Publications

2020

• Modèles Génératifs Profonds : l'échantillonnage en haute dimension revisité
  
  • Moulines Eric
  , 2020. Les modèles génératifs (GM) permettent d’inférer des modèles de loi pour des observations structurées de grande dimension, qui sont typiques de l'IA moderne. Les modèles génératifs peuvent également être utilisés pour échantillonner de nouveaux exemples, en reliant le problème d'inférence à l'échantillonnage.L'apprentissage de modèles génératifs profonds (MGD) capables de capturer les structures de dépendance complexe de lois à partir de grands ensemble de données dans un cadre non-ou semi-supervisé apparaît aujourd'hui comme l'un des principaux défis de l'IA. Les modèles génératifs profonds ont de nombreuses applications passionnantes pour résoudre la pénurie de données en générant de " nouveaux " exemples, pour préserver la confidentialité en diffusant le modèle génératif à laplace des données mais aussi pour détecter les observations aberrantes.Dans cette présentation, je vais couvrir trois directions de recherche sur lesquelles je travaille actuellement.Une première approche est basée sur la minimisation de l'entropie croisée (divergence de Kullback-Leibler) entre la distribution des observations et un modèle paramétré soit par des réseaux de neurones profonds, soit par des fonctions d’energies plus adaptées, reliant les modèles génératifs et les « energy based models » quiont été introduits pour l’apprentissage non-supervisé (mais dans un cadre non-probabiliste). Cette approche est séduisante mais elle pose des problèmes de calcul difficiles, liés à la nécessité d'estimer la constante de normalisation et son gradient.Une deuxième approche repose sur les méthodes d'entropie maximale. Cette approche trouve son origine dans les quantités de physique statistique pour apprendre une distribution maximisant l'entropie sous contrainte de moment, qui sont construites à partir d'unereprésentation issue d’un réseau de neurones profondsUne troisième approche consiste à utiliser des auto-encodeurs variationnels (Variational Autoencoder, VAE), un cas particulier d'inférence variationnelle. Les VAE apprennent conjointement un algorithme pour générer des échantillons à partir de la distribution ainsi qu'un espace latent qui résume la distribution des observations.J’illustrerai ces approches par des exemples et je discuterai des challenges théoriques et numériques que ces approches posent. <a href="https://videos-rennes.inria.fr/video/HJt6vEaXI" target="_blank">[Vidéo en ligne]</a>
• Comparative study of harmonic and Rayleigh-Ritz procedures with applications to deflated conjugate gradients
  
  • Venkovic Nicolas
  • Mycek Paul
  • Giraud Luc
  • Le Maitre Olivier
  , 2020. Harmonic Rayleigh-Ritz and Raleigh-Ritz projection techniques are compared in the context of iterative procedures to solve for small numbers of least dominant eigenvectors of large symmetric positive definite matrices. The procedures considered are (i) locally optimal conjugate gradient (CG) methods, i.e., LOBCG, (ii) thick-restart Lanczos methods, and (iii) recycled linear CG solvers, e.g., eigCG. Approaches based on principles of local optimality are adapted to enable the use of harmonic projection techniques. Upon investigating the search spaces generated by these methods, it is found that LOBCG and thick-restart Lanczos methods can be adapted, which is not the case of eigCG. Explanations are also given as to why eigCG works so well in comparison to other recycling strategies. Numerical experiments show that, while approaches based on harmonic projections consistently result in a faster convergence of eigen-residuals, they generally do not yield better convergence of the forward error of eigenvectors, until the Rayleigh quotients have converged. Then, the effect of recycling strategies is investigated on deflation for the resolution of sequences of linear systems. While non-locally optimal recycling strategies need to solve more linear systems in order to fully develop their effect on convergence, they eventually reach similar behaviors to those of locally optimal recycling procedures. While implementations based on Init-CG are robust for systems with multiple right-hand sides, this is not the case for multiple operators.
• Quantitative modeling links in vivo microstructural and macrofunctional organization of human and macaque insular cortex, and predicts cognitive control abilities
  
