Publications

2022

• Linear-sized independent sets in random cographs and increasing subsequences in separable permutations
  
  • Bassino Frédérique
  • Bouvel Mathilde
  • Drmota Michael
  • Feray Valentin
  • Gerin Lucas
  • Maazoun Mickaël
  • Pierrot Adeline
  Combinatorial Theory, eScholarship, 2022, 2 (3), pp.23340676. This paper is interested in independent sets (or equivalently, cliques) in uniform random cographs. We also study their permutation analogs, namely, increasing subsequences in uniform random separable permutations. First, we prove that, with high probability as $n$ gets large, the largest independent set in a uniform random cograph with $n$ vertices has size $o(n)$. This answers a question of Kang, McDiarmid, Reed and Scott. Using the connection between graphs and permutations via inversion graphs, we also give a similar result for the longest increasing subsequence in separable permutations. These results are proved using the self-similarity of the Brownian limits of random cographs and random separable permutations, and actually apply more generally to all families of graphs and permutations with the same limit. Second, and unexpectedly given the above results, we show that for $\beta >0$ sufficiently small, the expected number of independent sets of size $\beta n$ in a uniform random cograph with $n$ vertices grows exponentially fast with $n$. We also prove a permutation analog of this result. This time the proofs rely on singularity analysis of the associated bivariate generating functions. (10.5070/C62359179)
  DOI : 10.5070/C62359179
• Enriched nonconforming multiscale finite element method for Stokes flows in heterogeneous media based on high-order weighting functions
  
  • Feng Qingqing
  • Allaire Grégoire
  • Omnes Pascal
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2022, 20 (1). This paper addresses an enriched nonconforming Multiscale Finite Element Method (MsFEM) to solve viscous incompressible flow problems in genuine heterogeneous or porous media. In the work of [B. P. Muljadi, J. Narski, A. Lozinski, and P. Degond, Multiscale Modeling \& Simulation 2015 13:4, 1146-1172] and [G. Jankowiak and A. Lozinski, arXiv:1802.04389 [math.NA], 2018], a nonconforming MsFEM has been first developed for Stokes problems in such media. Based on these works, we propose an innovative enriched nonconforming MsFEM where the approximation space of both velocity and pressure are enriched by weighting functions which are defined by polynomials of higher-degree. Numerical experiments show that this enriched nonconforming MsFEM improves significantly the accuracy of the nonconforming MsFEMs. Theoretically, this method provides a general framework which allows to find a good compromise between the accuracy of the method and the computing costs, by varying the degrees of polynomials. (10.1137/21M141926X)
  DOI : 10.1137/21M141926X
• On scaling limits of random trees and maps with a prescribed degree sequence
  
  • Marzouk Cyril
  Annales Henri Lebesgue, UFR de Mathématiques - IRMAR, 2022, 5, pp.317-386. (10.5802/ahl.125)
  DOI : 10.5802/ahl.125
• Generating natural adversarial Remote Sensing Images
  
  • Burnel Jean-Christophe
  • Fatras Kilian
  • Flamary Rémi
  • Courty Nicolas
  IEEE Transactions on Geoscience and Remote Sensing, Institute of Electrical and Electronics Engineers, 2022, 60, pp.1-14. Over the last years, Remote Sensing Images (RSI) analysis have started resorting to using deep neural networks to solve most of the commonly faced problems, such as detection, land cover classification or segmentation. As far as critical decision making can be based upon the results of RSI analysis, it is important to clearly identify and understand potential security threats occurring in those machine learning algorithms. Notably, it has recently been found that neural networks are particularly sensitive to carefully designed attacks, generally crafted given the full knowledge of the considered deep network. In this paper, we consider the more realistic but challenging case where one wants to generate such attacks in the case of a black-box neural network. In this case, only the prediction score of the network is accessible, given a specific input. Examples that lure away the network's prediction, while being perceptually similar to real images, are called natural or unrestricted adversarial examples. We present an original method to generate such examples, based on a variant of the Wasserstein Generative Adversarial Network. We demonstrate its effectiveness on natural adversarial hyper-spectral image generation and image modification for fooling a state-of-the-art detector. Among others, we also conduct a perceptual evaluation with human annotators to better assess the effectiveness of the proposed method. (10.1109/TGRS.2021.3110601)
  DOI : 10.1109/TGRS.2021.3110601
• Multiply Accelerated Value Iteration for Non-Symmetric Affine Fixed Point Problems and application to Markov Decision Processes
  
