Publications

2018

• Day-ahead probabilistic forecast of solar irradiance: a Stochastic Differential Equation approach
  
  • Badosa Jordi
  • Gobet Emmanuel
  • Grangereau Maxime
  • Kim Daeyoung
  , 2018, 254, pp.73-93. In this work, we derive a probabilistic forecast of the solar irradiance during a day at a given location, using a stochastic differential equation (SDE for short) model. We propose a procedure that transforms a deterministic forecast into a proba-bilistic forecast: the input parameters of the SDE model are the Arome deterministic forecast computed at day D-1 for the day D. The model also accounts for the maximal irradiance from the clear sky model. The SDE model is mean-reverting towards the deterministic forecast and the instantaneous amplitude of the noise depends on the clear sky index, so that the fluctuations vanish as the index is close to 0 (cloudy) or 1 (sunny), as observed in practice. Our tests show a good adequacy of the confidence intervals of the model with the measurement. (10.1007/978-3-319-99052-1_4)
  DOI : 10.1007/978-3-319-99052-1_4
• Structured Feature Selection of Continuous Dynamical Systems for Aircraft Dynamics Identification
  
  • Rommel Cédric
  • Bonnans Frédéric
  • Gregorutti Baptiste
  • Martinon Pierre
  , 2018. This paper addresses the problem of identifying structured nonlinear dynamical systems, with the goal of using the learned dynamics in model-based reinforcement learning problems. We present in this setting a new class of scalable multi-task estimators which promote sparsity, while preserving the dynamics structure and leveraging available physical insight. An implementation leading to consistent feature selection is suggested, allowing to obtain accurate models. An additional regularizer is also proposed to help in recovering realistic hidden representations of the dynamics. We illustrate our method by applying it to an aircraft trajectory optimization problem. Our numerical results based on real flight data from 25 medium haul aircraft, totaling 8 millions observations, show that our approach is competitive with existing methods for this type of application.
• Routing Game on Parallel Networks: the Convergence of Atomic to Nonatomic
  
  • Jacquot Paulin
  • Wan Cheng
  , 2018, 1. We consider an instance of a nonatomic routing game. We assume that the network is parallel, that is, constituted of only two nodes, an origin and a destination. We consider infinitesimal players that have a symmetric network cost, but are heterogeneous through their set of feasible strategies and their individual utilities. We show that if an atomic routing game instance is correctly defined to approximate the nonatomic instance, then an atomic Nash Equilibrium will approximate the nonatomic Wardrop Equilibrium. We give explicit bounds on the distance between the equilibria according to the parameters of the atomic instance. This approximation gives a method to compute the Wardrop equilibrium at an arbitrary precision.
• Homogenization method for topology optmization of struc-tures built with lattice materials.
  
  • Geoffroy Donders Perle
  , 2018. Thanks to the recent developments of the additive manufacturing processes, structures built with modulated microstructures and featuring a complex topology are now manufacturable. This leads to a resurrection of the homogenization method for shape optimization, an approach developed in the 80’s but which progressively faded away because yielding too complex structures for manufacturing processes at this time.The goal of this thesis is to develop shape optimization methods for structures built with modulated locally periodic lattice microstructures.Three steps have been defined. The first consists in computing the homogenized, or effective, elastic properties of microstructures according to few parameters characterizing their geometry. In the second step, the geometric properties of the microstructure and its orientation are optimized in the working domain, yielding a homogenized optimized structure. Such a structure is nevertheless not straightforwardly manufacturable. Indeed, the homogenization is equivalent to have a structure featuring cells whose size is converging to zero. Hence, in the third and last step, a deshomogenization process is proposed. It consists in building a sequence of genuine structures converging to the homogenized optimal structures. The key point is to respect locally the orientation of the cells, which is performed thanks to a grid diffeomorphism.In this thesis, we present the details of the whole method, for isotropic and orthotropic microstructures, in 2D and in 3D.A coupling of this method with the level-set shape optimization method is also presented, thanks which the set of geometric constraints on the final structures may be enlarged.
• Dealing with missing data in model-based clustering through a MNAR model
  
  • Biernacki Christophe
  • Celeux Gilles
  • Josse Julie
  • Laporte Fabien
  , 2018.
• Discretization of processes at stopping times and Uncertainty quantification of stochastic approximation limits
  
