Publications

2019

• Order isomorphisms of complete order-unit spaces
  
  • Walsh Cormac
  , 2019. We investigate order isomorphisms, which are not assumed to be linear, between complete order unit spaces. We show that two such spaces are order isomorphic if and only if they are linearly order isomorphic. We then introduce a condition which determines whether all order isomorphisms on a complete order unit space are automatically affine. This characterisation is in terms of the geometry of the state space. We consider how this condition applies to several examples, including the space of bounded self-adjoint operators on a Hilbert space. Our techniques also allow us to show that in a unital C*-algebra there is an order isomorphism between the space of self-adjoint elements and the cone of positive invertible elements if and only if the algebra is commutative.
• Logistic Regression with Missing Covariates -- Parameter Estimation, Model Selection and Prediction
  
  • Jiang Wei
  • Josse Julie
  • Lavielle Marc
  Computational Statistics and Data Analysis, Elsevier, 2019, pp.106907. Logistic regression is a common classification method in supervised learning. Surprisingly , there are very few solutions for performing it and selecting variables in the presence of missing values. We develop a complete approach, including the estimation of parameters and variance of estimators, derivation of confidence intervals and a model selection procedure, for cases where the missing values can be anywhere in covariates. By well organizing different patterns of missingness in each observation , we propose a stochastic approximation version of the EM algorithm based on Metropolis-Hasting sampling, to perform statistical inference for logistic regression with incomplete data. We also tackle the problem of prediction for a new individual with missing values, which is never addressed. The methodology is computationally efficient, and its good coverage and variable selection properties are demonstrated in a simulation study where we contrast its performances to other methods. For instance, the popular multiple imputation by chained equation can lead to biased estimates while our method is unbiased. We then illustrate the method on a dataset of severely traumatized patients from Paris hospitals to predict the occurrence of hemorrhagic shock, a leading cause of early preventable death in severe trauma cases. The aim is to consolidate the current red flag procedure, a binary alert identifying patients with a high risk of severe hemorrhage. The methodology is implemented in the R package misaem. (10.1016/j.csda.2019.106907)
  DOI : 10.1016/j.csda.2019.106907
• On well-posedness of time-harmonic problems in an unbounded strip for a thin plate model
  
  • Bourgeois Laurent
  • Chesnel Lucas
  • Fliss Sonia
  Communications in Mathematical Sciences, International Press, 2019, 17 (6), pp.1487-1529. We study the propagation of elastic waves in the time-harmonic regime in a waveguide which is unbounded in one direction and bounded in the two other (transverse) directions. We assume that the waveguide is thin in one of these transverse directions, which leads us to consider a Kirchhoff-Love plate model in a locally perturbed 2D strip. For time harmonic scattering problems in unbounded domains, well-posedness does not hold in a classical setting and it is necessary to prescribe the behaviour of the solution at infinity. This is challenging for the model that we consider and constitutes our main contribution. Two types of boundary conditions are considered: either the strip is simply supported or the strip is clamped. The two boundary conditions are treated with two different methods. For the simply supported problem, the analysis is based on a result of Hilbert basis in the transverse section. For the clamped problem, this property does not hold. Instead we adopt the Kondratiev's approach, based on the use of the Fourier transform in the unbounded direction, together with techniques of weighted Sobolev spaces with detached asymptotics. After introducing radiation conditions, the corresponding scattering problems are shown to be well-posed in the Fredholm sense. We also show that the solutions are the physical (outgoing) solutions in the sense of the limiting absorption principle. (10.4310/CMS.2019.v17.n6.a2)
  DOI : 10.4310/CMS.2019.v17.n6.a2
• A practical guide for conducting calibration and decision-making optimisation with complex ecological models
  
