Touzi, we provide a multi-dimensional version of the quasi sure duality for the martingale optimal transport problem, thus extending the result of Beiglb\"ock, Nutz & Touzi. Similar, we also prove a disintegration result which states a natural decomposition of the martingale optimal transport problem on the irreducible components, with pointwise duality verified on each component. As another contribution, we extend the martingale monotonicity principle to the present multi-dimensional setting. Our results hold in dimensions 1, 2, and 3 provided that the target measure is dominated by the Lebesgue measure. More generally, our results hold in any dimension under an assumption which is implied by the Continuum Hypothesis. Finally, in contrast with the one-dimensional setting of Beiglb\"ock, Lim & Obloj, we provide an example which illustrates that the smoothness of the coupling function does not imply that pointwise duality holds for compactly supported measures.
• Entropic approximation for multi-dimensional martingale optimal transport
  
  • de March Hadrien
  , 2019. We study the existing algorithms that solve the multidimensional martingale optimal transport. Then we provide a new algorithm based on entropic regularization and Newton's method. Then we provide theoretical convergence rate results and we check that this algorithm performs better through numerical experiments. We also give a simple way to deal with the absence of convex ordering among the marginals. Furthermore, we provide a new universal bound on the error linked to entropy.