  • Stazhynski Uladzislau
  , 2018. This thesis consists of two parts which study two separate subjects. Chapters 1-4 are devoted to the problem of processes discretization at stopping times. In Chapter 1 we study the optimal discretization error of stochastic integrals, driven by a multidimensional continuous Brownian semimartingale. In this setting we establish a path wise lower bound for the renormalized quadratic variation of the error and we provide a sequence of discretization stopping times, which is asymptotically optimal. The latter is defined as hitting times of random ellipsoids by the semimartingale at hand. In comparison with previous available results, we allow a quite large class of semimartingales and we prove that the asymptotic lower bound is attainable. In Chapter 2 we study the model-adaptive optimal discretization error of stochastic integrals. In Chapter 1 the construction of the optimal strategy involved the knowledge about the diffusion coefficient of the semimartingale under study. In this work we provide a model-adaptive asymptotically optimal discretization strategy that does not require any prior knowledge about the model. In Chapter 3 we study the convergence in distribution of renormalized discretization errors of Ito processes for a concrete general class of random discretization grids given by stopping times. Previous works on the subject only treat the case of dimension 1. Moreover they either focus on particular cases of grids, or provide results under quite abstract assumptions with implicitly specified limit distribution. At the contrast we provide explicitly the limit distribution in a tractable form in terms of the underlying model. The results hold both for multidimensional processes and general multidimensional error terms. In Chapter 4 we study the problem of parametric inference for diffusions based on observations at random stopping times. We work in the asymptotic framework of high frequency data over a fixed horizon. Previous works on the subject consider only deterministic, strongly predictable or random, independent of the process, observation times, and do not cover our setting. Under mild assumptions we construct a consistent sequence of estimators, for a large class of stopping time observation grids. Further we carry out the asymptotic analysis of the estimation error and establish a Central Limit Theorem (CLT) with a mixed Gaussian limit. In addition, in the case of a 1-dimensional parameter, for any sequence of estimators verifying CLT conditions without bias, we prove a uniform a.s. lower bound on the asymptotic variance, and show that this bound is sharp. In Chapters 5-6 we study the problem of uncertainty quantification for stochastic approximation limits. In Chapter 5 we analyze the uncertainty quantification for the limit of a Stochastic Approximation (SA) algorithm. In our setup, this limit is defined as the zero of a function given by an expectation. The expectation is taken w.r.t. a random variable for which the model is assumed to depend on an uncertain parameter. We consider the SA limit as a function of this parameter. We introduce the so-called Uncertainty for SA (USA) algorithm, an SA algorithm in increasing dimension for computing the basis coefficients of a chaos expansion of this function on an orthogonal basis of a suitable Hilbert space. The almost-sure and Lp convergences of USA, in the Hilbert space, are established under mild, tractable conditions. In Chapter 6 we analyse the L2-convergence rate of the USA algorithm designed in Chapter 5.The analysis is non-trivial due to infinite dimensionality of the procedure. Moreover, our setting is not covered by the previous works on infinite dimensional SA. The obtained rate depends non-trivially on the model and the design parameters of the algorithm. Its knowledge enables optimization of the dimension growth speed in the USA algorithm, which is the key factor of its efficient performance.
• Verifiable Conditions for the Irreducibility and Aperiodicity of Markov Chains by Analyzing Underlying Deterministic Models
  
