Publications

2021

• A Universal 2-state n-action Adaptive Management Solver
  
  • Pascal Luz Valerie
  • Akian Marianne
  • Nicol Sam
  • Chades Iadine
  , 2021, 35 (17), pp.14884-14892. In poor data and urgent decision-making applications, managers need to make decisions without complete knowledge of the system dynamics. In biodiversity conservation, adaptive management (AM) is the principal tool for decision-making under uncertainty. AM can be solved using simplified Mixed Observable Markov Decision Processes called hidden model MDPs (hmMDPs) when the unknown dynamics are assumed stationary. hmMDPs provide optimal policies to AM problems by augmenting the MDP state space with an unobservable state variable representing a finite set of predefined models. A drawback in formalising an AM problem is that experts are often solicited to provide this predefined set of models by specifying the transition matrices. Expert elicitation is a challenging and time-consuming process that is prone to biases, and a key assumption of hmMDPs is that the true transition matrix will be included in the candidate model set. We propose an original approach to build a hmMDP with a universal set of predefined models that is capable of solving any 2-state n-action AM problem. Our approach uses properties of the transition matrices to build the model set and is independent of expert input, removing the potential for expert error in the optimal solution. We provide analytical formulations to derive the minimum set of models to include into an hmMDP to solve any AM problems with 2 states and n actions. We assess our universal AM algorithm on two species conservation case studies from Australia and randomly generated problems. (10.1609/aaai.v35i17.17747)
  DOI : 10.1609/aaai.v35i17.17747
• On stochastic gradient Langevin dynamics with dependent data streams in the logconcave case
  
  • Barkhagen M.
  • Chau N.H.
  • Moulines É.
  • Rásonyi M.
  • Sabanis S.
  • Zhang Y.
  Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2021, 27 (1). (10.3150/19-BEJ1187)
  DOI : 10.3150/19-BEJ1187
• Inference with selection, varying population size and evolving population structure: Application of ABC to a forward-backward coalescent process with interactions
  
  • Lepers Clotilde
  • Billiard Sylvain
  • Porte Matthieu
  • Méléard Sylvie
  • Tran Viet-Chi
  Heredity, Nature Publishing Group, 2021, 126, pp.335–350. Genetic data are often used to infer demographic history and changes or detect genes under selection. Inferential methods are commonly based on models making various strong assumptions: demography and population structures are supposed \textit{a priori} known, the evolution of the genetic composition of a population does not affect demography nor population structure, and there is no selection nor interaction between and within genetic strains. In this paper, we present a stochastic birth-death model with competitive interactions and asexual reproduction. We develop an inferential procedure for ecological, demographic and genetic parameters. We first show how genetic diversity and genealogies are related to birth and death rates, and to how individuals compete within and between strains. {This leads us to propose an original model of phylogenies, with trait structure and interactions, that allows multiple merging}. Second, we develop an Approximate Bayesian Computation framework to use our model for analyzing genetic data. We apply our procedure to simulated data from a toy model, and to real data by analyzing the genetic diversity of microsatellites on Y-chromosomes sampled from Central Asia human populations in order to test whether different social organizations show significantly different fertility. (10.1038/s41437-020-00381-x)
  DOI : 10.1038/s41437-020-00381-x
• The role of mode switching in a population of actin polymers with constraints
  
  • Robin François
  • van Gorp Anne
  • Véber Amandine
  Journal of Mathematical Biology, Springer, 2021, 82. In this paper, we introduce a stochastic model for the dynamics of actin polymers and their interactions with other proteins in the cellular envelop. Each polymer elongates and shortens, and can switch between several modes depending on whether it is bound to accessory proteins that modulate its behaviour as, for example, elongation-promoting factors. Our main aim is to understand the dynamics of a large population of polymers, assuming that the only limiting quantity is the total amount of monomers, set to be constant to some large N. We first focus on the evolution of a very long polymer, of size O(N), with a rapid switch between modes (compared to the timescale over which the macroscopic fluctuations in the polymer size appear). Letting N tend to infnity, we obtain a fluid limit in which the effect of the switching appears only through the fraction of time spent in each mode at equilibrium. We show in particular that, in our situation where the number of monomers is limiting, a rapid binding-unbinding dynamics may lead to an increased elongation rate compared to the case where the polymer is trapped in any of the modes. Next, we consider a large population of polymers and complexes, represented by a random measure on some appropriate type space. We show that as N tends to infinity, the stochastic system converges to a deterministic limit in which the switching appears as a flow between two categories of polymers. We exhibit some numerical examples in which the limiting behaviour of a single polymer differs from that of a population of competing (shorter) polymers for equivalent model parameters. Taken together, our results demonstrate that under conditions where the total number of monomers is limiting, the study of a single polymer is not sufficient to understand the behaviour of an ensemble of competing polymers. (10.1007/s00285-021-01551-z)
  DOI : 10.1007/s00285-021-01551-z
• Onset of energy equipartition among surface and body waves
  
