Publications

2021

• Propagation for KPP bulk-surface systems in a general cylindrical domain
  
  • Bogosel Beniamin
  • Giletti Thomas
  • Tellini Andrea
  Nonlinear Analysis: Theory, Methods and Applications, Elsevier, 2021, 213, pp.42. In this paper, we investigate propagation phenomena for KPP bulk-surface systems in a cylindrical domain with general section and heterogeneous coefficients. As for the scalar KPP equation, we show that the asymptotic spreading speed of solutions can be computed in terms of the principal eigenvalues of a family of self-adjoint elliptic operators. Using this characterization, we analyze the dependence of the spreading speed on various parameters, including diffusion rates and the size and shape of the section of the domain. In particular, we provide new theoretical results on several asymptotic regimes like small and high diffusion rates and sections with small and large sizes. These results generalize earlier ones which were available in the radial homogeneous case. Finally, we numerically investigate the issue of shape optimization of the spreading speed. By computing its shape derivative, we observe, in the case of homogeneous coefficients, that a disk either maximizes or minimizes the speed, depending on the parameters of the problem, both with or without constraints. We also show the results of numerical shape optimization with non homogeneous coefficients, when the disk is no longer an optimizer. (10.1016/j.na.2021.112528)
  DOI : 10.1016/j.na.2021.112528
• Size matters for OTC market makers: General results and dimensionality reduction techniques
  
  • Bergault Philippe
  • Guéant Olivier
  Mathematical Finance, Wiley, 2021, 31 (1), pp.279-322. (10.1111/mafi.12286)
  DOI : 10.1111/mafi.12286
• Dissipation-enhanced collapse singularity of a nonlocal fluid of light in a hot atomic vapor
  
  • Azam Pierre
  • Fusaro Adrien
  • Fontaine Quentin
  • Garnier Josselin
  • Bramati Alberto
  • Picozzi Antonio
  • Kaiser Robin
  • Glorieux Quentin
  • Bienaimé Tom
  Physical Review A, American Physical Society, 2021, 104 (1), pp.013515. We study the out-of-equilibrium dynamics of a two-dimensional paraxial fluid of light using a near-resonant laser propagating through a hot atomic vapor. We observe a double shock-collapse instability: a shock (gradient catastrophe) for the velocity, as well as an annular (ring-shaped) collapse singularity for the density. We find experimental evidence that this instability results from the combined effect of the nonlocal photon-photon interaction and the linear photon losses. The theoretical analysis based on the method of characteristics reveals the main result that dissipation (photon losses) is responsible for an unexpected enhancement of the collapse instability. Detailed analytical modeling makes it possible to evaluate the nonlocality range of the interaction. The nonlocality is controlled by adjusting the atomic vapor temperature and is seen to increase dramatically when the atomic density becomes much larger than one atom per cubic wavelength. Interestingly, such a large range of the nonlocal photon-photon interaction has not been observed in an atomic vapor so far and its microscopic origin is currently unknown. (10.1103/PhysRevA.104.013515)
  DOI : 10.1103/PhysRevA.104.013515
• Quantifying the closeness to a set of random curves via the mean marginal likelihood
  
  • Rommel Cédric
  • Bonnans Joseph-Frédéric
  • Gregorutti Baptiste
  • Martinon Pierre
  ESAIM: Probability and Statistics, EDP Sciences, 2021, 25, pp.1-30. In this paper, we tackle the problem of quantifying the closeness of a newly observed curve to a given sample of random functions, supposed to have been sampled from the same distribution. We define a probabilistic criterion for such a purpose, based on the marginal density functions of an underlying random process. For practical applications, a class of estimators based on the aggregation of multivariate density estimators is introduced and proved to be consistent. We illustrate the effectiveness of our estimators, as well as the practical usefulness of the proposed criterion, by applying our method to a dataset of real aircraft trajectories. (10.1051/ps/2020028)
  DOI : 10.1051/ps/2020028
• A semi-supervised method for the characterization of degradation of nuclear power plants steam generators
  
  • Pinciroli Luca
  • Baraldi Piero
  • Shokry Ahmed
  • Zio Enrico
  • Seraoui Redouane
  • Mai Carole
  Progress in Nuclear Energy, Elsevier, 2021, 131, pp.103580. (10.1016/j.pnucene.2020.103580)
  DOI : 10.1016/j.pnucene.2020.103580
• Infinite stable Boltzmann planar maps are subdiffusive
  
