Publications

2018

• Taking into account thermal residual stresses in topology optimization of structures built by additive manufacturing
  
  • Allaire Grégoire
  • Jakabčin Lukas
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2018, 28 (12), pp.2313-2366. We introduce a model and several constraints for shape and topology optimization of structures, built by additive manufacturing techniques. The goal of these constraints is to take into account the thermal residual stresses or the thermal deformations, generated by processes like Selective Laser Melting, right from the beginning of the structural design optimization. In other words, the structure is optimized concurrently for its final use and for its behavior during the layer by layer production process. It is well known that metallic additive manufacturing generates very high temperatures and heat fluxes, which in turn yield thermal deformations that may prevent the coating of a new powder layer, or thermal residual stresses that may hinder the mechanical properties of the final design. Our proposed constraints are targeted to avoid these undesired effects. Shape derivatives are computed by an adjoint method and are incorporated into a level set numerical optimization algorithm. Several 2-d and 3-d numerical examples demonstrate the interest and effectiveness of our approach. (10.1142/S0218202518500501)
  DOI : 10.1142/S0218202518500501
• Option pricing under fast-varying and rough stochastic volatility
  
  • Garnier Josselin
  • Solna Knut
  Annals of Finance, Springer Verlag, 2018, 14 (4), pp.489-516. (10.1007/s10436-018-0325-4)
  DOI : 10.1007/s10436-018-0325-4
• Some Results on Skorokhod Embedding and Robust Hedging with Local Time
  
  • Claisse Julien
  • Guo Gaoyue
  • Henry-Labordère Pierre
  Journal of Optimization Theory and Applications, Springer Verlag, 2018, 179 (2), pp.569-597. (10.1007/s10957-017-1201-5)
  DOI : 10.1007/s10957-017-1201-5
• Noninvasive Imaging Through Random Media
  
  • Garnier Josselin
  • Solna Knut
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2018, 78 (6), pp.3296-3315. (10.1137/18M1171977)
  DOI : 10.1137/18M1171977
• A bilevel optimization model for load balancing in mobile networks through price incentives
  
  • Akian Marianne
  • Bouhtou Mustapha
  • Eytard Jean-Bernard
  • Gaubert Stéphane
  , 2018. We propose a model of incentives for data pricing in large mobile networks, in which an operator wishes to balance the number of connections (active users) of different classes of users in the different cells and at different time instants, in order to ensure them a sufficient quality of service. We assume that each user has a given total demand per day for different types of applications, which he may assign to different time slots and locations, depending on his own mobility, on his preferences and on price discounts proposed by the operator. We show that this can be cast as a bilevel programming problem with a special structure allowing us to develop a polynomial time decomposition algorithm suitable for large networks. First, we determine the optimal number of connections (which maximizes a measure of balance); next, we solve an inverse problem and determine the prices generating this traffic. Our results exploit a recently developed application of tropical geometry methods to mixed auction problems, as well as algorithms in discrete convexity (minimization of discrete convex functions in the sense of Murota). We finally present an application on real data provided by Orange and we show the efficiency of the model to reduce the peaks of congestion.
• Model-adaptive optimal discretization of stochastic integrals
  
  • Gobet Emmanuel
  • Stazhynski Uladzislau
  Stochastics: An International Journal of Probability and Stochastic Processes, Taylor & Francis: STM, Behavioural Science and Public Health Titles, 2018, 91 (3), pp.321-351. (10.1080/17442508.2018.1539087)
  DOI : 10.1080/17442508.2018.1539087
• Data analysis for aircraft trajectory optimization
  