  • Borcea Liliana
  • Garnier Josselin
  • Sølna Knut
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2021, 477 (2246), pp.20200775. We derive a radiative transfer equation that accountsfor coupling from surface waves to body waves and the other way around. The model is the acoustic wave equation in a two-dimensional waveguide with reflecting boundary. The waveguide has a thin, weakly randomly heterogeneous layer near the top surface, and a thick homogeneous layer beneath it. There are two types of modes that propagate along the axis of the waveguide: those that are almost trapped in the thin layer, and thus model surface waves, and those that penetrate deep in the waveguide, and thus model body waves. The remaining modes are evanescent waves. We introduce a mathematical theory of mode coupling induced by scattering in the thin layer, and derive a radiative transfer equation which quantifies the mean mode power exchange. We study the solution of this equation in the asymptotic limit of infinite width of the waveguide. The mainresult is a quantification of the rate of convergence ofthe mean mode powers toward equipartition. (10.1098/rspa.2020.0775)
  DOI : 10.1098/rspa.2020.0775
• Intelligent Questionnaires Using Approximate Dynamic Programming
  
  • Logé Frédéric
  • Le Pennec Erwan
  • Amadou Boubacar Habiboulaye
  i-com, Oldenbourg Verlag, 2021, 19 (3), pp.227-237. Abstract Inefficient interaction such as long and/or repetitive questionnaires can be detrimental to user experience, which leads us to investigate the computation of an intelligent questionnaire for a prediction task. Given time and budget constraints (maximum q questions asked), this questionnaire will select adaptively the question sequence based on answers already given. Several use-cases with increased user and customer experience are given. The problem is framed as a Markov Decision Process and solved numerically with approximate dynamic programming, exploiting the hierarchical and episodic structure of the problem. The approach, evaluated on toy models and classic supervised learning datasets, outperforms two baselines: a decision tree with budget constraint and a model with q best features systematically asked. The online problem, quite critical for deployment seems to pose no particular issue, under the right exploration strategy. This setting is quite flexible and can incorporate easily initial available data and grouped questions. (10.1515/icom-2020-0022)
  DOI : 10.1515/icom-2020-0022
• Shape and topology optimization
  
  • Allaire Grégoire
  • Dapogny Charles
  • Jouve François
  , 2021, 22. This chapter is an introduction to shape and topology optimization, with a particular emphasis on the method of Hadamard for appraising the sensitivity of quantities of interest with respect to the domain, and on the level set method for the numerical representation of shapes and their evolutions. At the theoretical level, the method of Hadamard considers variations of a shape as "small" deformations of its boundary; this results in a mathematically convenient and versatile notion of differentiation with respect to the domain, which has historically often been associated with "body-fitted" geometric optimization methods. At the numerical level, the level set method features an implicit description of the shape, which arises as the negative subdomain of an auxiliary "level set function". This type of representation is well-known to be very efficient when it comes to describing dramatic evolutions of domains (including topological changes). The combination of these two ingredients is an ideal approach for optimizing both the geometry and the topology of shapes, and two related implementation frameworks are presented. The first and oldest one is a Eulerian shape capturing method, using a fixed mesh of a working domain in which the optimal shape is sought. The second and newest one is a Lagrangian shape tracking method, where the shape is exactly meshed at each iteration of the optimization process. In both cases, the level set algorithm is instrumental in updating the shapes, allowing for dramatic deformations between the iterations of the process, and even for topological changes. Most of our applicative examples stem from structural mechanics although some other physical contexts are briefly exemplified. Other topology optimization methods, like density-based algorithms or phase-field methods are also presented, at a lesser level of details, for comparison purposes. (10.1016/bs.hna.2020.10.004)
  DOI : 10.1016/bs.hna.2020.10.004
• A SIGEVO impact award for a paper arising from the COCO platform
  
  • Auger Anne
  • Hansen Nikolaus
  ACM SIGEVOlution, Association for Computing Machinery (ACM), 2021, 13 (4), pp.1-11. (10.1145/3447929.3447930)
  DOI : 10.1145/3447929.3447930
• High order homogenization of the Stokes system in a periodic porous medium
  