  • Menon Vinod
  • Gallardo Guillermo
  • Pinsk Mark
  • Nguyen Van-Dang
  • Li Jing-Rebecca
  • Cai Weidong
  • Wassermann Demian
  , 2020. (10.1101/662601)
  DOI : 10.1101/662601
• Computing bi-tangents for transmission belts
  
  • Chouly Franz
  • Loubani Jinan
  • Lozinski Alexei
  • Méjri Bochra
  • Merito Kamil
  • Passos Sébastien
  • Pineda Angie
  , 2020. In this note, we determine the bi-tangents of two rotated ellipses, and we compute the coordinates of their points of tangency. For these purposes, we develop two approaches. The first one is an analytical approach in which we compute analytically the equations of the bi-tangents. This approach is valid only for some cases. The second one is geometrical and is based on the determination of the normal vector to the tangent line. This approach turns out to be more robust than the first one and is valid for any configuration of ellipses.
• Adaptive Bayesian SLOPE—High-dimensional Model Selection with Missing Values
  
  • Jiang Wei
  • Bogdan Malgorzata
  • Josse Julie
  • Miasojedow Blazej
  • Rockova Veronika
  , 2020. We consider the problem of variable selection in high-dimensional settings with missing observations among the covariates. To address this relatively understudied problem, we propose a new synergistic procedure -- adaptive Bayesian SLOPE -- which effectively combines the SLOPE method (sorted l1 regularization) together with the Spike-and-Slab LASSO method. We position our approach within a Bayesian framework which allows for simultaneous variable selection and parameter estimation, despite the missing values. As with the Spike-and-Slab LASSO, the coefficients are regarded as arising from a hierarchical model consisting of two groups: (1) the spike for the inactive and (2) the slab for the active. However, instead of assigning independent spike priors for each covariate, here we deploy a joint "SLOPE" spike prior which takes into account the ordering of coefficient magnitudes in order to control for false discoveries. Through extensive simulations, we demonstrate satisfactory performance in terms of power, FDR and estimation bias under a wide range of scenarios. Finally, we analyze a real dataset consisting of patients from Paris hospitals who underwent a severe trauma, where we show excellent performance in predicting platelet levels. Our methodology has been implemented in C++ and wrapped into an R package ABSLOPE for public use.
• Nonparametric imputation by data depth
  
  • Mozharovskyi Pavlo
  • Josse Julie
  • Husson François
  Journal of the American Statistical Association, Taylor & Francis, 2020, 115 (529), pp.241-253. The presented methodology for single imputation of missing values borrows the idea from data depth --- a measure of centrality defined for an arbitrary point of the space with respect to a probability distribution or a data cloud. This consists in iterative maximization of the depth of each observation with missing values, and can be employed with any properly defined statistical depth function. On each single iteration, imputation is narrowed down to optimization of quadratic, linear, or quasiconcave function being solved analytically, by linear programming, or the Nelder-Mead method, respectively. Being able to grasp the underlying data topology, the procedure is distribution free, allows to impute close to the data, preserves prediction possibilities different to local imputation methods (k-nearest neighbors, random forest), and has attractive robustness and asymptotic properties under elliptical symmetry. It is shown that its particular case --- when using Mahalanobis depth --- has direct connection to well known treatments for multivariate normal model, such as iterated regression or regularized PCA. The methodology is extended to the multiple imputation for data stemming from an elliptically symmetric distribution. Simulation and real data studies positively contrast the procedure with existing popular alternatives. The method has been implemented as an R-package. (10.1080/01621459.2018.1543123)
  DOI : 10.1080/01621459.2018.1543123
• Variance reduction for Markov chains with application to MCMC
  