  • Curien Nicolas
  • Marzouk Cyril
  Probability and Mathematical Physics, MSP, 2021, 2 (1), pp.1-26. The infinite discrete stable Boltzmann maps are generalisations of the well-known Uniform Infinite Planar Quadrangulation in the case where large degree faces are allowed. We show that the simple random walk on these random lattices is always subdiffusive with exponent less than 1/3. Our method is based on stationarity and geometric estimates obtained via the peeling process which are of own interest. (10.2140/pmp.2021.2.1)
  DOI : 10.2140/pmp.2021.2.1
• Multipoint formulas for inverse scattering at high energies
  
  • Novikov Roman G
  Russian Mathematical Surveys, Turpion, 2021, 76 (4), pp.723–725. We consider inverse scattering for the multidimensional Schrödinger equation with smooth compactly supported potential v. We give explicit asymptotic formulas for the Fourier transform Fv(p) at fixed p in terms of the scattering amplitude f at n points at high energies. The precision of these formulas is proportional to n. To our knowledge these formulas are new for n≥ 2, whereas they reduce to the Born formula at high energies for n=1. (10.1070/RM9994)
  DOI : 10.1070/RM9994
• Topological sensitivity analysis with respect to a small idealized bolt
  
  • Rakotondrainibe Lalaina
  • Allaire Grégoire
  • Orval Patrick
  Engineering Computations, Emerald, 2021, 39 (1), pp.115-146. Purpose : This paper is devoted to the theoretical and numerical study of a new topological sensitivity concerning the insertion of a small bolt connecting two parts in a mechanical structure. First, an idealized model of bolt is proposed which relies on a non-local interaction between the two ends of the bolt (head and threads) and possibly featuring a pre-stressed state. Second, a formula for the topological sensitivity of such an idealized bolt is rigorously derived for a large class of objective functions. Third, numerical tests are performed in 2d and 3d to assess the efficiency of the bolt topological sensitivity in the case of no pre-stress. In particular, the placement of bolts (acting then as springs) is coupled to the further optimization of their location and to the shape and topology of the structure for volume minimization under compliance constraint. Design/methodology/approach : The methodology relies on the adjoint method and the variational formulation of the linearized elasticity equations in order to establish the topological sensitivity. Findings : The numerical results prove the influence of the number and locations of the bolts which strongly influence the final optimized design of the structure. Originality/value : This paper is the first one to study the topology optimization of bolted systems without a fixed prescribed number of bolts. (10.1108/EC-03-2021-0131)
  DOI : 10.1108/EC-03-2021-0131
• Incoherent localized structures and hidden coherent solitons from the gravitational instability of the Schrödinger-Poisson equation
  
  • Garnier Josselin
  • Baudin Kilian
  • Fusaro Adrien
  • Picozzi Antonio
  Physical Review E : Statistical, Nonlinear, and Soft Matter Physics [2001-2015], American Physical Society, 2021, 104 (5), pp.054205. The long-term behavior of a modulationally unstable conservative nonintegrable system is known to be characterized by the soliton turbulence self-organization process. We consider this problem in the presence of a long-range interaction in the framework of the Schrödinger-Poisson (or Newton-Schrödinger) equation accounting for the gravitational interaction. By increasing the amount of nonlinearity, the system self-organizes into a large-scale incoherent localized structure that contains “hidden” coherent soliton states: The solitons can hardly be identified in the usual spatial or spectral domains, but their existence can be unveiled in the phase-space representation (spectrogram). We develop a theoretical approach that provides the coupled description of the coherent soliton component [governed by the Schrödinger-Poisson equation (SPE)] and of the incoherent structure [governed by a wave turbulence Vlasov-Poisson equation (WT-VPE)]. We demonstrate theoretically and numerically that the incoherent structure introduces an effective trapping potential that stabilizes the hidden coherent soliton and we show that the incoherent structure belongs to a family of stationary solutions of the WT-VPE. The analysis reveals that the incoherent structure evolves in the strongly nonlinear regime and that it is characterized by a compactly supported spectral shape. By relating the analytical properties of the hidden soliton to those of the stationary incoherent structure, we clarify the quantum-to-classical (i.e., SPE-to-VPE) correspondence in the limit <math><mrow><mi>ℏ</mi><mo>/</mo><mi>m</mi><mo>→</mo><mn>0</mn></mrow></math>: The hidden soliton appears as the latest residual quantum correction preceding the classical limit described by the VPE. This study is of potential interest for self-gravitating Boson models of fuzzy dark matter. Although we focus our paper on the Schrödinger-Poisson equation, we show that the regime of hidden solitons stabilized by an incoherent structure is general for long-range wave systems featured by an algebraic decay of the interacting potential. This work should stimulate nonlinear optics experiments in highly nonlocal nonlinear (thermal) media that mimic the long-range nature of gravitational interactions. (10.1103/PhysRevE.104.054205)
  DOI : 10.1103/PhysRevE.104.054205
• Variance Reduction for Dependent Sequences with Applications to Stochastic Gradient MCMC
  