  • Mahévas Stéphanie
  • Picheny Victor
  • Lambert Patrick
  • Dumoulin Nicolas
  • Rouan Lauriane
  • Soulié Christophe
  • Brockhoff Dimo
  • Lehuta Sigrid
  • Le Riche Rodolphe
  • Faivre Robert
  • Drouineau Hilaire
  , 2019. Calibrating ecological models or making decisions with them is an optimisation problem with challenging methodological issues. Depending on the optimisation formulation, there may be a large variety of optimisation configurations (e.g. multiple objectives, constraints, stochastic criteria) and finding a single acceptable solution may be difficult. The challenges are exacerbated by the high computational cost and the non linear or elusive mathematical properties that increased with the complexity of numerical models. From the feedbacks of practitioners, the need for a guideline for conducting optimisation of complex models has emerged. In this context, we propose a practical guide for the complex model optimisation process, covering both calibration and decision-making. The guide sets out the workflow with recommendations for each step based on existing tools and methods usually scattered throughout the literature. This guide is accompanied with an ODDO template (Overview, Design, Details of Optimisation) to standardise the published description of model-based optimisation and suggests research directions. (10.20944/preprints201912.0249.v1)
  DOI : 10.20944/preprints201912.0249.v1
• Contrôle optimal, apprentissage statistique et modélisation du carnet d'ordres
  
  • Mounjid Othmane
  , 2019. L'objectif principal de cette thèse est de comprendre les interactions entre les agents financiers et le carnet d'ordres. Elle se compose de six chapitres inter-connectés qui peuvent toutefois être lus indépendamment.Nous considérons dans le premier chapitre le problème de contrôle d'un agent cherchant à prendre en compte la liquidité disponible dans le carnet d'ordres afin d'optimiser le placement d'un ordre unitaire. Notre stratégie permet de réduire le risque de sélection adverse. Néanmoins, la valeur ajoutée de cette approche est affaiblie en présence de temps de latence: prédire les mouvements futurs des prix est peu utile si le temps de réaction des agents est lent.Dans le chapitre suivant, nous étendons notre étude à un problème d'exécution plus général où les agents traitent des quantités non unitaires afin de limiter leur impact sur le prix. Notre tactique permet d'obtenir de meilleurs résultats que les stratégies d'exécution classiques.Dans le troisième chapitre, on s'inspire de l'approche précédente pour résoudre cette fois des problèmes de market making plutôt que des problèmes d'exécution. Ceci nous permet de proposer des stratégies pertinentes compatibles avec les actions typiques des market makers. Ensuite, nous modélisons les comportements des traders haute fréquence directionnels et des brokers institutionnels dans le but de simuler un marché où nos trois types d'agents interagissent de manière optimale les uns avec les autres.Nous proposons dans le quatrième chapitre un modèle d'agents où la dynamique des flux dépend non seulement de l'état du carnet d'ordres mais aussi de l'historique du marché. Pour ce faire, nous utilisons des généralisations des processus de Hawkes non linéaires. Dans ce cadre, nous sommes en mesure de calculer en fonction de flux individuels plusieurs indicateurs pertinents. Il est notamment possible de classer les market makers en fonction de leur contribution à la volatilité.Pour résoudre les problèmes de contrôle soulevés dans la première partie de la thèse, nous avons développé des schémas numériques. Une telle approche est possible lorsque la dynamique du modèle est connue. Lorsque l'environnement est inconnu, on utilise généralement les algorithmes itératifs stochastiques. Dans le cinquième chapitre, nous proposons une méthode permettant d'accélérer la convergence de tels algorithmes.Les approches considérées dans les chapitres précédents sont adaptées pour des marchés liquides utilisant le mécanisme du carnet d'ordres. Cependant, cette méthodologie n'est plus nécessairement pertinente pour des marchés régis par des règles de fonctionnement spécifiques. Pour répondre à cette problématique, nous proposons, dans un premier temps, d'étudier le comportement des prix sur le marché très particulier de l'électricité.
• A PHASE TRANSITION FOR LARGE VALUES OF BIFURCATING AUTOREGRESSIVE MODELS
  