  • Belomestny D
  • Iosipoi L
  • Moulines E
  • Naumov A
  • Samsonov S
  Statistics and Computing, Springer Verlag (Germany), 2020. In this paper we propose a novel variance reduction approach for additive functionals of Markov chains based on minimization of an estimate for the asymptotic variance of these functionals over suitable classes of control variates. A distinctive feature of the proposed approach is its ability to significantly reduce the overall finite sample variance. This feature is theoretically demonstrated by means of a deep non asymptotic analysis of a variance reduced functional as well as by a thorough simulation study. In particular we apply our method to various MCMC Bayesian estimation problems where it favourably compares to the existing variance reduction approaches.
• Multiscale population dynamics in reproductive biology: singular perturbation reduction in deterministic and stochastic models
  
  • Bonnet Celine
  • Chahour Keltoum
  • Clément Frédérique
  • Postel Marie
  • Yvinec Romain
  ESAIM: Proceedings and Surveys, EDP Sciences, 2020, 67, pp.72-99. In this study, we describe different modeling approaches for ovarian follicle population dynamics, based on either ordinary (ODE), partial (PDE) or stochastic (SDE) differential equations, and accounting for interactions between follicles. We put a special focus on representing the populationlevel feedback exerted by growing ovarian follicles onto the activation of quiescent follicles. We take advantage of the timescale difference existing between the growth and activation processes to apply model reduction techniques in the framework of singular perturbations. We first study the linear versions of the models to derive theoretical results on the convergence to the limit models. In the nonlinear cases, we provide detailed numerical evidence of convergence to the limit behavior. We reproduce the main semi-quantitative features characterizing the ovarian follicle pool, namely a bimodal distribution of the whole population, and a slope break in the decay of the quiescent pool with aging. (10.1051/proc/202067006)
  DOI : 10.1051/proc/202067006
• On the Turnpike Property and the Receding-Horizon Method for Linear-Quadratic Optimal Control Problems
  
  • Breiten Tobias
  • Pfeiffer Laurent
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2020, 58 (2), pp.26. Optimal control problems with a very large time horizon can be tackled with the Receding Horizon Control (RHC) method, which consists in solving a sequence of optimal control problems with small prediction horizon. The main result of this article is the proof of the exponential convergence (with respect to the prediction horizon) of the control generated by the RHC method towards the exact solution of the problem. The result is established for a class of infinite-dimensional linear-quadratic optimal control problems with time-independent dynamics and integral cost. Such problems satisfy the turnpike property: the optimal trajectory remains most of the time very close to the solution to the associated static optimization problem. Specific terminal cost functions, derived from the Lagrange multiplier associated with the static optimization problem, are employed in the implementation of the RHC method. (10.1137/18M1225811)
  DOI : 10.1137/18M1225811
• Accuracy assessment of the Non-Ideal Computational Fluid Dynamics model for siloxane MDM from the open-source SU2 suite
  
  • Gori Giulio
  • Zocca Marta
  • Cammi Giorgia
  • Spinelli Andrea
  • Congedo Pietro Marco
  • Guardone Alberto
  European Journal of Mechanics - B/Fluids, Elsevier, 2020, 79, pp.109-120. The first-ever accuracy assessment of a computational model for Non-Ideal Compressible-Fluid Dynamics (NICFD) flows is presented. The assessment relies on a comparison between numerical predictions, from the open-source suite SU2, and pressure and Mach number measurements of compressible fluid flows in the non-ideal regime. Namely, measurements regard supersonic flows of siloxane MDM (Octamethyltrisiloxane, C 8 H 24 O 2 Si 3) vapor expanding along isentropes in the close proximity of the liquid-vapor saturation curve. The model accuracy assessment takes advantage of an Uncertainty Quantification (UQ) analysis, to compute the variability of the numerical solution with respect the uncertainties affecting the test-rig operating conditions. This allows for an uncertainty-based assessment of the accuracy of numerical predictions. The test set is representative of typical operating conditions of Organic Rankine Cycle systems and it includes compressible flows expanding through a converging-diverging nozzle in mildly-to-highly non-ideal conditions. All the considered flows are well represented by the computational model. Therefore, the reliability of the numerical implementation and the predictiveness of the NICFD model are confirmed. (10.1016/j.euromechflu.2019.08.014)
  DOI : 10.1016/j.euromechflu.2019.08.014
• On Invariance and Linear Convergence of Evolution Strategies with Augmented Lagrangian Constraint Handling
  