  • Feppon Florian
  , 2021. We derive high order homogenized models for the incompressible Stokes system in a cubic domain filled with periodic obstacles. These models have the potential to unify the three classical limit problems (namely the ``unchanged' Stokes system, the Brinkman model, and the Darcy's law) corresponding to various asymptotic regimes of the ratio $\eta\equiv a_{\epsilon}/\epsilon$ between the radius $a_{\epsilon}$ of the holes and the size $\epsilon$ of the periodic cell. What is more, a novel, rather surprising feature of our higher order effective equations is the occurrence of odd order differential operators when the obstacles are not symmetric. Our derivation relies on the method of two-scale power series expansions and on the existence of a ``criminal' ansatz, which allows to reconstruct the oscillating velocity and pressure $(\u_{\epsilon},p_{\epsilon})$ as a linear combination of the derivatives of their formal average $(\u_{\epsilon}^{*},p_{\epsilon}^{*})$ weighted by suitable corrector tensors. The formal average $(\u_\epsilon^{*},p_{\epsilon}^{*})$ is itself the solution to a formal, infinite order homogenized equation, whose truncation at any finite order is in general ill-posed. Inspired by the variational truncation method of \cite{smyshlyaev2000rigorous,cherednichenko2016full}, we derive, for any $K\in\N$, a well-posed model of order $2K+2$ which yields approximations of the original solutions with an error of order $O(\epsilon^{K+3})$ in the $L^{2}$ norm. Furthermore, the error improves up to the order $O(\epsilon^{2K+4})$ if a slight modification of this model remains well-posed. Finally, we find asymptotics of all homogenized tensors in the low volume fraction limit $\eta\rightarrow 0$ and in dimension $d\>3$. This allows us to obtain that our effective equations converge coefficient-wise to either of the Brinkman or Darcy regimes which arise when $\eta$ is respectively equivalent, or greater than the critical scaling $\eta_{\mathrm{crit}}\sim\epsilon^{2/(d-2)}$
• Federated stochastic control of numerous heterogeneous energy storage systems
  
  • Gobet Emmanuel
  • Grangereau Maxime
  , 2021. We propose a stochastic control problem to control cooperatively Thermostatically Controlled Loads (TCLs) to promote power balance in electricity networks. We develop a method to solve this stochastic control problem with a decentralized architecture, in order to respect privacy of individual users and to reduce both the telecommunications and the computational burden compared to the setting of an omniscient central planner. This paradigm is called federated learning in the machine learning community, see [YFY20], therefore we refer to this problem as a federated stochastic control problem. The optimality conditions are expressed in the form of a high-dimensional Forward-Backward Stochastic Differential Equation (FBSDE), which is decomposed into smaller FBSDEs modeling the optimal behaviors of the aggregate population of TCLs of individual agents. In particular, we show that these FBSDEs fully characterize the Nash equilibrium of a stochastic Stackelberg differential game. In this game, a coordinator (the leader) aims at controlling the aggregate behavior of the population, by sending appropriate signals, and agents (the followers) respond to this signal by optimizing their storage system locally. A mean-field-type approximation is proposed to circumvent telecommunication constraints and privacy issues. Convergence results and error bounds are obtained for this approximation depending on the size of the population of TCLs. A numerical illustration is provided to show the interest of the control scheme and to exhibit the convergence of the approximation. An implementation which answers practical industrial challenges to deploy such a scheme is presented and discussed.
• Bayesian Inference of Model Error in Imprecise Models
  
  • Leoni Nicolas
  • Congedo Pietro Marco
  • Le Maitre Olivier
  • Rodio Maria-Giovanna
  , 2021. Modern science makes use of computer models to reproduce and predict complex physical systems. Every model involves parameters, which can be measured experimentally (e.g., mass of a solid), or not (e.g., coefficients in the k − ε turbulence model). The latter parameters can be inferred from experimental data, through a procedure called calibration of the computer model. However, some models may not be able to represent reality accurately, due to their limited structure : this is the definition of model error. The "best value" of the parameters of a model is traditionnally defined as the best fit to the data. It depends on the experiment, the quantities of interest considered, and also on the supposed underlying statistical structure of the error. Bayesian methods allow the calibration of the model by taking into account its error. The fit to the data is balanced with the complexity of the model, following Occam's principle. Kennedy and O'Hagan's innovative method [1] to represent model error with a Gaussian process is a reference in this field. Recently, Tuo and Wu [3] proposed a frequentist addition to this method, to deal with the identifiability problem between model error and calibration error. Plumlee [2] applied the method to simple situations and demonstrated the potential of the approach. In this work, we compare Kennedy and O'Hagan's method with its frequentist version, which involves an optimization problem, on several numerical examples with varying degrees of model error. The calibration provides estimates of the model parameters and model predictions, while also inferring model error within observed and not observed parts of the experimental design space. The case of non-linear costly computer models is also considered, and we propose a new algorithm to reduce the numerical complexity of Bayesian calibration techniques.
• Classification and feature selection using a primal-dual method and projection on structured constraints
  