  • Chotard Alexandre
  • Auger Anne
  Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2018, 25 (1), pp.112-147. We consider Markov chains that obey the following general non-linear state space model: $\Phi_{k+1} = F(\Phi_k, \alpha(\Phi_k, U_{k+1}))$ where the function $F$ is $C^1$ while $\alpha$ is typically discontinuous and $\{U_k: k \in \mathbb{Z}_{>0} \}$ is an independent and identically distributed process. We assume that for all $x$, the random variable $\alpha(x, U_1)$ admits a density $p_x$ such that $(x, w) \mapsto p_x(w)$ is lower semi-continuous. We generalize and extend previous results that connect properties of the underlying deterministic control model to provide conditions for the chain to be $\varphi$-irreducible and aperiodic. By building on those results, we show that if a rank condition on the controllability matrix is satisfied for all $x$, there is equivalence between the existence of a globally attracting state for the control model and $\varphi$-irreducibility of the Markov chain. Additionally, under the same rank condition on the controllability matrix, we prove that there is equivalence between the existence of a steadily attracting state and the $\varphi$-irreducibility and aperiodicity of the chain. The notion of steadily attracting state is new. We additionally derive practical conditions by showing that the rank condition on the controllability matrix needs to be verified only at a globally attracting state (resp.\ steadily attracting state) for the chain to be a $\varphi$-irreducible T-chain (resp.\ $\varphi$-irreducible aperiodic T-chain). Those results hold under considerably weaker assumptions on the model than previous ones that would require $(x,u) \mapsto F(x,\alpha(x,u))$ to be $C^\infty$ (while it can be discontinuous here). Additionally the establishment of a \emph{necessary and sufficient} condition for the $\varphi$-irreducibility and aperiodicity without a structural assumption on the control set is novel---even for Markov chains where $(x,u) \mapsto F(x,\alpha(x,u))$ is $C^\infty$. We illustrate that the conditions are easy to verify on a non-trivial and non-artificial example of Markov chain arising in the context of adaptive stochastic search algorithms to optimize continuous functions in a black-box scenario. (10.3150/17-BEJ970)
  DOI : 10.3150/17-BEJ970
• Convergence rate of strong approximations of compound random maps
  
  • Gobet Emmanuel
  • Mrad Mohamed
  Discrete & Continuous Dynamical Systems- Series-B, American Institute of Mathematical Sciences, 2018. We consider a random map x → F (ω, x) and a random variable Θ(ω), and we denote by F^N (ω, x) and Θ^N (ω) their approximations: We establish a strong convergence result, in Lp-norms, of the compound approximation F^N (ω, Θ^N (ω)) to the compound variable F (ω, Θ(ω)), in terms of the approximations of F and Θ. We then apply this result to the composition of two Stochastic Differential Equations through their initial conditions, which can give a way to solve some Stochastic Partial Differential Equations.
• Impact of Geometric, Operational, and Model Uncertainties on the Non-ideal Flow Through a Supersonic ORC Turbine Cascade
  
  • Razaaly Nassim
  • Persico Giacomo
  • Congedo Pietro Marco
  Energy, Elsevier, 2018. Typical energy sources for Organic Rankine Cycle (ORC) power systems feature variable heat load and turbine in-let/outlet thermodynamic conditions. The use of organic compounds with heavy molecular weight introduces uncertainties in the fluid thermodynamic modeling. In addition, the peculiarities of organic fluids typically leads to supersonic turbine configurations featuring supersonic flows and shocks, which grow in relevance in the aforemen-tioned off-design conditions; these features also depends strongly on the local blade shape, which can be influenced by the geometric tolerances of the blade manufacturing. This study presents an Uncertainty Quantification (UQ) analysis on a typical supersonic nozzle cascade for ORC applications, by considering a two-dimensional high-fidelity turbulent Computational Fluid Dynamic (CFD) model. Kriging-based techniques are used in order to take into account at a low computational cost, the combined effect of uncertainties associated to operating conditions, fluid parameters, and geometric tolerances. The geometric variability is described by a finite Karhunen-Loeve expansion representing a non-stationary Gaussian random field, entirely defined by a null mean and its autocorrelation function. Several results are illustrated about the ANOVA decomposition of several quantities of interest for different operating conditions, showing the importance of geometric uncertainties on the turbine performances. (10.1016/j.energy.2018.11.100)
  DOI : 10.1016/j.energy.2018.11.100
• Spectraèdres tropicaux : application à la programmation semi-définie et aux jeux à paiement moyen
  