  • Belomestny Denis
  • Iosipoi Leonid
  • Moulines Eric
  • Naumov Alexey
  • Samsonov Sergey
  SIAM/ASA Journal on Uncertainty Quantification, ASA, American Statistical Association, 2021, 9, pp.507 - 535. In this paper we propose a novel and practical variance reduction approach for additive functionals of dependent sequences. Our approach combines the use of control variates with the minimization of an empirical variance estimate. We analyze finite sample properties of the proposed method and derive finite-time bounds of the excess asymptotic variance to zero. We apply our methodology to stochastic gradient Markov chain Monte Carlo (SGMCMC) methods for Bayesian inference on large data sets and combine it with existing variance reduction methods for SGMCMC. We present empirical results carried out on a number of benchmark examples showing that our variance reduction method achieves significant improvement as compared to state-of-the-art methods at the expense of a moderate increase of computational overhead. (10.1137/19m1301199)
  DOI : 10.1137/19m1301199
• Multipoint formulas for phase recovering from phaseless scattering data
  
  • Novikov Roman
  The Journal of Geometric Analysis, Springer, 2021, 31 (2), pp.1965–1991. We give formulas for phase recovering from appropriate monochromatic phaseless scattering data at 2n points in dimension d = 3 and in dimension d = 2. These formulas are recurrent and explicit and their precision is proportional to n. By this result we continue studies of [Novikov, Bull.Sci.Math. 139, 923-936, 2015], where formulas of such a type were given for n = 1, d ≥ 2. (10.1007/s12220-019-00329-6)
  DOI : 10.1007/s12220-019-00329-6
• A coherent framework for learning spatiotemporal piecewise-geodesic trajectories from longitudinal manifold-valued data
  
  • Chevallier Juliette
  • Debavelaere Vianney
  • Allassonnière Stéphanie
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2021, 14 (1), pp.349-388. This paper provides a coherent framework for studying longitudinal manifold-valued data. We introduce a Bayesian mixed-effects model which allows estimating both a group-representative piecewise-geodesic trajectory in the Riemannian space of shape and inter-individual variability. We prove the existence of the maximum a posteriori estimate and its asymptotic consistency under reasonable assumptions. Due to the non-linearity of the proposed model, we use a stochastic version of the Expectation-Maximization algorithm to estimate the model parameters. Our simulations show that our model is not noise-sensitive and succeeds in explaining various paths of progression. (10.1137/20M1328026)
  DOI : 10.1137/20M1328026
• Dissipative boundary conditions and entropic solutions in dynamical perfect plasticity
  
  • Babadjian Jean-François
  • Crismale Vito
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2021, 148 (9), pp.75-127. We prove the well-posedness of a dynamical perfect plasticity model under general assumptions on the stress constraint set and on the reference configuration. The problem is studied by combining both calculus of variations and hyperbolic methods. The hyperbolic point of view enables one to derive a class of dissipative boundary conditions, somehow intermediate between homogeneous Dirichlet and Neumann ones. By using variational methods, we show the existence and uniqueness of solutions. Then we establish the equivalence between the original variational solutions and generalized entropic-dissipative ones, derived from a weak hyperbolic formulation for initial-boundary value Friedrichs' systems with convex constraints. (10.1016/j.matpur.2021.02.001)
  DOI : 10.1016/j.matpur.2021.02.001
• Homogenization of Maxwell's equations and related scalar problems with sign-changing coefficients
  