  • Atamna Asma
  • Auger Anne
  • Hansen Nikolaus
  Theoretical Computer Science, Elsevier, 2020, 832, pp.68-97. In the context of numerical constrained optimization, we investigate stochastic algorithms, in particular evolution strategies, handling constraints via augmented Lagrangian approaches. In those approaches, the original constrained problem is turned into an unconstrained one and the function optimized is an augmented Lagrangian whose parameters are adapted during the optimization. The use of an augmented Lagrangian however breaks a central invariance property of evolution strategies, namely invariance to strictly increasing transformations of the objective function. We formalize nevertheless that an evolution strategy with augmented Lagrangian constraint handling should preserve invariance to strictly increasing affine transformations of the objective function and the scaling of the constraints—a subclass of strictly increasing transformations. We show that this invariance property is important for the linear convergence of these algorithms and show how both properties are connected. (10.1016/j.tcs.2018.10.006)
  DOI : 10.1016/j.tcs.2018.10.006
• Multimode communication through the turbulent atmosphere
  
  • Borcea Liliana
  • Garnier Josselin
  • Sølna Knut
  Journal of the Optical Society of America. A Optics, Image Science, and Vision, Optical Society of America, 2020, 37 (5), pp.720. A central question in free-space optical communications is how to improve the transfer of information between a transmitter and a receiver. The capacity of the communication channel can be increased by multiplexing of independent modes using either: (1) the multiple-input–multiple-output (MIMO) approach, where communication is done with modes obtained from the singular value decomposition of the transfer matrix from the transmitter array to the receiver array, or (2) the orbital angular momentum (OAM) approach, which uses vortex beams that carry angular momenta. In both cases, the number of usable modes is limited by the finite aperture of the transmitter and receiver, and the effect of the turbulent atmosphere. The goal of this paper is twofold: first, we show that the MIMO and OAM multiplexing schemes are closely related. Specifically, in the case of circular apertures, the leading singular vectors of the transfer matrix, which are useful for communication, are essentially the same as the commonly used Laguerre–Gauss vortex beams, provided these have a special radius that depends on the wavelength, the distance from the transmitter to the receiver, and the ratio of the radii of their apertures. Second, we characterize the effect of atmospheric turbulence on the communication modes using the phase screen method put in the mathematical framework of beam propagation in random media. (10.1364/JOSAA.384007)
  DOI : 10.1364/JOSAA.384007
• The boundary of random planar maps via looptrees
  
  • Kortchemski Igor
  • Richier Loïc
  Annales de la Faculté des Sciences de Toulouse. Mathématiques., Université Paul Sabatier _ Cellule Mathdoc, 2020, 29 (2), pp.391-430. (10.5802/afst.1636)
  DOI : 10.5802/afst.1636
• Parametric inference for diffusions observed at stopping times
  
  • Gobet Emmanuel
  • Stazhynski Uladzislau
  Electronic Journal of Statistics, Shaker Heights, OH : Institute of Mathematical Statistics, 2020, 14 (1). In this paper we study the problem of parametric inference for multidimensional 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 (such as [Doh87, GJ93, Gob01, AM04] among others) 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 (studied in [GL14, GS18]). 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. (10.1214/20-EJS1708)
  DOI : 10.1214/20-EJS1708
• A new McKean-Vlasov stochastic interpretation of the parabolic-parabolic Keller-Segel model: The one-dimensional case
  