  • Skomra Mateusz
  , 2018. Semidefinite programming (SDP) is a fundamental tool in convex and polynomial optimization. It consists in minimizing the linear functions over the spectrahedra (sets defined by linear matrix inequalities). In particular, SDP is a generalization of linear programming.The purpose of this thesis is to study the nonarchimedean analogue of SDP, replacing the field of real numbers by the field of Puiseux series. Our methods rely on tropical geometry and, in particular, on the study of tropicalization of spectrahedra.In the first part of the thesis, we analyze the images by valuation of general semialgebraic sets defined over the Puiseux series. We show that these images have a polyhedral structure, giving the real analogue of the Bieri--Groves theorem. Subsequently, we introduce the notion of tropical spectrahedra and show that, under genericity conditions, these objects can be described explicitly by systems of polynomial inequalities of degree 2 in the tropical semifield. This generalizes the result of Yu on the tropicalization of the SDP cone.One of the most important questions about real SDPs is to characterize the sets that arise as projections of spectrahedra. In this context, Helton and Nie conjectured that every semialgebraic convex set is a projected spectrahedron. This conjecture was disproved by Scheiderer. However, we show that the conjecture is true ''up to taking the valuation'': 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.In the second part of the thesis, we study the algorithmic questions related to SDP. The basic computational problem associated with SDP over real numbers is to decide whether a spectrahedron is nonempty. It is unknown whether this problem belongs to NP in the Turing machine model, and the state-of-the-art algorithms that certify the (in)feasibility of spectrahedra are based on cylindrical decomposition or the critical points method. We show that, in the nonarchimedean setting, generic tropical spectrahedra can be described by Shapley operators associated with stochastic mean payoff games. This provides a tool to solve nonarchimedean semidefinite feasibility problems using combinatorial algorithms designed for stochastic games.In the final chapters of the thesis, we provide new complexity bounds for the value iteration algorithm, exploiting the correspondence between stochastic games and tropical convexity. We show that the number of iterations needed to solve a game is controlled by a condition number, which is related to the inner radius of the associated tropical spectrahedron. We provide general upper bounds on the condition number. To this end, we establish optimal bounds on the bit-length of stationary distributions of Markov chains. As a corollary, our estimates show that value iteration can solve ergodic mean payoff games in pseudopolynomial time, provided that the number of random positions of the game is fixed. Finally, we apply our approach to large scale random nonarchimedean SDPs.
• On the Factorization Method for a Far Field Inverse Scattering Problem in the Time Domain
  
  • Cakoni Fioralba
  • Haddar Houssem
  • Lechleiter Armin
  , 2018.
• The promises and pitfalls of Stochastic Gradient Langevin Dynamics
  
  • Brosse Nicolas
  • Durmus Alain
  • Moulines Éric
  , 2018. Stochastic Gradient Langevin Dynamics (SGLD) has emerged as a key MCMC algorithm for Bayesian learning from large scale datasets. While SGLD with decreasing step sizes converges weakly to the posterior distribution, the algorithm is often used with a constant step size in practice and has demonstrated successes in machine learning tasks. The current practice is to set the step size inversely proportional to N where N is the number of training samples. As N becomes large, we show that the SGLD algorithm has an invariant probability measure which significantly departs from the target posterior and behaves like Stochastic Gradient Descent (SGD). This difference is inherently due to the high variance of the stochastic gradients. Several strategies have been suggested to reduce this effect; among them, SGLD Fixed Point (SGLDFP) uses carefully designed control variates to reduce the variance of the stochastic gradients. We show that SGLDFP gives approximate samples from the posterior distribution, with an accuracy comparable to the Langevin Monte Carlo (LMC) algorithm for a computational cost sublinear in the number of data points. We provide a detailed analysis of the Wasserstein distances between LMC, SGLD, SGLDFP and SGD and explicit expressions of the means and covariance matrices of their invariant distributions. Our findings are supported by limited numerical experiments.
• Low-rank Interaction with Sparse Additive Effects Model for Large Data Frames
  
  • Robin Geneviève
  • Wai Hoi-To
  • Josse Julie
  • Klopp Olga
  • Moulines Éric
  , 2018. Many applications of machine learning involve the analysis of large data frames-matrices collecting heterogeneous measurements (binary, numerical, counts, etc.) across samples-with missing values. Low-rank models, as studied by Udell et al. [30], are popular in this framework for tasks such as visualization, clustering and missing value imputation. Yet, available methods with statistical guarantees and efficient optimization do not allow explicit modeling of main additive effects such as row and column, or covariate effects. In this paper, we introduce a low-rank interaction and sparse additive effects (LORIS) model which combines matrix regression on a dictionary and low-rank design, to estimate main effects and interactions simultaneously. We provide statistical guarantees in the form of upper bounds on the estimation error of both components. Then, we introduce a mixed coordinate gradient descent (MCGD) method which provably converges sub-linearly to an optimal solution and is computationally efficient for large scale data sets. We show on simulated and survey data that the method has a clear advantage over current practices, which consist in dealing separately with additive effects in a preprocessing step.
• A partition of unity finite element method for computational diffusion MRI
  