  • Akian Marianne
  • Gaubert Stéphane
  • Qu Zheng
  • Saadi Omar
  SIAM Journal on Matrix Analysis and Applications, Society for Industrial and Applied Mathematics, 2022, 43 (1). We analyze a modified version of Nesterov accelerated gradient algorithm, which applies to affine fixed point problems with non self-adjoint matrices, such as the ones appearing in the theory of Markov decision processes with discounted or mean payoff criteria. We characterize the spectra of matrices for which this algorithm does converge with an accelerated asymptotic rate. We also introduce a $d$th-order algorithm, and show that it yields a multiply accelerated rate under more demanding conditions on the spectrum. We subsequently apply these methods to develop accelerated schemes for non-linear fixed point problems arising from Markov decision processes. This is illustrated by numerical experiments. (10.1137/20M1367192)
  DOI : 10.1137/20M1367192
• DAMPING OPTIMIZATION OF VISCOELASTIC CANTILEVER BEAMS AND PLATES UNDER FREE VIBRATION
  
  • Joubert A
  • Allaire G
  • Amstutz S
  • Diani J
  Computers & Structures, Elsevier, 2022. The goal of this work is to significantly enhance the damping of linear viscoelastic structures under free vibration by relying on optimal design. Homogeneous cantilever slender beams and plates satisfying, respectively, the Euler-Bernoulli and Kirchhoff-Love assumptions are considered. A sizing optimization of the beam or plate thickness is proposed, as well as a coupled optimization of the thickness and geometry of the plate applying Hadamard's boundary variation method. The isotropic linear viscoelastic material is modeled by a classical generalized Maxwell model, well suited for polymers. Gradients of the objective functions are computed by an adjoint approach. Optimization is performed by a projected gradient algorithm and the mechanical models are evaluated by the finite element method. Numerical tests indicate that the optimal designs, as well as their damping properties, strongly depend on the material parameters.
• Multidimensional inverse scattering for the Schrödinger equation
  
  • Novikov Roman
  , 2022, 385, pp.75-98. We give a short review of old and recent results on the multidimensional inverse scattering problem for the Schrödinger equation. A special attention is paid to efficient reconstructions of the potential from scattering data which can be measured in practice. In this connection our considerations include reconstructions from non-overdetermined monochromatic scattering data and formulas for phase recovering from phaseless scattering data. Potential applications include phaseless inverse X-ray scattering, acoustic tomography and tomographies using elementary particles. This paper is based, in particular, on results going back to M. Born (1926), L. Faddeev (1956, 1974), S. Manakov (1981), R.Beals, R. Coifman (1985), G. Henkin, R. Novikov (1987), and on more recent results of R. Novikov ( 1998 - 2019), A. Agaltsov, T. Hohage, R. Novikov (2019). This paper is an extended version of the talk given at the 12th ISAAC Congress, Aveiro, Portugal, 29 July - 2 August, 2019. (10.1007/978-3-030-97127-4_3)
  DOI : 10.1007/978-3-030-97127-4_3
• QLSD: Quantised Langevin Stochastic Dynamics for Bayesian Federated Learning
  
  • Vono Maxime
  • Plassier Vincent
  • Durmus Alain
  • Dieuleveut Aymeric
  • Moulines Eric
  , 2022. The objective of Federated Learning (FL) is to perform statistical inference for data which are decentralised and stored locally on networked clients. FL raises many constraints which include privacy and data ownership, communication overhead, statistical heterogeneity, and partial client participation. In this paper, we address these problems in the framework of the Bayesian paradigm. To this end, we propose a novel federated Markov Chain Monte Carlo algorithm, referred to as Quantised Langevin Stochastic Dynamics which may be seen as an extension to the FL setting of Stochastic Gradient Langevin Dynamics, which handles the communication bottleneck using gradient compression. To improve performance, we then introduce variance reduction techniques, which lead to two improved versions coined QLSD and QLSD ++. We give both non-asymptotic and asymptotic convergence guarantees for the proposed algorithms. We illustrate their performances using various Bayesian Federated Learning benchmarks.
• Using hospital data for monitoring the dynamics of COVID 19 in France
  