  • Tomasevic Milica
  • Talay Denis
  Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2020, 26 (2), pp.1323-1353. In this paper we analyze a stochastic interpretation of the one-dimensional parabolic-parabolic Keller-Segel system without cut-off. It involves an original type of McKean-Vlasov interaction kernel. At the particle level, each particle interacts with all the past of each other particle by means of a time integrated functional involving a singular kernel. At the mean-field level studied here, the McKean-Vlasov limit process interacts with all the past time marginals of its probability distribution in a similarly singular way. We prove that the parabolic-parabolic Keller-Segel system in the whole Euclidean space and the corresponding McKean-Vlasov stochastic differential equation are well-posed for any values of the parameters of the model. (10.3150/19-BEJ1158)
  DOI : 10.3150/19-BEJ1158
• A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors
  
  • Akian Marianne
  • Gaubert Stéphane
  • Hochart Antoine
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2020, 40, pp.207--231. We establish a generalized Perron-Frobenius theorem, based on a combinatorial criterion which entails the existence of an eigenvector for any nonlinear order-preserving and positively homogeneous map $f$ acting on the open orthant $\mathbb{R}_{\scriptscriptstyle >0}^n$. This criterion involves dominions, i.e., sets of states that can be made invariant by one player in a two-person game that only depends on the behavior of $f$ "at infinity". In this way, we characterize the situation in which for all $\alpha, \beta > 0$, the "slice space" $\mathcal{S}_\alpha^\beta := \{ x \in \mathbb{R}_{\scriptscriptstyle >0}^n \mid \alpha x \leq f(x) \leq \beta x \}$ is bounded in Hilbert's projective metric, or, equivalently, for all uniform perturbations $g$ of $f$, all the orbits of $g$ are bounded in Hilbert's projective metric. This solves a problem raised by Gaubert and Gunawardena (Trans. AMS, 2004). We also show that the uniqueness of an eigenvector is characterized by a dominion condition, involving a different game depending now on the local behavior of $f$ near an eigenvector. We show that the dominion conditions can be verified by directed hypergraph methods. We finally illustrate these results by considering specific classes of nonlinear maps, including Shapley operators, generalized means and nonnegative tensors. (10.3934/dcds.2020009)
  DOI : 10.3934/dcds.2020009
• Commentaires sur le rapport de surveillance de culture du MON 810 en 2018. Paris, le 25 février 2020
  
  • Du Haut Conseil Des Biotechnologies Comité Scientifique
  • Angevin Frédérique
  • Bagnis Claude
  • Bar-Hen Avner
  • Barny Marie-Anne
  • Boireau Pascal
  • Brévault Thierry
  • Chauvel Bruno B.
  • Collonnier Cécile
  • Couvet Denis
  • Dassa Elie
  • Demeneix Barbara
  • Franche Claudine
  • Guerche Philippe
  • Guillemain Joël
  • Hernandez Raquet Guillermina
  • Khalife Jamal
  • Klonjkowski Bernard
  • Lavielle Marc
  • Le Corre Valérie
  • Lefèvre François
  • Lemaire Olivier
  • Lereclus Didier D.
  • Maximilien Rémy
  • Meurs Eliane
  • Naffakh Nadia
  • Négre Didier
  • Noyer Jean-Louis
  • Ochatt Sergio
  • Pages Jean-Christophe
  • Raynaud Xavier
  • Regnault-Roger Catherine
  • Renard Michel M.
  • Renault Tristan
  • Saindrenan Patrick
  • Simonet Pascal
  • Troadec Marie-Bérengère
  • Vaissière Bernard
  • de Verneuil Hubert
  • Vilotte Jean-Luc
  , 2020, pp.35 p.. Les analyses contenues dans le rapport de surveillance de Bayer Agriculture BVBA ne font apparaître aucun problème majeur associé à la culture de maïs MON 810 en 2018. Toutefois, le CS du HCB identifie encore certaines faiblesses et limites méthodologiques concernant la surveillance de la sensibilité des ravageurs ciblés à la toxine Cry1Ab, remettant en question les conclusions du rapport. Le HCB estime notamment que l’utilisation d’une dose diagnostic présente certaines limites pour la détection précoce de l’évolution de la résistance, tant dans son principe intrinsèque que dans sa mise en oeuvre par Bayer, et recommande une méthode alternative de type F2 screen permettant de déterminer la fréquence des allèles de résistance au sein d’une population de ravageurs cibles. Par ailleurs, le HCB formule des recommandations destinées à renforcer la mise en oeuvre des zones refuges pour prévenir ou retarder le développement de résistance à la toxine Cry1Ab chez les ravageurs ciblés. Concernant la surveillance générale, le CS du HCB relève un problème de pertinence méthodologique quant aux questions étudiées, avec des règles de décision arbitraires, des conclusions incorrectement justifiées et un possible biais associé au format d’enquête auprès du panel d’agriculteurs qui ont accepté de répondre au questionnaire. Enfin, le CS du HCB recommande que le rapport de surveillance considère la présence de téosinte dans des zones de culture du maïs MON 810 en Espagne et les risques potentiels associés à une éventuelle introgression de gènes de maïs MON 810 chez le téosinte.
• Tropical planar networks
  