  • Bansaye Vincent
  • Bitseki Penda Siméon Valère
  , 2019. We describe the asymptotic behavior of the number Zn[an, ∞) of individuals with a large value in a stable bifurcating autoregressive process. The study of the associated first moment E(Zn[an, ∞)) is equivalent to the annealed large deviation problem P(Yn ≥ an), where Y is an autoregressive process in a random environment and an → ∞. The population with large values and the trajectorial behavior of Zn[an, ∞) is obtained from the ancestral paths associated to the large deviations of Y together with its environment. The study of large deviations of autoregressive processes in random environment is of independent interest and achieved first in this paper. The proofs of trajectorial estimates for bifurcating autoregressive process involves then a law of large numbers for non-homogenous trees. Two regimes appear in the stable case, depending on the fact that one of the autoregressive parameter is greater than one or not. It yields two different asymptotic behaviors for the large local densities and maximal value of the bifurcating autoregressive process.
• On the use of sampling methods and spectral signatures to identify defects in inhomogeneous media
  
  • Napal Kevish
  , 2019. This thesis is a contribution to inverse scattering theory. We are more specifically interested in the non-destructive testing of heterogeneous materials such as composite materials by using acoustic waves. Monitoring this type of materials in an industrial environment is of major importance, but their complex structure makes this task difficult. The so-called sampling methods seem very promising to address this issue. We develop these techniques to detect the appearance of defects from far field data. The defects considered are impenetrable Neumann obstacles. We distinguish two categories of them, each requiring a specific treatment: cracks and obstacles with non empty interior.Thanks to the two complementary factorizations of the far field operator that we establish, we show that it is possible to approach the solution of the Interior Transmission Problem (ITP) from the data. The ITP is a system of partial differential equations that takes into account the physical parameters of the material being surveyed. We show that it is then possible to detect an anomaly by comparing the solutions of two different ITPs, one associated with measurements made before the defect appeared and the other one associated with measurements made after. The validity of the described method requires avoiding particular frequencies, which are the elements of the ITP spectrum for which this problem is not well posed. We show that this spectrum is an infinite set, countable and without finite accumulation points.In the last chapter, we use the recent notion of artificial backgrounds to image crack networks embedded in a homogeneous background. This approach allows us to design a transmission problem with the choice of the artificial background, for instance made of an obstacle. The associated spectrum is then sensitive to the presence of cracks inside the artificial obstacle. This allows to quantify locally the crack density. However, the computation of the spectrum requires data at several frequencies and is expensive in terms of calculations. We propose an alternative method using only data at fixed frequency and which consists in working with the solutions of the ITP instead of it's spectrum.
• Optimisation topologique de systèmes multiphysiques
  
  • Feppon Florian
  , 2019. This work is devoted to shape and topology optimization of multiphysics systemsmotivated by aeronautic industrial applications. Shape derivatives of arbitraryobjective functionals are computed for a weakly coupled thermal fluid-structuremodel. A novel gradient flow type algorithm is then developed for solving genericconstrained shape optimization problems without the need for tuning non-physicalmetaparameters. Motivated by the need for enforcing non-mixing constraints in thedesign of liquid-liquid heat exchangers, a variational method is developed in orderto simplify the numerical evaluation of geometric constraints: it allows to computeline integrals on a mesh by solving a variational problem without requiring theexplicit knowledge of these lines on the spatial discretization. All theseingredients allowed us to implement a variety of 2-d and 3-d multiphysics shapeoptimization test cases: from single, double or three physics problems in 2-d, tomoderately large-scale 3-d test cases for structural design, thermal conduction,aerodynamic design and a fluid-structure interacting system. A final opening chapterderives high order homogenized equations for perforated elliptic systems. These highorder equations encompass the three classical regimes of homogenized modelsassociated with different obstacle's size scalings. They could allow, in futureworks, to develop new topology optimization methods for fluid systems characterizedby multi-scale patterns as commonly encountered in industrial heat exchanger designs.
• Stochastic optimal control for the energy management of hybrid electric vehicles under traffic constraints
  