  • Barlaud Michel
  • Chambolle Antonin
  • Caillau Jean-Baptiste
  , 2021, pp.6538-6545. This paper concerns feature selection using supervised classification on high dimensional datasets. The classical approach is to project data onto a low dimensional space and classify by minimizing an appropriate quadratic cost. We first introduced a matrix of centers in the definition of this cost. Moreover, as quadratic costs are not robust to outliers, we propose instead to use an 1 cost (or Huber loss to mitigate overfitting issues). While control on sparsity is commonly obtained by adding an 1 constraint on the vectorized matrix of weights used for projecting the data, we propose to enforce structured sparsity. To this end we used constraints that take into account the matrix structure of the data, based either on the nuclear norm, on the 2,1 norm, or on the 1,2 norm for which we provide a new projection algorithm. We optimize simultaneously the projection matrix and the matrix of centers with a new tailored constrained primaldual method. The primal-dual framework is general enough to encompass the various robust losses and structured constraints we use, and allows a convergence analysis. We demonstrate the effectiveness of this approach on three biological datasets. Our primal-dual method with robust losses, adaptive centers and structured constraints does significantly better than classical methods, both in terms of accuracy and computational time. (10.1109/ICPR48806.2021.9412873)
  DOI : 10.1109/ICPR48806.2021.9412873
• Analyses de modèles et de mécanismes incitatifs pour la régulation financière et le suivi des populations
  
  • Mastrolia Thibaut
  , 2021.
• SIRUS: Stable and Interpretable RUle Set for Classification
  
  • Bénard Clément
  • Biau Gérard
  • Da Veiga Sébastien
  • Scornet Erwan
  Electronic Journal of Statistics, Shaker Heights, OH : Institute of Mathematical Statistics, 2021, 15 (1), pp.427 - 505. State-of-the-art learning algorithms, such as random forests or neural networks, are often qualified as "black-boxes" because of the high number and complexity of operations involved in their prediction mechanism. This lack of interpretability is a strong limitation for applications involving critical decisions, typically the analysis of production processes in the manufacturing industry. In such critical contexts, models have to be interpretable, i.e., simple, stable, and predictive. To address this issue, we design SIRUS (Stable and Interpretable RUle Set), a new classification algorithm based on random forests, which takes the form of a short list of rules. While simple models are usually unstable with respect to data perturbation, SIRUS achieves a remarkable stability improvement over cutting-edge methods. Furthermore, SIRUS inherits a predictive accuracy close to random forests, combined with the simplicity of decision trees. These properties are assessed both from a theoretical and empirical point of view, through extensive numerical experiments based on our R/C++ software implementation sirus available from CRAN. (10.1214/20-EJS1792)
  DOI : 10.1214/20-EJS1792
• Stochastic homogenization of the Landau-Lifshitz-Gilbert equation
  
  • Alouges François
  • de Bouard Anne
  • Merlet Benoît
  • Nicolas Léa
  Stochastics and Partial Differential Equations: Analysis and Computations, Springer US, 2021, 9 (4), pp.789–818. Following the ideas of V. V. Zhikov and A. L. Pyatnitski, and more precisely the stochastic two-scale convergence, this paper establishes a homogenization theorem in a stochastic setting for two nonlinear equations : the equation of harmonic maps into the sphere and the Landau-Lifschitz equation. These equations have strong nonlinear features, in particular, in general their solutions are not unique. (10.1007/s40072-020-00185-4)
  DOI : 10.1007/s40072-020-00185-4
• Algorithmic market making for options
  
  • Baldacci Bastien
  • Bergault Philippe
  • Guéant Olivier
  Quantitative Finance, Taylor & Francis (Routledge), 2021, 21 (1), pp.85-97. In this article, we tackle the problem of a market maker in charge of a book of options on a single liquid underlying asset. By using an approximation of the portfolio in terms of its vega, we show that the seemingly high-dimensional stochastic optimal control problem of an option market maker is in fact tractable. More precisely, when volatility is modeled using a classical stochastic volatility model—e.g. the Heston model—the problem faced by an option market maker is characterized by a low-dimensional functional equation that can be solved numerically using a Euler scheme along with interpolation techniques, even for large portfolios. In order to illustrate our findings, numerical examples are provided. (10.1080/14697688.2020.1766099)
  DOI : 10.1080/14697688.2020.1766099
• 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
• Practical computation of the diffusion MRI signal based on Laplace eigenfunctions: permeable interfaces
  