  • Rommel Cédric
  , 2018. This thesis deals with the use of flight data for the optimization of climb trajectories with relation to fuel consumption.We first focus on methods for identifying the aircraft dynamics, in order to plug it in the trajectory optimization problem. We suggest a static formulation of the identification problem, which we interpret as a structured multi-task regression problem. In this framework, we propose parametric models and use different maximum likelihood approaches to learn the unknown parameters.Furthermore, polynomial models are considered and an extension to the structured multi-task setting of the bootstrap Lasso is used to make a consistent selection of the monomials despite the high correlations among them.Next, we consider the problem of assessing the optimized trajectories relatively to the validity region of the identified models. For this, we propose a probabilistic criterion for quantifying the closeness between an arbitrary curve and a set of trajectories sampled from the same stochastic process. We propose a class of estimators of this quantity and prove their consistency in some sense. A nonparemetric implementation based on kernel density estimators, as well as a parametric implementation based on Gaussian mixtures are presented. We introduce the later as a penalty term in the trajectory optimization problem, which allows us to control the trade-off between trajectory acceptability and consumption reduction.
• Nonsmooth Aggregative Games with Coupling Constraints and Infinitely Many Classes of Players
  
  • Jacquot Paulin
  • Wan Cheng
  , 2018. After defining a pure-action profile in a nonatomic aggregative game, where players have specific compact convex pure-action sets and nonsmooth convex cost functions, as a square-integrable function, we characterize a Wardrop equilibrium as a solution to an infinite-dimensional generalized variational inequality. We show the existence of Wardrop equilibrium and variational Wardrop equilibrium, a concept of equilibrium adapted to the presence of coupling constraints, in monotone nonatomic aggregative games. The uniqueness of (variational) Wardrop equilibrium is proved for strictly or aggregatively strictly monotone nonatomic aggregative games. We then show that, for a sequence of finite-player aggregative games with aggregative constraints, if the players' pure-action sets converge to those of a strongly (resp. aggregatively strongly) monotone nonatomic aggregative game, and the aggregative constraints in the finite-player games converge to the aggregative constraint of the nonatomic game, then a sequence of so-called variational Nash equilibria in these finite-player games converge to the variational Wardrop equilibrium in pure-action profile (resp. aggregate-action profile). In particular, it allows the construction of an auxiliary sequence of games with finite-dimensional equilibria to approximate the infinite-dimensional equilibrium in such a nonatomic game. Finally, we show how to construct auxiliary finite-player games for two general classes of nonatomic games.
• Probabilistic modeling of space object controlled reentry and ground risk estimation
  
  • Sanson Francois
  • Bertorello Charles
  • Bouilly Jean-Marc
  • Congedo Pietro Marco
  , 2018.
• A relaxation scheme for two-phase multi-component flows
  
  • Baudin Michaël
  • Coquel Frédéric
  • Tran Quang Huy
  , 2018. Pursuing the program launched our previous works [Numer. Math. 99 (2005), 411-440] and [SIAM J. Sci. Comput. 27 (2005), 914-936], we propose a relaxation scheme for the numerical simulation of one-dimensional two-phase multi-component flows governed by a drift-flux model, the main features of which are a large number of components and a high degree of nonlinearity in the closure laws. In the explicit setting, the relaxation approach allows to ensure positivity for the densities and the mass fractions. The relaxation method is worked out further so as to fit into a hybrid explicit-implicit setting, where fast acoustic waves are treated implicitly to save computational time while slow kinematic waves are treated explicitly in order to maintain accuracy on the transportation of materials.
• Efficient Computation of Rare Events: Failure Probability and Quantile
  
  • Razaaly Nassim
  • Congedo Pietro Marco
  , 2018.
• Efficient uncertainty propagation in systems of solvers
  
  • Sanson Francois
  • Le Maitre Olivier
  • Congedo Pietro Marco
  , 2018.
• FEM-BEM Coupling for Electromagnetism with the Sparse Cardinal Sine Decomposition,
  