  • Bunoiu Renata
  • Chesnel Lucas
  • Ramdani Karim
  • Rihani Mahran
  Annales de la Faculté des Sciences de Toulouse. Mathématiques., Université Paul Sabatier _ Cellule Mathdoc, 2021, 30 (5), pp.1075-1119. In this work, we are interested in the homogenization of time-harmonic Maxwell's equations in a composite medium with periodically distributed small inclusions of a negative material. Here a negative material is a material modelled by negative permittivity and permeability. Due to the sign-changing coefficients in the equations, it is not straightforward to obtain uniform energy estimates to apply the usual homogenization techniques. The goal of this article is to explain how to proceed in this context. The analysis of Maxwell's equations is based on a precise study of two associated scalar problems: one involving the sign-changing permittivity with Dirichlet boundary conditions, another involving the sign-changing permeability with Neumann boundary conditions. For both problems, we obtain a criterion on the physical parameters ensuring uniform invertibility of the corresponding operators as the size of the inclusions tends to zero. In the process, we explain the link existing with the so-called Neumann-Poincaré operator, complementing the existing literature on this topic. Then we use the results obtained for the scalar problems to derive uniform energy estimates for Maxwell's system. At this stage, an additional difficulty comes from the fact that Maxwell's equations are also sign-indefinite due to the term involving the frequency. To cope with it, we establish some sort of uniform compactness result. (10.5802/afst.1694)
  DOI : 10.5802/afst.1694
• On the influence of cross-diffusion in pattern formation
  
  • Breden Maxime
  • Kuehn Christian
  • Soresina Cinzia
  Journal of Computational Dynamics, American Institute of Mathematical Sciences, 2021, 8 (2), pp.213. In this paper we consider the Shigesada-Kawasaki-Teramoto (SKT) model to account for stable inhomogeneous steady states exhibiting spatial segregation, which describe a situation of coexistence of two competing species. We provide a deeper understanding on the conditions required on both the cross-diffusion and the reaction coefficients for non-homogeneous steady states to exist, by combining a detailed linearized analysis with advanced numerical bifurcation methods via the continuation software pde2path. We report some numerical experiments suggesting that, when cross-diffusion is taken into account, there exist positive and stable non-homogeneous steady states outside of the range of parameters for which the coexistence homogeneous steady state is positive. Furthermore, we also analyze the case in which self-diffusion terms are considered. (10.3934/jcd.2021010)
  DOI : 10.3934/jcd.2021010
• Finite Volume approximation of a two-phase two fluxes degenerate Cahn-Hilliard model
  
  • Cancès Clément
  • Nabet Flore
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2021, 55 (3), pp.969--1003. We study a time implicit Finite Volume scheme for degenerate Cahn-Hilliard model proposed in [W. E and P. Palffy-Muhoray. Phys. Rev. E, 55:R3844R3846, 1997] and studied mathematically by the authors in [C. Cancès, D. Matthes, and F. Nabet. Arch. Ration. Mech. Anal., 233(2):837-866, 2019]. The scheme is shown to preserve the key properties of the continuous model, namely mass conservation, positivity of the concentrations, the decay of the energy and the control of the entropy dissipation rate. This allows to establish the existence of a solution to the nonlinear algebraic system corresponding to the scheme. Further, we show thanks to compactness arguments that the approximate solution converges towards a weak solution of the continuous problems as the discretization parameters tend to 0. Numerical results illustrate the behavior of the numerical model. (10.1051/m2an/2021002)
  DOI : 10.1051/m2an/2021002
• Computational Challenges in Sampling and Representation of Uncertain Reaction Kinetics in Large Dimensions
  
  • Almohammadi Saja M
  • Le Maitre Olivier
  • Knio Omar M
  International Journal for Uncertainty Quantification, Begell House Publishers, 2021, 12 (1), pp.1-24. This work focuses on constructing functional representations of quantities of interest (QoIs) of an uncertain system in high dimension. Attention is focused on the ignition delay time of an iso-octane air mixture, using a detailed chemical mechanism with 3,811 elementary reactions. Uncertainty in all reaction rates is directly accounted for using associated uncertainty factors, assuming independent log-uniform priors. A Latin hypercube sample (LHS) of the ignition delay times was first generated, and the resulting database was then exploited to assess the possibility of constructing polynomial chaos (PC) representations in terms of the canonical random variables parametrizing the uncertain rates. We explored two avenues, namely sparse regression (SR) using LASSO, and a coordinate transform (CT) approach. Preconditioned variants of both approaches were also considered, namely using the logarithm of the ignition delay time as QoI. Both approaches resulted in representations of the ignition delay with similar representation errors. However, the CT approach was able to reproduce better the empirical distribution of the underlying LHS ensemble, and also preserved the positivity of the ignition delay time. When preconditioned representations were considered, however, similar performances were obtained using CT and SR representations. The results also revealed that both the CT and SR representations yield consistent global sensitivity estimates. The results were finally used to test a reduced dimension representation, and to outline potential extensions of the work. (10.1615/Int.J.UncertaintyQuantification.2021035691)
  DOI : 10.1615/Int.J.UncertaintyQuantification.2021035691
• Decoupled Greedy Learning of CNNs for Synchronous and Asynchronous Distributed Learning
  