  • Gaubert Stéphane
  • Niv Adi
  Linear Algebra and its Applications, Elsevier, 2020, 595, pp.123-144. We show that every tropical totally positive matrix can be uniquely represented as the transfer matrix of a canonical totally connected weighted planar network. We deduce a uniqueness theorem for the factorization of a tropical totally positive in terms of elementary Jacobi matrices. (10.1016/j.laa.2020.02.019)
  DOI : 10.1016/j.laa.2020.02.019
• Regression Monte Carlo methods for HJB-type equations: which approximation space?
  
  • Barrera David
  • Gobet Emmanuel
  • Lopez-Salas Jose
  • Turkedjiev Plamen
  • Vasquez Carlos
  • Zubelli Jorge
  , 2020.
• Optimal Hedging Under Fast-Varying Stochastic Volatility
  
  • Garnier Josselin
  • Sølna Knut
  SIAM Journal on Financial Mathematics, Society for Industrial and Applied Mathematics, 2020, 11 (1), pp.274-325. In a market with a rough or Markovian mean-reverting stochastic volatility thereis no perfect hedge. Here it is shown how various delta-type hedging strategies perform and canbe evaluated in such markets in the case of European options.A precise characterization of thehedging cost, the replication cost caused by the volatilityfluctuations, is presented in an asymptoticregime of rapid mean reversion for the volatility fluctuations. The optimal dynamic asset basedhedging strategy in the considered regime is identified as the so-called “practitioners” delta hedgingscheme. It is moreover shown that the performances of the delta-type hedging schemes are essentiallyindependent of the regularity of the volatility paths in theconsidered regime and that the hedgingcosts are related to a Vega risk martingale whose magnitude is proportional to a new market riskparameter. It is also shown via numerical simulations that the proposed hedging schemes whichderive from option price approximations in the regime of rapid mean reversion, are robust: the“practitioners” delta hedging scheme that is identified as being optimal by our asymptotic analysiswhen the mean reversion time is small seems to be optimal witharbitrary mean reversion times. (10.1137/18M1221655)
  DOI : 10.1137/18M1221655
• Kinetic derivation of diffuse-interface fluid models
  
  • Giovangigli Vincent
  Physical Review E, American Physical Society (APS), 2020, 102. We present a full derivation of capillary fluid equations from the kinetic theory of dense gases. These equations involve van der Waals' gradient energy, Korteweg's tensor, and Dunn and Serrin's heat flux as well as viscous and heat dissipative fluxes. Starting from macroscopic equations obtained from the kinetic theory of dense gases, we use a second-order expansion of the pair distribution function in order to derive the diffuse interface model. The capillary extra terms and the capillarity coefficient are then associated with intermolecular forces and the pair interaction potential. (10.1103/physreve.102.012110)
  DOI : 10.1103/physreve.102.012110
• The mean field Schrödinger problem: ergodic behavior, entropy estimates and functional inequalities
  