  • Alouges Francois
  • Aussal Matthieu
  • Parolin Emile
  ESAIM: Proceedings and Surveys, EDP Sciences, 2018, 63, pp.44-59. This paper presents a FEM-BEM coupling method suitable for the numerical simulation of the electromagnetic scattering of objects composed of dielectric materials and perfect electric conduc- tors. The originality of the approach lies in part in the use of the newly proposed Sparse Cardinal Sine Decomposition SCSD) method for the BEM part of the computation and the fact that the simulation software is almost entirely written in MATLAB. The performance of the method is illustrated by the computation of the electromagnetic scattering by an UAV-like object with two RAM regions proposed in the workshop ISAE EM 2016. (10.1051/proc/201863044)
  DOI : 10.1051/proc/201863044
• Breakup prediction under uncertainty: application to Upper Stage controlled reentries from GTO orbit
  
  • Sanson Francois
  • Bertorello Charles
  • Bouilly Jean-Marc
  • Congedo Pietro Marco
  , 2018. More and more human-made space objects re-enter the atmosphere, and yet the risk for human population remains often unknown because predicting their reentry trajectories is formidably complex. While falling back on Earth, the space object absorbs large amounts of thermal energy that affects its structural integrity.It undergoes strong aerodynamic forces that lead to one or several breakups. Breakup events have a critical influence on the rest of the trajectory are extremely challenging to predict and subject to uncertainties. In this work, we present an original model for robustly predicting the breakup of a reentering space object. This model is composed of a set of individual solvers that are coupled together such as each solver resolves a specific aspect of this multiphysics problem. This paper deals with two levels of uncertainties. The first level is the stochastic modelling of the breakup while the second level is the statistical characterization of the model input uncertainties. The framework provides robust estimates of the quantities of interest and quantitative sensitivity analysis. The objective is twofold: first to compute a robust estimate of the breakup distribution and secondly to identify the main uncertainties in the quantities of interest. Due to the significant computational cost, we use an efficient framework par-* Corresponding author. ticularly suited to multiple solver predictions for the uncertainty quantification analysis. Then, we illustrate the breakup model for the controlled reentry of an upper stage deorbited from a Geo Transfer Orbit (GTO), which is a classical Ariane mission.
• Surrogate-Assisted Bounding-Box Approach Applied to Constrained Multi-Objective Optimisation Under Uncertainty
  
  • Rivier Mickael
  • Congedo Pietro Marco
  , 2018, pp.1-37. This paper is devoted to tackling constrained multi-objective optimisation under uncertainty problems. A Surrogate-Assisted Bounding-Box approach (SABBa) is formulated here to deal with robustness and reliability measures, which can be computed with tunable and refinable fidelity. A Bounding-Box is defined as a multi-dimensional product of intervals centred on the estimated objectives and constraints that contains the true underlying values. The fidelity of these estimations can be tuned throughout the optimisation so as to reach high accuracy only on promising designs, which allows quick convergence toward the optimal area. In SABBa, this approach is supplemented with a Surrogate-Assisting (SA) strategy, which is very useful to reduce the overall computational cost. The adaptive refinement within the Bounding-Box approach is based on the computation of a Pareto Optimal Probability (POP) for each box. We first assess the proposed method on several analytical uncertainty-based optimisation test-cases with respect to an \textit{a priori} metamodel approach in terms of a probabilistic modified Hausdorff distance to the true Pareto optimal set. The method is then applied to two engineering applications: the design of two-bar truss in structural mechanics and the design of a thermal protection system for atmospheric reentry.
• The Tamed Unadjusted Langevin Algorithm
  
  • Brosse Nicolas
  • Durmus Alain
  • Moulines Éric
  • Sabanis Sotirios
  Stochastic Processes and their Applications, Elsevier, 2018. In this article, we consider the problem of sampling from a probability measure π having a density on R d known up to a normalizing constant, $x → e −U (x) / R d e −U (y) dy$. The Euler discretization of the Langevin stochastic differential equation (SDE) is known to be unstable in a precise sense, when the potential U is superlinear, i.e. lim inf $x→+∞ U (x) / x = +∞$. Based on previous works on the taming of superlinear drift coefficients for SDEs, we introduce the Tamed Unadjusted Langevin Algorithm (TULA) and obtain non-asymptotic bounds in V-total variation norm and Wasserstein distance of order 2 between the iterates of TULA and π, as well as weak error bounds. Numerical experiments are presented which support our findings. (10.1016/j.spa.2018.10.002)
  DOI : 10.1016/j.spa.2018.10.002
• Asymptotic optimal pricing with asymmetric risk and applications in finance
  