  • Belilovsky Eugene
  • Leconte Louis
  • Caccia Lucas
  • Eickenberg Michael
  • Oyallon Edouard
  , 2021. A commonly cited inefficiency of neural network training using back-propagation is the update locking problem: each layer must wait for the signal to propagate through the full network before updating. Several alternatives that can alleviate this issue have been proposed. In this context, we consider a simple alternative based on minimal feedback, which we call Decoupled Greedy Learning (DGL). It is based on a classic greedy relaxation of the joint training objective, recently shown to be effective in the context of Convolutional Neural Networks (CNNs) on large-scale image classification. We consider an optimization of this objective that permits us to decouple the layer training, allowing for layers or modules in networks to be trained with a potentially linear parallelization. With the use of a replay buffer we show that this approach can be extended to asynchronous settings, where modules can operate and continue to update with possibly large communication delays. To address bandwidth and memory issues we propose an approach based on online vector quantization. This allows to drastically reduce the communication bandwidth between modules and required memory for replay buffers. We show theoretically and empirically that this approach converges and compare it to the sequential solvers. We demonstrate the effectiveness of DGL against alternative approaches on the CIFAR-10 dataset and on the large-scale ImageNet dataset.
• On the sensitivity of structural turbulence uncertainty estimates to time and space resolution
  
  • Gori Giulio
  • Le Maître O P
  • Congedo P M
  Computers and Fluids, Elsevier, 2021. This paper presents a sensitivity analysis of structural turbulence uncertainty estimates to time and space resolution of numerical computations. Turbulence uncertainty estimates are obtained by means of the Eigenspace Perturbation Method (EPM). Results show that, in general, one can not expect the turbulence uncertainty estimates to be mesh and time-step independent based on the sole sensitivity analysis of the baseline solution. The recommendation is to carry out independent sensitivity studies, to guarantee that the confidence uncertainty estimates are well-predicted regardless of the space and time resolution. (10.1016/j.compfluid.2021.105081)
  DOI : 10.1016/j.compfluid.2021.105081
• Super-relaxation of space–time-quantized ensemble of energy loads to curtail their synchronization after demand response perturbation
  
  • Luchnikov Ilia
  • Métivier David
  • Ouerdane Henni
  • Chertkov Michael
  Applied Energy, Elsevier, 2021, 285, pp.116419. Ensembles of thermostatically controlled loads (TCL) provide a significant demand response reserve for the system operator to balance power grids. However, this also results in the parasitic synchronization of individual devices within the ensemble leading to long post-demand-response oscillations in the integrated energy consumption of the ensemble. The synchronization is eventually destructed by fluctuations, thus leading to the (pre-demand response) steady state; however, this natural desynchronization, or relaxation to a statistically steady state, is too long. A resolution of this problem consists in measuring the ensemble's instantaneous consumption and using it as a feedback to stochastic switching of the ensemble's devices between on- and off- states. A simplified continuous-time model showed that carefully tuned nonlinear feedback results in a fast (super-) relaxation of the ensemble energy consumption. Since both state information and control signals are discrete, the actual TCL devices operation is space-time quantized, and this must be considered for realistic TCL ensemble modelling. Here, assuming that states are characterized by indoor temperature (quantifying comfort) and air conditioner regime (on, off), we construct a discrete model based on the probabilistic description of state transitions. We demonstrate that super-relaxation holds in such a more realistic setting, and that while it is stable against randomness in the stochastic matrix of the quantized model, it remains sensitive to the time discretization scheme. Aiming to achieve a balance between super-relaxation and customer's comfort, we analyze the dependence of super-relaxation on details of the space-time quantization, and provide a simple analytical criterion to avoid undesirable oscillations in consumption. (10.1016/j.apenergy.2020.116419)
  DOI : 10.1016/j.apenergy.2020.116419
• Stochastic analysis of emergence of evolutionary cyclic behavior in population dynamics with transfer
  