  • Nguyen Van-Dang
  • Jansson Johan
  • Hoffman Johan
  • Li Jing-Rebecca
  Journal of Computational Physics, Elsevier, 2018, 375, pp.271-290.
• The spectral analysis of the interior transmission eigenvalue problem for Maxwell's equations
  
  • Haddar Houssem
  • Meng Shixu
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2018, 120, pp.1-32. In this paper we consider the transmission eigenvalue problem for Maxwell's equations corresponding to nonmagnetic inhomogeneities with contrast in electric permittivity that has fixed sign (only) in a neighborhood of the boundary. Following the analysis made by Robbiano in the scalar case we study this problem in the framework of semiclassical analysis and relate the transmission eigenvalues to the spectrum of a Hilbert-Schmidt operator. Under the additional assumption that the contrast is constant in a neighborhood of the boundary, we prove that the set of transmission eigenvalues is discrete, infinite and without finite accumulation points. A notion of generalized eigenfunctions is introduced and a denseness result is obtained in an appropriate solution space.
• A ghost imaging modality in a random waveguide
  
  • Borcea Liliana
  • Garnier Josselin
  Inverse Problems, IOP Publishing, 2018, 34 (12), pp.124007. (10.1088/1361-6420/aaea37)
  DOI : 10.1088/1361-6420/aaea37
• Tropical totally positive matrices
  
  • Gaubert Stéphane
  • Niv Adi
  Journal of Algebra, Elsevier, 2018, 515, pp.511-544. We investigate the tropical analogues of totally positive and totally nonnegative matrices. These arise when considering the images by the nonarchimedean valuation of the corresponding classes of matrices over a real nonarchimedean valued field, like the field of real Puiseux series. We show that the nonarchimedean valuation sends the totally positive matrices precisely to the Monge matrices. This leads to explicit polyhedral representations of the tropical analogues of totally positive and totally nonnegative matrices. We also show that sign-nonsingular tropical totally nonnegative matrix can be factorized in terms of elementary matrices. We finally determine the eigenvalues of tropical totally positive matrices, and relate them with the eigenvalues of totally positive matrices over nonarchimedean fields. (10.1016/j.jalgebra.2018.07.005)
  DOI : 10.1016/j.jalgebra.2018.07.005
• Incoherent Shock and Collapse Singularities in Non-Instantaneous Nonlinear Media
  
  • Xu Gang
  • Fusaro Adrien
  • Garnier Josselin
  • Picozzi Antonio
  Applied Sciences, Multidisciplinary digital publishing institute (MDPI), 2018, 8 (12), pp.2559. (10.3390/app8122559)
  DOI : 10.3390/app8122559
• Pathological Interactions Between Mutant Thyroid Hormone Receptors and Corepressors and Their Modulation by a Thyroid Hormone Analogue with Therapeutic Potential
  