  • Santa Brigida Pimentel Isaque
  , 2018. This thesis is constituted by two parts that can be read independently.In the first part, we study several problems of hedging and pricing of options related to a risk measure. Our main approach is the use of an asymmetric risk function and an asymptotic framework in which we obtain optimal solutions through nonlinear partial differential equations (PDE).In the first chapter, we focus on pricing and hedging European options. We consider the optimization problem of the residual risk generated by a discrete-time hedging in the presence of an asymmetric risk criterion. Instead of analyzing the asymptotic behavior of the solution to the associated discrete problem, we study the integrated asymmetric measure of the residual risk in a Markovian framework. In this context, we show the existence of the asymptotic risk measure. Thus, we describe an asymptotically optimal hedging strategy via the solution to a fully nonlinear PDE.The second chapter is an application of the hedging method to the valuation problem of the power plant. Since the power plant generates maintenance costs whether it is on or off, we are interested in reducing the risk associated with its uncertain revenues by hedging with forwards contracts. We study the impact of a maintenance cost depending on the electricity price into the hedging strategy.In the second part, we consider several control problems associated with economy and finance.The third chapter is dedicated to the study of a McKean-Vlasov (MKV) problem class with common noise, called polynomial conditional MKV. We reduce this polynomial class by a Markov embedding to finite-dimensional control problems.We compare three different probabilistic techniques for numerical resolution of the reduced problem: quantization, control randomization and regress later.We provide numerous numerical examples, such as the selection of a portfolio under drift uncertainty.In the fourth chapter, we solve dynamic programming equations associated with financial valuations in the energy market. We consider that a calibrated underlying model is not available and that a limited sample of historical data is accessible.In this context, we suppose that forward contracts are governed by hidden factors modeled by Markov processes. We propose a non-intrusive method to solve these equations through empirical regression techniques using only the log price history of observable futures contracts.
• Transmit Strategies for Massive Machine-Type Communications based on Mean Field Games
  
  • Bertucci Charles
  • Vassilaras Spyridon
  • Lasry Jean-Michel
  • Paschos Georgios
  • Debbah Merouane
  • Lions Pierre-Louis
  , 2018, pp.1-5. Massive Machine Type Communications are one of the three main type of communication applications in upcoming 5G wireless networks. In this type of communication, the network is required to handle a huge number of devices transmitting information to the same base station receiver in an uncoordinated manner. In this setting, the problem of minimizing energy usage while achieving QoS requirements is a very complex stochastic control problem with a very large number of optimizing agents. In this paper, we propose a Mean Field Games model for this problem that reduces the complexity by a great deal and is thus amenable to numerical solution. Our model is general enough to include generic rate functions, arbitrary energy and QoS requirements per user, different channel fading models, and design knobs for determining the importance of different performance goals. We provide details of the proposed numerical solution and present numerical results that illustrate the characteristics of the obtained control policy. (10.1109/ISWCS.2018.8491236)
  DOI : 10.1109/ISWCS.2018.8491236
• Uncertainty Quantification Framework for Complex Systems of Solvers
  
  • Sanson Francois
  • Le Maitre Olivier
  • Congedo Pietro Marco
  , 2018.
• GENERALIZED CRYSTALLINE EVOLUTIONS AS LIMITS OF FLOWS WITH SMOOTH ANISOTROPIES
  
  • Chambolle Antonin
  • Morini Massimiliano
  • Novaga Matteo
  • Ponsiglione Marcello
  Analysis & PDE, Mathematical Sciences Publishers, 2018, 12 (3), pp.789–813. We prove existence and uniqueness of weak solutions to anisotropic and crystalline mean curvature flows, obtained as limit of the viscosity solutions to flows with smooth anisotropies.
• Analysis and optimisation of a variational model for mixed Gaussian and Salt & Pepper noise removal
  