  • Lavielle Marc
  Journal of Data Science, Statistics, and Visualisation, International Association for Statistical Computing, 2022. The aim of this article is to show how daily hospital data can be used to track the evolution of the COVID-19 epidemic in France. A piecewise defined dynamic model allows a very good fit of the available data on hospital admissions, deaths and discharges. The change-points detected correspond to moments when the dynamics of the epidemic changed abruptly. Although the proposed model is relatively simple, it can serve several purposes: It is an analytical tool to better understand what has happened so far by relating observed changes to changes in health policy or the evolution of the virus. It is also a surveillance tool that can be used effectively to warn of a resurgence of epidemic activity, and finally a short-term forecasting tool if conditions remain unchanged. The model, data and fits are implemented in an interactive web application. (10.52933/jdssv.v2i7.48)
  DOI : 10.52933/jdssv.v2i7.48
• Coupled topology optimization of structure and connections for bolted mechanical systems
  
  • Rakotondrainibe Lalaina
  • Desai Jeet
  • Orval Patrick
  • Allaire Grégoire
  European Journal of Mechanics - A/Solids, Elsevier, 2022. This work introduces a new coupled topology optimization approach for a structural assembly. Considering several parts connected by bolts, the shape and topology of potentially each part, as well as the position and number of bolts are simultaneously optimized. The main ingredients of our optimization approach are the level-set method for structural optimization, a new notion of topological derivative of an idealized model of bolt in order to decide where it is advantageous to add a new bolt, coupled with a parametric gradient-based algorithm for its position optimization. Both idealized bolt and its topological derivative handle prestressed state complexity. Several 3d numerical test cases are performed to demonstrate the efficiency of the proposed strategy for mass minimization, considering Von Mises and fatigue constraints for the bolts and compliance constraint for the structure. In particular, a simplified but industrially representative example of an accessories bracket for car engines demonstrates significant benefits. Optimizing both the structure and its connections reduces the mass by 24% compared to classical "structure-only" optimization.
• Leveraging Local Variation in Data: Sampling and Weighting Schemes for Supervised Deep Learning
  
  • Novello Paul
  • Poëtte Gaël
  • Lugato David
  • Congedo Pietro Marco
  Journal of Machine Learning for Modeling and Computing, Begell House, 2022, 3 (1). In the context of supervised learning of a function by a neural network, we claim and empirically verify that the neural network yields better results when the distribution of the data set focuses on regions where the function to learn is steep. We first traduce this assumption in a mathematically workable way using Taylor expansion and emphasize a new training distribution based on the derivatives of the function to learn. Then, theoretical derivations allow construction of a methodology that we call variance based samples weighting (VBSW). VBSW uses labels' local variance to weight the training points. This methodology is general, scalable, cost-effective, and significantly increases the performances of a large class of neural networks for various classification and regression tasks on image, text, and multivariate data. We highlight its benefits with experiments involving neural networks from linear models to ResNet and BERT. (10.1615/JMachLearnModelComput.2022041819)
  DOI : 10.1615/JMachLearnModelComput.2022041819
• Time reversal of Markov processes with jumps under a finite entropy condition
  
  • Conforti Giovanni
  • Léonard Christian
  Stochastic Processes and their Applications, Elsevier, 2022, 144, pp.85-124. Motivated by entropic optimal transport, time reversal of Markov jump processes in Rn is investigated. Relying on an abstract integration by parts formula for the carré du champ of a Markov process recently obtained by Cattiaux, Gentil and the auhors, and using an entropic improvement strategy discovered by Föllmer in the eighties, we compute the semimartingale characteristics of the time reversed process for a wide class of jump processes with possibly unbounded variation sample paths and singular intensities of jump.
• Spectral inequality for Schrödinger's equation with multipoint potential
  
  • Grinevich Piotr
  • Novikov Roman
  Russian Mathematical Surveys, Turpion, 2022, 77 (6), pp.1021–1028. Schrödinger's equation with potential that is a sum of a regular function and a finite set of point scatterers of Bethe–Peierls type is under consideration. For this equation the spectral problem with homogeneous linear boundary conditions is considered, which covers the Dirichlet, Neumann, and Robin cases. It is shown that when the energy E is an eigenvalue with multiplicity m, it remains an eigenvalue with multiplicity at least m−n after adding n<m point scatterers. As a consequence, because for the zero potential all values of the energy are transmission eigenvalues with infinite multiplicity, this property also holds for n-point potentials, as discovered originally in a recent paper by the authors. (10.4213/rm10080e)
  DOI : 10.4213/rm10080e
• Provably Convergent Working Set Algorithm for Non-Convex Regularized Regression
  