  • Harrus Déborah
  • Démèné Héléne
  • Vasquez Edwin
  • Boulahtouf Abdelhay
  • Germain Pierre
  • Figueira Ana Carolina
  • Privalsky Martin
  • Bourguet William
  • Le Maire Albane
  Thyroid, Mary Ann Liebert, 2018, 28 (12), pp.1708-1722. BACKGROUND:Thyroid hormone receptors (TRs) are tightly regulated by the corepressors nuclear receptor corepressor (NCoR) and silencing mediator of retinoic acid and thyroid hormone receptors. Three conserved corepressor/NR signature box motifs (CoRNR1-3) forming the nuclear receptor interaction domain have been identified in these corepressors. Whereas TRs regulate multiple normal physiological and developmental pathways, mutations in TRs can result in endocrine diseases and be associated with cancers due to impairment of corepressor release. Three mutants that are located in helix H11 of TRs are of special interest: TRα-M388I, a mutant associated with the development of renal clear cell carcinomas (RCCCs), and TRβ-Δ430 and TRβ-Δ432, two deletion mutants causing resistance to thyroid hormone syndrome.METHODS:Several cell-based and biophysical methods were used to measure the affinity between wild-type and mutant TRα and TRβ and all the CoRNR motifs from corepressors to quantify the effects of different thyroid hormone analogues on these interactions. This study was coupled with the measurement of interactions between wild-type and mutant TRs in the context of a heterodimer with RXR to a NCoR fragment in the presence of the same ligands. Structural insights into the binding mode of corepressors to TRs were assessed in parallel by nuclear magnetic resonance spectroscopy.RESULTS:The study shows that TRs interact more avidly with the silencing mediator of retinoic acid and thyroid hormone receptors than with NCoR peptides, and that TRα binds most avidly to S-CoRNR3, whereas TRβ binds preferentially to S-CoRNR2. In the studied TR mutants, a transfer of the CoRNR-specificity toward CoRNR1 was observed, coupled with a significant increase in the binding strength. In contrast to 3,5,3'-triiodothyronine (T3), the agonist TRIAC and the antagonist NH-3 were very efficient at dissociating the abnormally strong interactions between mutant TRβs and corepressors. A strong impairment of T3-binding for TRβ mutants was shown compared to TRIAC and NH-3 and could explain the different efficiencies of the different ligands in releasing corepressors from the studied TRβ mutants. Consequently, TRIAC was found to be more effective than T3 in facilitating coactivator recruitment and decreasing the dominant activity of TRβ-Δ430.CONCLUSION:This study helps to clarify the specific interaction surfaces involved in the pathologic phenotype of TR mutants and demonstrates that TRIAC is a potential therapeutic agent for patients suffering from resistance to thyroid hormone syndromes. (10.1089/thy.2017.0551)
  DOI : 10.1089/thy.2017.0551
• Martingales in self-similar growth-fragmentations and their connections with random planar maps
  
  • Bertoin Jean
  • Budd Timothy
  • Curien Nicolas
  • Kortchemski Igor
  Probability Theory and Related Fields, Springer Verlag, 2018, 172 (3-4), pp.663-724. (10.1007/s00440-017-0818-5)
  DOI : 10.1007/s00440-017-0818-5
• Selection-mutation dynamics with age structure : long-time behaviour and application to the evolution of life-history traits
  
  • Roget Tristan
  , 2018. This thesis is divided into two parts connected by the same thread. It concerns the theoretical study and the application of mathematical models describing population dynamics. The individuals reproduce and die at rates which depend on age a and phenotypic trait. The trait is fixed duringthe life of the individual. It is modified over generations by mutations appearing during reproduction. Natural selection is modeled by introducing a density-dependent mortality rate describing competition for resources.In the first part, we study the long-term behavior of a selection-mutation partial differential equation with age structure describing such a large population. By studying the spectral properties of a family of positive operators on a measures space, we show the existence of stationary measures that can admit Dirac masses in traits maximizing fitness. When these measures admit a continuous density, we show the convergence of the solutions towards this (unique) stationary state.The second part of this thesis is motivated by a problem from the biology of aging. We want to understand the appearance and maintenance during evolution of a senescence marker observed in the species Drosophila melanogaster. For this, we introduce an individual-based model describing the dynamics of a population structured by age and by the following life history trait: the age of reproduction ending and the one where the mortality becomes non-zero. We also model the Lansing effect, which is the effect through which the “progeny of old parents do not live as long as those of young parents”. We show, under large population and rare mutation assumptions, that the evolution brings these two traits to coincide. For this, we are led to extend the canonical equation of adaptive dynamics to a situation where the fitness gradient does not admit sufficient regularity properties. The evolution of the trait is no longer described by the (unique) trajectory of an ordinary differential equation but by a set of trajectories solutions of a differential inclusion.
• Relationship between biodiversity and agricultural production
  
  • Brunetti Ilaria
  • Tidball Mabel
  • Couvet Denis
  , 2018. Agriculture is one of the main causes of biodiversity loss. In this work we model the interdependent relationship between biodiversity and agriculture on a farmed land, supposing that, while agriculture has a negative impact on biodiversity, the latter can increase agricultural production. Farmers act as myopic agents, who maximize their instantaneous profit without considering the negative effects of their practice on the evolution of biodiversity. We find that a tax on inputs can have a positive effect on yield since it can be considered as a social signal helping farmers to avoid myopic behavior in regards to the positive effect of biodiversity on yield. We also prove that, by increasing biodiversity productivity the level of biodiversity at equilibrium decreases, since when biodiversity is more productive farmers can maintain lower biodiversity to get the same yield
• Diffusion approximations and control variates for MCMC
  