  • Calatroni Luca
  • Papafitsoros Kostas
  , 2018. We analyse a variational regularisation problem for mixed noise removal that was recently proposed in [14]. The data discrepancy term of the model combines $L^1$ and $L^2$ terms in an infimal convolution fashion and it is appropriate for the joint removal of Gaussian and Salt & Pepper noise. In this work we perform a finer analysis of the model which emphasises on the balancing effect of the two parameters appearing in the discrepancy term. Namely, we study the asymptotic behaviour of the model for large and small values of these parameters and we compare it to the corresponding variational models with $L^1$ and $L^2$ data fidelity. Furthermore, we compute exact solutions for simple data functions taking the total variation as regulariser. Using these theoretical results, we then analytically study a bilevel optimisation strategy for automatically selecting the parameters of the model by means of a training set. Finally, we report some numerical results on the selection of the optimal noise model via such strategy which confirm the validity of our analysis and the use of popular data models in the case of "blind" model selection.
• Phase-field approximation for some branched transportation problems
  
  • Ferrari Luca Alberto Davide
  , 2018. In this thesis we devise phase field approximations of some Branched Transportation problems. Branched Transportation is a mathematical framework for modeling supply-demand distribution networks which exhibit tree like structures. In particular the network, the supply factories and the demand location are modeled as measures and the problem is cast as a constrained optimization problem. The transport cost of a mass m along an edge with length L is h(m)xL and the total cost of a network is defined as the sum of the contribution on all its edges. The branched transportation case consists with the specific choice h(m)=|m|^α where α is a value in [0,1). The sub-additivity of the cost function ensures that transporting two masses jointly is cheaper than doing it separately. In this work we introduce various variational approximations of the branched transport optimization problem. The approximating functionals are based on a phase field representation of the network and are smoother than the original problem which allows for efficient numerical optimization methods. We introduce a family of functionals inspired by the Ambrosio and Tortorelli one to model an affine transport cost functions. This approach is firstly used to study the problem any affine cost function h in the ambient space R². For this case we produce a full Γ-convergence result and correlate it with an alternate minimization procedure to obtain numerical approximations of the minimizers. We then generalize this approach to any ambient space and obtain a full Γ-convergence result in the case of k-dimensional surfaces. In particular, we obtain a variational approximation of the Plateau problem in any dimension and co-dimension. In the last part of the thesis we propose two models for general concave cost functions. In the first one we introduce a multiphase field approach and recover any piecewise affine cost function. Finally we propose and study a family of functionals allowing to recover in the limit any concave cost function h.
• Robust Optimization of a Supersonic ORC Turbine Cascade under a probabilistic constraint: a Quantile Formulation
  
  • Razaaly Nassim
  • Persico Giacomo
  • Congedo Pietro Marco
  , 2018.
• Robust Optimization of a Supersonic ORC Turbine Cascade: a Quantile-based Approach
  
  • Razaaly Nassim
  • Persico Giacomo
  • Gori Giulio
  • Congedo Pietro Marco
  , 2018.
• STOCHASTIC PRIMAL-DUAL HYBRID GRADIENT ALGORITHM WITH ARBITRARY SAMPLING AND IMAGING APPLICATIONS
  
  • Chambolle Antonin
  • Ehrhardt Matthias J
  • Richtarik Peter
  • Schönlieb Carola-Bibiane
  SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2018, 28 (4), pp.2783-2808. We propose a stochastic extension of the primal-dual hybrid gradient algorithm studied by Chambolle and Pock in 2011 to solve saddle point problems that are separable in the dual variable. The analysis is carried out for general convex-concave saddle point problems and problems that are either partially smooth / strongly convex or fully smooth / strongly convex. We perform the analysis for arbitrary samplings of dual variables, and obtain known deterministic results as a special case. Several variants of our stochastic method significantly outperform the deterministic variant on a variety of imaging tasks. (10.1137/17M1134834)
  DOI : 10.1137/17M1134834