  • Le Rhun Arthur
  , 2019. The focus of this PhD thesis is to design an optimal Energy Management System (EMS) for a Hybrid Electric Vehicle (HEV) following traffic constraints.In the current state of the art, EMS are typically divided between real-time designs relying on local optimization methods, and global optimization that is only suitable for off-line use due to computational constraints.The starting point of the thesis is that in terms of energy consumption, the stochastic aspect of the traffic conditions can be accurately modelled thanks to (speed,acceleration) probability distributions.In order to reduce the data size of the model, we use clustering techniques based on the Wasserstein distance, the corresponding barycenters being computed by either a Sinkhorn or Stochastic Alternate Gradient method.Thanks to this stochastic traffic model, an off-line optimization can be performed to determine the optimal control (electric motor torque) that minimizes the fuel consumption of the HEV over a certain road segment.Then, a bi-level algorithm takes advantage of this information to optimize the consumption over a whole travel, the upper level optimization being deterministic and therefore fast enough for real-time implementation.We illustrate the relevance of the traffic model and the bi-level optimization, using both traffic data generated by a simulator, as well as some actual traffic data recorded near Lyon (France).Finally, we investigate the extension of the bi-level algorithm to the eco-routing problem, using an augmented graph to track the state of charge information over the road network.
• Solving Ergodic Markov Decision Processes and Perfect Information Zero-sum Stochastic Games by Variance Reduced Deflated Value Iteration
  
  • Akian Marianne
  • Gaubert Stéphane
  • Qu Zheng
  • Saadi Omar
  , 2019. Recently, Sidford, Wang, Wu and Ye (2018) developed an algorithm combining variance reduction techniques with value iteration to solve discounted Markov decision processes. This algorithm has a sublinear complexity when the discount factor is fixed. Here, we extend this approach to mean-payoff problems, including both Markov decision processes and perfect information zero-sum stochastic games. We obtain sublinear complexity bounds, assuming there is a distinguished state which is accessible from all initial states and for all policies. Our method is based on a reduction from the mean payoff problem to the discounted problem by a Doob h-transform, combined with a deflation technique. The complexity analysis of this algorithm uses at the same time the techniques developed by Sidford et al. in the discounted case and non-linear spectral theory techniques (Collatz-Wielandt characterization of the eigenvalue).
• Second order conditions for a control problem with discontinuous cost
  
  • Pfeiffer Laurent
  • Bayen Térence
  , 2019. In this paper, we consider the problem of minimizing the total time spent by a controlled dynamics outside a constraint set K. Also known as time of crisis, one essential feature of this problem is the discontinuity of the involved integral cost with respect to the state. We first relate this optimal control problem to a mixed initial-final problem with smooth data. Applying the classical theory of optimality conditions to the auxiliary (smooth) problem, we obtain as a main result second order necessary optimality conditions for the time of crisis. Considering the partition of the state space made out of K and its complementary, we notice that the problem can be seen as a particular case of a hybrid problem. Our analysis is thus a first step toward a second order analysis for the more general class of hybrid problems.
• A Min-plus-SDDP Algorithm for Deterministic Multistage Convex Programming
  
  • Akian Marianne
  • Chancelier Jean-Philippe
  • Tran Benoît
  , 2019. We consider discrete time optimal control problems with finite horizon involving continuous states and possibly both continuous and discrete controls, subject to non-stationary linear dynamics and convex costs. In this general framework, we present a stochastic algorithm which generates monotone approximations of the value functions as a pointwise supremum or infimum of basic functions (for example affine or quadratic) which are randomly selected. We give sufficient conditions on the way basic functions are selected in order to ensure almost sure convergence of the approximations to the value function on a set of interest. Then we study a linear-quadratic optimal control problem with one control constraint. On this toy example we show how to use our algorithm in order to build lower approximations, like the SDDP algorithm, as supremum of affine cuts and upper approximations, by min-plus techniques, as infimum of quadratic fonctions. (10.1109/cdc40024.2019.9028935)
  DOI : 10.1109/cdc40024.2019.9028935
• A Privacy-preserving Disaggregation Algorithm for Non-intrusive Management of Flexible Energy
  