  • Champagnat Nicolas
  • Méléard Sylvie
  • Tran Viet Chi
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2021, 31 (4), pp.1820-1867. Horizontal gene transfer consists in exchanging genetic materials between microorganisms during their lives. This is a major mechanism of bacterial evolution and is believed to be of main importance in antibiotics resistance. We consider a stochastic model for the evolution of a discrete population structured by a trait taking finitely many values, with density-dependent competition. Traits are vertically inherited unless a mutation occurs, and can also be horizontally transferred by unilateral conjugation with frequency dependent rate. Our goal is to analyze the trade-off between natural evolution to higher birth rates and transfer, which drives the population towards lower birth rates. Simulations show that evolutionary outcomes include evolutionary suicide or cyclic re-emergence of small populations with well-adapted traits. We focus on a parameter scaling where individual mutations are rare but the global mutation rate tends to infinity. This implies that negligible sub-populations may have a strong contribution to evolution. Our main result quantifies the asymptotic dynamics of subpopulation sizes on a logarithmic scale. We characterize the possible evolutionary outcomes with explicit criteria on the model parameters. An important ingredient for the proofs lies in comparisons of the stochastic population process with linear or logistic birth-death processes with immigration. For the latter processes, we derive several results of independent interest. (10.1214/20-AAP1635)
  DOI : 10.1214/20-AAP1635
• Imaging in Random Media by Two-Point Coherent Interferometry
  
  • Garnier Josselin
  • Borcea Liliana
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2021, 14 (4), pp.1635-1668. (10.1137/21M142068X)
  DOI : 10.1137/21M142068X
• Weak solutions for potential mean field games of controls
  
  • Graber Jameson
  • Mullenix Alan
  • Pfeiffer Laurent
  Nonlinear Differential Equations and Applications, Springer Verlag, 2021, 28 (5), pp.Paper No 50, 34 pages. We analyze a system of partial differential equations that model a potential mean field game of controls, briefly MFGC. Such a game describes the interaction of infinitely many negligible players competing to optimize a personal value function that depends in aggregate on the state and, most notably, control choice of all other players. A solution of the system corresponds to a Nash Equilibrium, a group optimal strategy for which no one player can improve by altering only their own action. We investigate the second order, possibly degenerate, case with non-strictly elliptic diffusion operator and local coupling function. The main result exploits potentiality to employ variational techniques to provide a unique weak solution to the system, with additional space and time regularity results under additional assumptions. New analytical subtleties occur in obtaining a priori estimates with the introduction of an additional coupling that depends on the state distribution as well as feedback. (10.1007/s00030-021-00712-9)
  DOI : 10.1007/s00030-021-00712-9
• Economic Modelling of the Bitcoin Mining Industry
  
  • Bertucci Charles
  • Bertucci Louis
  • Lasry Jean-Michel
  • Lions Pierre-Louis
  SSRN Electronic Journal, Elsevier, 2021, pp.3907822. We propose a parsimonious homogenous framework for analyzing the production industry of Bitcoin. Despite a constant growth environment, the revenue per hashrate unit follows a mean reverting process. Empirically, our model fits the data well. We quantify the stability and the strength of the bitcoin transactional system which is the public good created by the Bitcoin protocol. Shocks can have a lasting effect in the medium run, but in the long run the mining equilibrium, and therefore the Blockchain security, is shown to be highly resilient even in extreme scenarios. (10.2139/ssrn.3907822)
  DOI : 10.2139/ssrn.3907822
• Piecewise Affine Dynamical Models of Timed Petri Nets -- Application to Emergency Call Centers
  
  • Allamigeon Xavier
  • Boyet Marin
  • Gaubert Stéphane
  Fundamenta Informaticae, Polskie Towarzystwo Matematyczne, 2021, 183 (3-4), pp.169-201. We study timed Petri nets, with preselection and priority routing. We represent the behavior of these systems by piecewise affine dynamical systems. We use tools from the theory of nonexpansive mappings to analyze these systems. We establishan equivalence theorem between priority-free fluid timed Petri nets and semi-Markov decision processes, from which we derive the convergence to a periodic regime and the polynomial-time computability of the throughput. More generally, we develop an approach inspired by tropical geometry, characterizing the congestion phases as the cells of a polyhedral complex. We illustrate these results by a current application to the performance evaluation of emergency call centers in the Paris area. (10.3233/FI-2021-2086)
  DOI : 10.3233/FI-2021-2086