  • Agdestein Syver Døving
  • Tran Try Nguyen
  • Li Jing‐rebecca
  NMR in Biomedicine, Wiley, 2021. (10.1002/nbm.4646)
  DOI : 10.1002/nbm.4646
• 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
• Some EM-type algorithms for incomplete data model building
  
  • Lavielle Marc
  , 2021. We propose an extension of the EM algorithm and its stochastic versions for the construction of incomplete data models when the selected model minimizes a penalized likelihood criterion. This optimization problem is particularly challenging in the context of incomplete data, even when the model is relatively simple. However, by completing the data, the E-step of the algorithm allows us to simplify this problem of complete model selection into a classical problem of complete model selection that does not pose any major difficulties. We then show that the criterion to be minimized decreases with each iteration of the algorithm. Examples of the use of these algorithms are presented for the identification of regression mixture models and the construction of nonlinear mixed-effects models.
• EXISTENCE, UNIQUENESS AND REGULARITY FOR THE STOCHASTIC ERICKSEN-LESLIE EQUATION
  
  • de Bouard Anne
  • Hocquet Antoine
  • Prohl Andreas
  Nonlinearity, IOP Publishing, 2021. We investigate existence and uniqueness for the liquid crystal flow driven by colored noise on the two-dimensional torus. After giving a natural uniqueness criterion, we prove local solvability in L p-based spaces, for every p > 2. Thanks to a bootstrap principle together with a Gyöngy-Krylov-type compactness argument, this will ultimately lead us to prove the existence of a particular class of global solutions which are partially regular, strong in the probabilistic sense, and taking values in the "critical space" L 2 × H 1 . (10.1088/1361-6544/ac022e)
  DOI : 10.1088/1361-6544/ac022e
• 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
• Optimal control techniques based on infection age for the study of the COVID-19 epidemic
  
  • Bonnans Joseph Frédéric
  • Gianatti Justina
  Mathematical Modelling of Natural Phenomena, EDP Sciences, 2021. We propose a model for the COVID-19 epidemic where the population is partitioned into classes corresponding to ages (that remain constant during the epidemic). The main feature is to take into account the infection age of the infected population. This allows to better simulate the infection propagation that crucially depend on the infection age. We discuss how to compute the coefficients from data available in the future, and introduce a confinement variable as control. The cost function is a compromise between confinement cost, hospitalization peak and the death toll. Our numerical experiments allow to evaluate the interest of confinement varying with age classes. (10.1051/mmnp/2020035)
  DOI : 10.1051/mmnp/2020035
• Numerical simulation of rigid particles in Stokes flow: lubrication correction for general shapes of particles
  
  • Lefebvre-Lepot Aline
  • Nabet Flore
  Mathematical Modelling of Natural Phenomena, EDP Sciences, 2021, 16, pp.45. We address the problem of numerical simulation of suspensions of rigid particles in a Stokes flow. We focus on the inclusion of the singular short range interaction effects (lubrication effects) in the simulations when the particles come close one to another. The problem is solved without introducing new hypothesis nor model. As in Lefebvre-Lepot et al. [J. Fluid Mech. 769 (2015) 369–386], the key idea is to decompose the velocity and pressure flows in a sum of a singular and a regular part. In this article, the singular part is computed using an explicit asymptotic expansion of the solution when the distance goes to zero. This expansion is similar to the asymptotic expansion proposed in Hillairet and Kelai [Asymptotic Anal. 95 (2015) 187–241] but is more appropriate for numerical simulations of suspensions. It can be computed for any locally convex (that is the particles have to be convex close to the contact point) and regular shape of particles. Using Hillairet and Kelai [Asymptotic Anal. 95 (2015) 187–241] as an intermediate result, we prove that the remaining part is regular in the sense that it is bounded independently of the distance. As a consequence, only a small number of degrees of freedom are necessary to obtain accurate results. The method is tested in dimension 2 for clusters of two or three aligned particles with general rigid velocities. We show that, as expected, the convergence is independent of the distance. (10.1051/mmnp/2021037)
  DOI : 10.1051/mmnp/2021037
• 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