  • Jacquot Paulin
  • Beaude Olivier
  • Benchimol Pascal
  • Gaubert Stéphane
  • Oudjane Nadia
  , 2019. We consider a resource allocation problem involving a large number of agents with individual constraints subject to privacy, and a central operator whose objective is to optimizing a global, possibly non-convex, cost while satisfying the agents' constraints. We focus on the practical case of the management of energy consumption flexibilities by the operator of a microgrid. This paper provides a privacy-preserving algorithm that does compute the optimal allocation of resources, avoiding each agent to reveal her private information (constraints and individual solution profile) neither to the central operator nor to a third party. Our method relies on an aggregation procedure: we maintain a global allocation of resources, and gradually disaggregate this allocation to enforce the satisfaction of private constraints, by a protocol involving the generation of polyhedral cuts and secure multiparty computations (SMC). To obtain these cuts, we use an alternate projections method à la Von Neumann, which is implemented locally by each agent, preserving her privacy needs. Our theoretical and numerical results show that the method scales well as the number of agents gets large, and thus can be used to solve the allocation problem in high dimension, while addressing privacy issues.
• On the compatibility between the adiabatic and the rotating wave approximations in quantum control
  
  • Augier Nicolas
  • Boscain Ugo
  • Sigalotti Mario
  , 2019. In this paper, we discuss the compatibility between the rotating-wave and the adiabatic approximations for controlled quantum systems. Although the paper focuses on applications to two-level quantum systems, the main results apply in higher dimension. Under some suitable hypotheses on the time scales, the two approximations can be combined. As a natural consequence of this, it is possible to design control laws achieving transitions of states between two energy levels of the Hamiltonian that are robust with respect to inhomogeneities of the amplitude of the control input.
• Learning from Both Experts and Data
  
  • Besson Rémi
  • Le Pennec Erwan
  • Allassonnière Stéphanie
  Entropy, MDPI, 2019, 21 (12), pp.1208. In this work, we study the problem of inferring a discrete probability distribution using both expert knowledge and empirical data. This is an important issue for many applications where the scarcity of data prevents a purely empirical approach. In this context, it is common to rely first on an a priori from initial domain knowledge before proceeding to an online data acquisition. We are particularly interested in the intermediate regime, where we do not have enough data to do without the initial a priori of the experts, but enough to correct it if necessary. We present here a novel way to tackle this issue, with a method providing an objective way to choose the weight to be given to experts compared to data. We show, both empirically and theoretically, that our proposed estimator is always more efficient than the best of the two models (expert or data) within a constant. (10.3390/e21121208)
  DOI : 10.3390/e21121208
• A Taylor Based Sampling Scheme for Machine Learning in Computational Physics
  
  • Novello Paul
  • Poëtte Gaël
  • Lugato David
  • Congedo Pietro Marco
  , 2019. Machine Learning (ML) is increasingly used to construct surrogate models for physical simulations. We take advantage of the ability to generate data using numerical simulations programs to train ML models better and achieve accuracy gain with no performance cost. We elaborate a new data sampling scheme based on Taylor approximation to reduce the error of a Deep Neural Network (DNN) when learning the solution of an ordinary differential equations (ODE) system.
• Unsupervised Scalable Representation Learning for Multivariate Time Series
  
  • Franceschi Jean-Yves
  • Dieuleveut Aymeric
  • Jaggi Martin
  , 2019, 32.
• Unsupervised Scalable Representation Learning for Multivariate Time Series
  
  • Franceschi Jean-Yves
  • Dieuleveut Aymeric
  • Jaggi Martin
  , 2019, 32, pp.4650--4661. Time series constitute a challenging data type for machine learning algorithms, due to their highly variable lengths and sparse labeling in practice. In this paper, we tackle this challenge by proposing an unsupervised method to learn universal embeddings of time series. Unlike previous works, it is scalable with respect to their length and we demonstrate the quality, transferability and practicability of the learned representations with thorough experiments and comparisons. To this end, we combine an encoder based on causal dilated convolutions with a novel triplet loss employing time-based negative sampling, obtaining general-purpose representations for variable length and multivariate time series.
• On the Global Convergence of (Fast) Incremental Expectation Maximization Methods
  