  • Brosse Nicolas
  • Durmus Alain
  • Meyn Sean
  • Moulines Éric
  , 2018. A new methodology is presented for the construction of control variates to reduce the variance of additive functionals of Markov Chain Monte Carlo (MCMC) samplers. Our control variates are defined as linear combinations of functions whose coefficients are obtained by minimizing a proxy for the asymptotic variance. The construction is theoretically justified by two new results. We first show that the asymptotic variances of some well-known MCMC algorithms, including the Random Walk Metropolis and the (Metropolis) Unadjusted/Adjusted Langevin Algorithm, are close to the asymptotic variance of the Langevin diffusion. Second, we provide an explicit representation of the optimal coefficients minimizing the asymptotic variance of the Langevin diffusion. Several examples of Bayesian inference problems demonstrate that the corresponding reduction in the variance is significant, and that in some cases it can be dramatic.
• OPTIMAL INPUT POTENTIAL FUNCTIONS IN THE INTERACTING PARTICLE SYSTEM METHOD
  
  • Chraibi H
  • Dutfoy A
  • Galtier T
  • Garnier J.
  , 2018. The assessment of the probability of a rare event with a naive Monte-Carlo method is computationally intensive, so faster estimation methods, such as variance reduction methods, are needed. We focus on one of these methods which is the interacting particle (IPS) system method. The method requires to specify a set of potential functions. The choice of these functions is crucial, because it determines the magnitude of the variance reduction. So far, little information was available on how to choose the potential functions. To remedy this, we provide the expression of the optimal potential functions minimizing the asymptotic variance of the estimator of the IPS method.
• A tropical geometry and discrete convexity approach to bilevel programming : application to smart data pricing in mobile telecommunication networks
  
  • Eytard Jean-Bernard
  , 2018. Bilevel programming deals with nested optimization problems involving two players. A leader annouces a decision to a follower, who responds by selecting a solution of an optimization problem whose data depend on this decision (low level problem). The optimal decision of the leader is the solution of another optimization problem whose data depend on the follower's response (high level problem). When the follower's response is not unique, one distinguishes between optimistic and pessimistic bilevel problems, in which the leader takes into account the best or worst possible response of the follower.Bilevel problems are often used to model pricing problems.We are interested in applications in which the leader is a seller who announces a price, and the follower models the behavior of a large number of customers who determine their consumptions depending on this price.Hence, the dimension of the low-level is large. However, most bilevel problems are NP-hard, and in practice, there is no general method to solve efficiently large-scale bilevel problems.In this thesis, we introduce a new approach to tackle bilevel programming. We assume that the low level problem is a linear program, in continuous or discrete variables, whose cost function is determined by the leader. Then, the follower responses correspond to the cells of a special polyhedral complex, associated to a tropical hypersurface. This is motivated by recent applications of tropical geometry to model the behavior of economic agents.We use the duality between this polyhedral complex and a regular subdivision of an associated Newton polytope to introduce a decomposition method, in which one solves a series of subproblems associated to the different cells of the complex. Using results about the combinatorics of subdivisions, we show thatthis leads to an algorithm to solve a wide class of bilevel problemsin a time that is polynomial in the dimension of the low-level problem when the dimension of the high-level problem is fixed.Then, we identify special structures of bilevel problems forwhich this complexity bound can be improved.This is the case when the leader's cost function depends only on the follower's response. Then, we showthe optimistic bilevel problem can be solved in polynomial time.This applies in particular to high dimensional instances in which the datasatisfy certain discrete convexity properties. We also show that the solutions of such bilevel problems are limits of competitive equilibria.In the second part of this thesis, we apply this approach to a price incentive problem in mobile telecommunication networks.The aim for Internet service providers is to use pricing schemes to encourage the different users to shift their data consumption in time(and so, also in space owing to their mobility),in order to reduce the congestion peaks.This can be modeled by a large-scale bilevel problem.We show that a simplified case can be solved in polynomial time by applying the previous decomposition approach together with graph theory and discrete convexity results. We use these ideas to develop an heuristic method which applies to the general case. We implemented and validated this method on real data provided by Orange.