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Sub-Riemannian Ricci curvatures and universal diameter bounds for 3-Sasakian manifolds
No full textInternational audienceFor a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev-Zelenko-Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet-Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian. As an application, we consider the sub-Riemannian structure of 3-Sasakian manifolds, for which we provide explicit curvature formulas. We prove that any complete 3-Sasakian structure of dimension 4d + 3, with d > 1, has sub-Riemannian diameter bounded by π. When d = 1, a similar statement holds under additional Ricci bounds. These results are sharp for the natural sub-Riemannian structure of the quaternionic Hopf fibrations on the 4d+3 dimensional sphere, whose exact sub-Riemannian diameter is π, for all d ≥ 1
Convergence rate of strong approximations of compound random maps
No full textInternational audienceWe 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
Control of radiation and evaporation on temperature variability in a WRF regional climate simulation: comparison with colocated long term ground based observations near Paris
No full textInternational audienceThe objective of this paper is to understand how large-scale processes, cloud cover and surface fluxes affect the temperature variability over the SIRTA site, near Paris, and in a regional climate simulation performed in the frame of HyMeX/Med-CORDEX programs. This site is located in a climatic transitional area where models usually show strong dispersions despite the significant influence of large scale on interannual variability due to its western location. At seasonal time scale, the temperature is mainly controlled by surface fluxes. In the model, the transition from radiation to soil moisture limited regime occurs earlier than in observations leading to an overestimate of summertime temperature. An overestimate of shortwave radiation (SW), consistent with a lack of low clouds, enhances the soil dryness. A simulation with a wet soil is used to better analyse the relationship between dry soil and clouds but while the wetter soil leads to colder temperature, the cloud cover during daytime is not increased due to the atmospheric stability. At shorter time scales, the control of surface radiation becomes higher. In the simulation, higher temperatures are associated with higher SW. A wet soil mitigates the effect of radiation due to modulation by evaporation. In observations, the variability of clouds and their effect on SW is stronger leading to a nearly constant mean SW when sorted by temperature quantile but a stronger impact of cloud cover on day-to-day temperature variability. Impact of cloud albedo effect on precipitation is also compared
Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling
No full textInternational audienceWe relate some basic constructions of stochastic analysis to differential geometry , via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral curves associated to orthonormal frames, and we give particular attention to walks where the choice of step is influenced by a volume on the manifold. A primary motivation is to explore how one can pass, in the parabolic scaling limit, from geodesics, orthonormal frames, and/or volumes to diffusions, and hence their infinitesimal generators , on sub-Riemannian manifolds, which is interesting in light of the fact that there is no completely canonical notion of sub-Laplacian on a general sub-Riemannian mani-fold. However, even in the Riemannian case, this random walk approach illuminates the geometric significance of Ito and Stratonovich stochastic differential equations as well as the role played by the volume
Hilbert and Thompson geometries isometric to infinite-dimensional Banach spaces
No full textSee also arXiv:1610.07508International audienceWe study the horofunction boundaries of Hilbert and Thompson geome-tries, and of Banach spaces, in arbitrary dimension. By comparing the boundaries of these spaces, we show that the only Hilbert and Thompson geometries that are isometric to Banach spaces are the ones defined on the cone of positive continuous functions on a compact space
Study of new rare event simulation schemes and their application to extreme scenario generation
No full textInternational audienceThis is a companion paper based on our previous work [ADGL15] on rare event simulation methods. In this paper, we provide an alternative proof for the ergodicity of shaking transformation in the Gaussian case and propose two variants of the existing methods with comparisons of numerical performance. In numerical tests, we also illustrate the idea of extreme scenario generation based on the convergence of marginal distributions of the underlying Markov chains and show the impact of the discretization of continuous time models on rare event probability estimation
Continuous Optimal Control Approaches to Microgrid Energy Management
No full textInternational audience—We propose a novel method for the microgrid energy management problem by introducing a continuous-time, rolling horizon formulation. The energy management problem is formulated as a deterministic optimal control problem (OCP). We solve (OCP) with two classical approaches: the direct method [1], and Bellman's Dynamic Programming Principle (DPP) [2]. In both cases we use the optimal control toolbox BOCOP [3] for the numerical simulations. For the DPP approach we implement a semi-Lagrangian scheme [4] adapted to handle the optimization of switching times for the on/off modes of the diesel generator. The DPP approach allows for an accurate modeling and is computationally cheap. It finds the global optimum in less than 3 seconds, a CPU time similar to the Mixed Integer Linear Programming (MILP) approach used in [5]. We achieve this performance by introducing a trick based on the Pontryagin Maximum Principle (PMP). The trick increases the computation speed by several orders and also improves the precision of the solution. For validation purposes, simulation are performed using datasets from an actual isolated microgrid located in northern Chile. Results show that DPP method is very well suited for this type of problem when compared with the MILP approach
L'automate Epidémie et le modèle d'Eden face à l'irrégularité
No full textWe present some well-known and less well-known properties of the Probabilistic Cellular Automaton \emph{Epidemics} on a finite grid and its analogous on the infinite square lattice: the Eden model.Nous présentons dans cet article de survol quelques propriétés, plus ou moins bien connues, de l'automate cellulaire probabiliste Épidémie sur une grille finie, et de son analogue sur la grille infinie : le modèle d'Eden en temps discret
A NON-INTRUSIVE STRATIFIED RESAMPLER FOR REGRESSION MONTE CARLO: APPLICATION TO SOLVING NON-LINEAR EQUATIONS
No full textInternational audienceOur goal is to solve certain dynamic programming equations associated to a given Markov chain X, using a regression-based Monte Carlo algorithm. More specifically, we assume that the model for X is not known in full detail and only a root sample X1, . . . , XM of such process is available. By a stratification of the space and a suitable choice of a probability measure ν, we design a new resampling scheme that allows to compute local regressions (on basis functions) in each stratum. The combination of the stratification and the resampling allows to compute the solution to the dynamic programming equation (possibly in large dimensions) using only a relatively small set of root paths. To assess the accuracy of the algorithm, we establish non-asymptotic error estimates in L2(ν). Our numerical experiments illustrate the good performance, even with M = 20 − 40 root paths