  • Karimi Belhal
  • Wai Hoi-To
  • Moulines Éric
  • Lavielle Marc
  , 2019. The EM algorithm is one of the most popular algorithm for inference in latent data models. The original formulation of the EM algorithm does not scale to large data set, because the whole data set is required at each iteration of the algorithm. To alleviate this problem, Neal and Hinton [1998] have proposed an incremental version of the EM (iEM) in which at each iteration the conditional expectation of the latent data (E-step) is updated only for a mini-batch of observations. Another approach has been proposed by Cappé and Moulines [2009] in which the E-step is replaced by a stochastic approximation step, closely related to stochastic gradient. In this paper, we analyze incremental and stochastic version of the EM algorithm as well as the variance reduced-version of [Chen et al., 2018] in a common unifying framework. We also introduce a new version incremental version, inspired by the SAGA algorithm by Defazio et al. [2014]. We establish non-asymptotic convergence bounds for global convergence. Numerical applications are presented in this article to illustrate our findings.
• Méthodes d'Optimisation et de Théorie des Jeux Appliquées aux Systèmes Électriques Décentralisés
  
  • Jacquot Paulin
  , 2019. In the context of smart grid and in the transition to decentralized electric systems, we address the problem of the management of distributed electric consumption flexibilities. We develop different methods based on distributed optimization and game theory approaches.We start by adopting the point of view of a centralized operator in charge of the management of flexibilities for several agents. We provide a distributed and privacy-preserving algorithm to compute consumption profiles for agents that are optimal for the operator.In the proposed method, the individual constraints as well as the individual consumption profile of each agent are never revealed to the operator or the other agents.Then, in a second model, we adopt a more decentralized vision and consider a game theoretic framework for the management of consumption flexibilities.This approach enables, in particular, to take into account the strategic behavior of consumers.Individual objectives are determined by dynamic billing mechanisms, which is motivated by the modeling of congestion effects occurring on time periods receiving a high electricity load from consumers.A relevant class of games in this framework is given by atomic splittable congestion games.We obtain several theoretical results on Nash equilibria for this class of games, and we quantify the efficiency of those equilibria by providing bounds on the price of anarchy.We address the question of the decentralized computation of equilibria in this context by studying the conditions and rates of convergence of the best response and projected gradients algorithms.In practice an operator may deal with a very large number of players, and evaluating the equilibria in a congestion game in this case will be difficult.To address this issue, we give approximation results on the equilibria in congestion and aggregative games with a very large number of players, in the presence of coupling constraints.These results, obtained in the framework of variational inequalities and under some monotonicity conditions, can be used to compute an approximate equilibrium, solution of a small dimension problem.In line with the idea of modeling large populations, we consider nonatomic congestion games with coupling constraints, with an infinity of heterogeneous players: these games arise when the characteristics of a population are described by a parametric density function.Under monotonicity hypotheses, we prove that Wardrop equilibria of such games, given as solutions of an infinite dimensional variational inequality, can be approximated by symmetric Wardrop equilibria of auxiliary games, solutions of low dimension variational inequalities.Again, those results can be the basis of tractable methods to compute an approximate Wardrop equilibrium in a nonatomic infinite-type congestion game.Last, we consider a game model for the study of decentralized peer-to-peer energy exchanges between a community of consumers with renewable production sources.We study the generalized equilibria in this game, which characterize the possible energy trades and associated individual consumptions.We compare the equilibria with the centralized solution minimizing the social cost, and evaluate the efficiency of equilibria through the price of anarchy.
• Feedback Stabilization of the Two-Dimensional Navier–Stokes Equations by Value Function Approximation
  
  • Breiten Tobias
  • Kunisch Karl
  • Pfeiffer Laurent
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2019, 80 (3), pp.599-641. The value function associated with an optimal control problem subject to the Navier–Stokes equations in dimension two is analyzed. Its smoothness is established arounda steady state, moreover, its derivatives are shown to satisfy a Riccati equation at theorder two and generalized Lyapunov equations at the higher orders. An approximationof the optimal feedback law is then derived from the Taylor expansion of the valuefunction. A convergence rate for the resulting controls and closed-loop systems isdemonstrated. (10.1007/s00245-019-09586-x)
  DOI : 10.1007/s00245-019-09586-x
• The Wellposedness of Path-dependent Multidimensional Forward-backward SDE
  