  • Backhoff Julio
  • Conforti Giovanni
  • Gentil Ivan
  • Léonard Christian
  Probability Theory and Related Fields, Springer Verlag, 2020, 178, pp.475-530. (10.1007/s00440-020-00977-8)
  DOI : 10.1007/s00440-020-00977-8
• A quantitative McDiarmid’s inequality for geometrically ergodic Markov chains
  
  • Havet Antoine
  • Lerasle Matthieu
  • Moulines Éric
  • Vernet Elodie
  Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2020, 25. (10.1214/20-ECP286)
  DOI : 10.1214/20-ECP286
• Null space gradient flows for constrained optimization with applications to shape optimization
  
  • Feppon Florian
  • Allaire Grégoire
  • Dapogny Charles
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2020, 26, pp.90. The purpose of this article is to introduce a gradient-flow algorithm for solving equality and inequality constrained optimization problems, which is particularly suited for shape optimization applications. We rely on a variant of the Ordinary Differential Equation (ODE) approach proposed by Yamashita (Math. Program. 18 (1980) 155–168) for equality constrained problems: the search direction is a combination of a null space step and a range space step, aiming to decrease the value of the minimized objective function and the violation of the constraints, respectively. Our first contribution is to propose an extension of this ODE approach to optimization problems featuring both equality and inequality constraints. In the literature, a common practice consists in reducing inequality constraints to equality constraints by the introduction of additional slack variables. Here, we rather solve their local combinatorial character by computing the projection of the gradient of the objective function onto the cone of feasible directions. This is achieved by solving a dual quadratic programming subproblem whose size equals the number of active or violated constraints. The solution to this problem allows to identify the inequality constraints to which the optimization trajectory should remain tangent. Our second contribution is a formulation of our gradient flow in the context of – infinite-dimensional – Hilbert spaces, and of even more general optimization sets such as sets of shapes, as it occurs in shape optimization within the framework of Hadamard’s boundary variation method. The cornerstone of this formulation is the classical operation of extension and regularization of shape derivatives. The numerical efficiency and ease of implementation of our algorithm are demonstrated on realistic shape optimization problems. (10.1051/cocv/2020015)
  DOI : 10.1051/cocv/2020015
• Computing invariant sets of random differential equations using polynomial chaos
  
  • Breden Maxime
  • Kuehn Christian
  SIAM Journal on Applied Dynamical Systems, Society for Industrial and Applied Mathematics, 2020, 19 (1), pp.577–618. Differential equations with random parameters have gained significant prominence in recent years due to their importance in mathematical modelling and data assimilation. In many cases, random ordinary differential equations (RODEs) are studied by using Monte-Carlo methods or by direct numerical simulation techniques using polynomial chaos (PC), i.e., by a series expansion of the random parameters in combination with forward integration. Here we take a dynamical systems viewpoint and focus on the invariant sets of differential equations such as steady states, stable/unstable manifolds, periodic orbits, and heteroclinic orbits. We employ PC to compute representations of all these different types of invariant sets for RODEs. This allows us to obtain fast sampling, geometric visualization of distributional properties of invariants sets, and uncertainty quantification of dynamical output such as periods or locations of orbits. We apply our techniques to a predator-prey model, where we compute steady states and stable/unstable manifolds. We also include several benchmarks to illustrate the numerical efficiency of adaptively chosen PC depending upon the random input. Then we employ the methods for the Lorenz system, obtaining computational PC representations of periodic orbits, stable/unstable manifolds and heteroclinic orbits. (10.1137/18M1235818)
  DOI : 10.1137/18M1235818