Publications

2021

• Analysis of the SORAS domain decomposition preconditioner for non-self-adjoint or indefinite problems
  
  • Bonazzoli Marcella
  • Claeys Xavier
  • Nataf Frédéric
  • Tournier Pierre-Henri
  Journal of Scientific Computing, Springer Verlag, 2021, 89. We analyze the convergence of the one-level overlapping domain decomposition preconditioner SORAS (Symmetrized Optimized Restricted Additive Schwarz) applied to a generic linear system whose matrix is not necessarily symmetric/self-adjoint nor positive definite. By generalizing the theory for the Helmholtz equation developed in [I.G. Graham, E.A. Spence, and J. Zou, SIAM J.Numer.Anal., 2020], we identify a list of assumptions and estimates that are sufficient to obtain an upper bound on the norm of the preconditioned matrix, and a lower bound on the distance of its field of values from the origin. We stress that our theory is general in the sense that it is not specific to one particular boundary value problem. Moreover, it does not rely on a coarse mesh whose elements are sufficiently small. As an illustration of this framework, we prove new estimates for overlapping domain decomposition methods with Robin-type transmission conditions for the heterogeneous reaction-convection-diffusion equation (to prove the stability assumption for this equation we consider the case of a coercive bilinear form, which is non-symmetric, though). (10.1007/s10915-021-01631-8)
  DOI : 10.1007/s10915-021-01631-8
• Regenerative properties of the linear Hawkes process with unbounded memory
  
  • Graham Carl
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2021, 31 (6), pp.2844-2863. We prove regenerative properties for the linear Hawkes process under minimal assumptions on the transfer function, which may have unbounded support. These results are applicable to sliding window statistical estimators. We exploit independence in the Poisson cluster point process decomposition, and the regeneration times are not stopping times for the Hawkes process. The regeneration time is interpreted as the renewal time at zero of a M/G/infinity queue, which yields a formula for its Laplace transform. When the transfer function admits some exponential moments, we stochastically dominate the cluster length by exponential random variables with parameters expressed in terms of these moments. This yields explicit bounds on the Laplace transform of the regeneration time in terms of simple integrals or special functions yielding an explicit negative upper-bound on its abscissa of convergence. These regenerative results allow, e.g., to systematically derive long-time asymptotic results in view of statistical applications. This is illustrated on a concentration inequality previously obtained with coauthors. (10.1214/21-AAP1664)
  DOI : 10.1214/21-AAP1664
• Outliers Detection in Networks with Missing Links
  
  • Gaucher Solenne
  • Klopp Olga
  • Robin Geneviève
  Computational Statistics and Data Analysis, Elsevier, 2021, 164, pp.107308. Outliers arise in networks due to different reasons such as fraudulent behavior of malicious users or default in measurement instruments and can significantly impair network analyses. In addition, real-life networks are likely to be incompletely observed, with missing links due to individual non-response or machine failures. Identifying outliers in the presence of missing links is therefore a crucial problem in network analysis. In this work, we introduce a new algorithm to detect outliers in a network that simultaneously predicts the missing links. The proposed method is statistically sound: we prove that, under fairly general assumptions, our algorithm exactly detects the outliers, and achieves the best known error for the prediction of missing links with polynomial computation cost. It is also computationally efficient: we prove sub-linear convergence of our algorithm. We provide a simulation study which demonstrates the good behavior of the algorithm in terms of outliers detection and prediction of the missing links. We also illustrate the method with an application in epidemiology, and with the analysis of a political Twitter network. The method is freely available as an R package on the Comprehensive R Archive Network. (10.1016/j.csda.2021.107308)
  DOI : 10.1016/j.csda.2021.107308
• Design of an acoustic energy distributor using thin resonant slits
  
  • Chesnel Lucas
  • Nazarov Sergei A
  Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences, Royal Society, The, 2021. We consider the propagation of time harmonic acoustic waves in a device made of three unbounded channels connected by thin slits. The wave number is chosen such that only one mode can propagate. The main goal of this work is to present a device which can serve as an energy distributor. More precisely, the geometry is first designed so that for an incident wave coming from one channel, the energy is almost completely transmitted in the two other channels. Additionally, adjusting slightly two geometrical parameters, we can control the ratio of energy transmitted in the two channels. The approach is based on asymptotic analysis for thin slits around resonance lengths. We also provide numerical results to illustrate the theory. (10.1098/rspa.2020.0896)
  DOI : 10.1098/rspa.2020.0896
• A Non-Nested Infilling Strategy for Multi-Fidelity based Efficient Global Optimization
  
  • Sacher Matthieu
  • Le Maitre Olivier
  • Duvigneau Régis
  • Hauville Frédéric
  • Durand Mathieu
  • Lothode C.
  International Journal for Uncertainty Quantification, Begell House Publishers, 2021, 11 (1), pp.1-30. Efficient Global Optimization (EGO) has become a standard approach for the global optimization of complex systems with high computational costs. EGO uses a training set of objective function values computed at selected input points to construct a statistical surrogate model, with low evaluation cost, on which the optimization procedure is applied. The training set is sequentially enriched, selecting new points, according to a prescribed infilling strategy, in order to converge to the optimum of the original costly model. Multi-fidelity approaches combining evaluations of the quantity of interest at different fidelity levels have been recently introduced to reduce the computational cost of building a global surrogate model. However, the use of multi-fidelity approaches in the context of EGO is still a research topic. In this work, we propose a new effective infilling strategy for multi-fidelity EGO. Our infilling strategy has the particularity of relying on non-nested training sets, a characteristic that comes with several computational benefits. For the enrichment of the multi-fidelity training set, the strategy selects the next input point together with the fidelity level of the objective function evaluation. This characteristic is in contrast with previous nested approaches, which require estimation all lower fidelity levels and are more demanding to update the surrogate. The resulting EGO procedure achieves a significantly reduced computational cost, avoiding computations at useless fidelity levels whenever possible, but it is also more robust to low correlations between levels and noisy estimations. Analytical problems are used to test and illustrate the efficiency of the method. It is finally applied to the optimization of a fully nonlinear fluid-structure interaction system to demonstrate its feasibility on real large-scale problems, with fidelity levels mixing physical approximations in the constitutive models and discretization refinements. (10.1615/Int.J.UncertaintyQuantification.2020032982)
  DOI : 10.1615/Int.J.UncertaintyQuantification.2020032982
• 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
• 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
• 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
• 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
• 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
• 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
• 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, American Physical Society (APS), 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
• 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
• 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
• 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
• 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
• 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
• Handling Hard Affine SDP Shape Constraints in RKHSs
  
  • Aubin-Frankowski Pierre-Cyril
  • Szabó Zoltán
  , 2021. Shape constraints, such as non-negativity, monotonicity, convexity or supermodularity, play a key role in various applications of machine learning and statistics. However, incorporating this side information into predictive models in a hard way (for example at all points of an interval) for rich function classes is a notoriously challenging problem. We propose a unified and modular convex optimization framework, relying on second-order cone (SOC) tightening, to encode hard affine SDP constraints on function derivatives, for models belonging to vector-valued reproducing kernel Hilbert spaces (vRKHSs). The modular nature of the proposed approach allows to simultaneously handle multiple shape constraints, and to tighten an infinite number of constraints into finitely many. We prove the consistency of the proposed scheme and that of its adaptive variant, leveraging geometric properties of vRKHSs. The efficiency of the approach is illustrated in the context of shape optimization, safety-critical control and econometrics.
• 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
• 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
• 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
• 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.
• 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