  • Hu Kaitong
  , 2019. We study in this paper the wellposedness of path-dependent multidimensional forward-backward stochastic differential equations (FBSDE). By path-dependent we mean that the coefficients of the forward-backward SDE at time t can depend on the whole path of the forward process up to time t. These kinds of forward-backward SDE appear when solving path-dependent stochastic control problem by means of variational calculus. At the heart of our analysis is the construction of a decoupling random field on the path space. We first prove the existence and the uniqueness of decoupling field on small time interval. Then by introducing the characteristic BSDE, we show that a global decoupling field can be constructed by patching local solutions together as long as the solution of the characteristic BSDE remains bounded. Finally, we show that the solution of a path-dependent forward-backward SDE is stable.
• Continuous-Time Principal-Agent Problem in Degenerate Systems
  
  • Hu Kaitong
  • Ren Zhenjie
  • Touzi Nizar
  , 2019. In this paper we present a variational calculus approach to Principal-Agent problem with a lump-sum payment on finite horizon in degenerate stochastic systems, such as filtered partially observed linear systems. Our work extends the existing methodologies in the Principal-Agent literature using dynamic programming and BSDE representation of the contracts in the non-degenerate controlled stochastic systems. We first solve the Principal's problem in an enlarged set of contracts defined by a forward-backward SDE system given by the first order condition of the Agent's problem using variational calculus. Then we use the sufficient condition of the Agent's problem to verify that the optimal contract that we obtain by solving the Principal's problem is indeed implementable (i.e. belonging to the admissible contract set). Importantly we consider the control problem in a weak formulation. Finally, we give explicit solution of the Principal-Agent problem in partially observed linear systems and extend our results to some mean field interacting Agents case.
• Portable simulation framework for diffusion MRI
  
  • Nguyen Van-Dang
  • de Leoni Massimiliano
  • Dancheva Tamara
  • Jansson Johan
  • Hoffman Johan
  • Wassermann Demian
  • Li Jing-Rebecca
  Journal of Magnetic Resonance, Elsevier, 2019, 309, pp.106611. The numerical simulation of the diffusion MRI signal arising from complex tissue micro-structures is helpful for understanding and interpreting imaging data as well as for designing and optimizing MRI sequences. The discretization of the Bloch-Torrey equation by finite elements is a more recently developed approach for this purpose, in contrast to random walk simulations, which has a longer history. While finite elements discretization is more difficult to implement than random walk simulations, the approach benefits from a long history of theoreticaland numerical developments by the mathematical and engineering communities. In particular, software packages for the automated solutions of partial differential equations using finite elements discretization, such as FEniCS, are undergoing active support and development. However, because diffusion MRI simulation is a relatively new application area, there is still a gap between thesimulation needs of the MRI community and the available tools provided by finite elements software packages. In this paper, we address two potential difficulties in using FEniCS for diffusion MRI simulation. First, we simplified software installation by the use of FEniCS containers that are completely portable across multiple platforms. Second, we provide a portable simulation framework based on Python and whose code is open source. This simulation framework can be seamlessly integrated with cloud computing resources such as Google. Colaboratory notebooks working on a web browser or with Google Cloud Platform with MPI parallelization. We show examples illustrating the accuracy, the computational times, and parallel computing capabilities. The framework contributes to reproducible science and open-source software in computational diffusion MRI with the hope that it will help to speed up method developments and stimulate research collaborations. (10.1016/j.jmr.2019.106611)
  DOI : 10.1016/j.jmr.2019.106611
• Optimal make-take fees for market making regulation
  
  • Euch Omar El
  • Mastrolia Thibaut
  • Rosenbaum Mathieu
  • Touzi Nizar
  , 2020, pp.109-148. We address the mechanism design problem of an exchange setting suitable make-take fees to attract liquidity on its platform. Using a principal-agent approach, we provide the optimal compensation scheme of a market maker in quasi-explicit form. This contract depends essentially on the market maker inventory trajectory and on the volatility of the asset. We also provide the optimal quotes that should be displayed by the market maker. The simplicity of our formulas allows us to analyze in details the effects of optimal contracting with an exchange, compared to a situation without contract. We show in particular that it improves liquidity and reduces trading costs for investors. We extend our study to an oligopoly of symmetric exchanges and we study the impact of such common agency policy on the system. (10.1111/mafi.12295)
  DOI : 10.1111/mafi.12295