  • Rakotomamonjy Alain
  • Flamary Rémi
  • Gasso Gilles
  • Salmon Joseph
  , 2022. Owing to their statistical properties, non-convex sparse regularizers have attracted much interest for estimating a sparse linear model from high dimensional data. Given that the solution is sparse, for accelerating convergence, a working set strategy addresses the optimization problem through an iterative algorithm by incre-menting the number of variables to optimize until the identification of the solution support. While those methods have been well-studied and theoretically supported for convex regularizers, this paper proposes a working set algorithm for non-convex sparse regularizers with convergence guarantees. The algorithm, named FireWorks, is based on a non-convex reformulation of a recent primal-dual approach and leverages on the geometry of the residuals. Our theoretical guarantees derive from a lower bound of the objective function decrease between two inner solver iterations and shows the convergence to a stationary point of the full problem. More importantly, we also show that convergence is preserved even when the inner solver is inexact, under sufficient decay of the error across iterations. Our experimental results demonstrate high computational gain when using our working set strategy compared to the full problem solver for both block-coordinate descent or a proximal gradient solver.
• A consistent approximation of the total perimeter functional for topology optimization algorithms
  
  • Amstutz Samuel
  • Dapogny Charles
  • Ferrer Alex
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2022, 28, pp.18:1-71. This article revolves around the total perimeter functional, one particular version of the perimeter of a shape Ω contained in a fixed computational domain D measuring the total area of its boundary ∂Ω, as opposed to its relative perimeter, which only takes into account the regions of ∂Ω strictly inside D. We construct and analyze approximate versions of the total perimeter which make sense for general “density functions” u, as generalized characteristic functions of shapes. Their use in the context of density-based topology optimization is particularly convenient insofar as they do not involve the gradient of the optimized function u. Two different constructions are proposed: while the first one involves the convolution of the function u with a smooth mollifier, the second one is based on the resolution of an elliptic boundary-value problem featuring Robin boundary conditions. The “consistency” of these approximations with the original notion of total perimeter is appraised from various points of view. At first, we prove the pointwise convergence of our approximate functionals, then the convergence of their derivatives, as the level of smoothing tends to 0, when the considered density function u is the characteristic function of a “regular enough” shape Ω ⊂ D. Then, we focus on the Γ-convergence of the second type of approximate total perimeter functional, that based on elliptic regularization. Several numerical examples are eventually presented in two and three space dimensions to validate our theoretical findings and demonstrate the efficiency of the proposed functionals in the context of structural optimization. (10.1051/cocv/2022005)
  DOI : 10.1051/cocv/2022005
• TIME DEPENDENT SCANNING PATH OPTIMIZATION FOR THE POWDER BED FUSION ADDITIVE MANUFACTURING PROCESS
  
  • Boissier Mathilde
  • Allaire G
  • Tournier Christophe
  Computer-Aided Design, Elsevier, 2022, 142, pp.103122. In this paper, scanning paths optimization for the powder bed fusion additive manufacturing process is investigated. The path design is a key factor of the manufacturing time and for the control of residual stresses arising during the building, since it directly impacts the temperature distribution. In the literature, the scanning paths proposed are mainly based on existing patterns, the relevance of which is not related to the part to build. In this work, we propose an optimization algorithm to determine the scanning path without a priori restrictions. Taking into account the time dependence of the source, the manufacturing time is minimized under two constraints: melting the required structure and avoiding any overheating causing thermally induced residual stresses. The results illustrate how crucial the part's shape and topology is in the path quality and point out promising leads to define path and part design constraints. (10.1016/j.cad.2021.103122)
  DOI : 10.1016/j.cad.2021.103122
• Anytime Performance Assessment in Blackbox Optimization Benchmarking
  
  • Hansen Nikolaus
  • Auger Anne
  • Brockhoff Dimo
  • Tusar Tea
  IEEE Transactions on Evolutionary Computation, Institute of Electrical and Electronics Engineers, 2022, 26 (6), pp.1293--1305. We present concepts and recipes for the anytime performance assessment when benchmarking optimization algorithms in a blackbox scenario. We consider runtime-oftentimes measured in number of blackbox evaluations needed to reach a target quality-to be a universally measurable cost for solving a problem. Starting from the graph that depicts the solution quality versus runtime, we argue that runtime is the only performance measure with a generic, meaningful, and quantitative interpretation. Hence, our assessment is solely based on runtime measurements. We discuss proper choices for solution quality indicators in single-and multiobjective optimization, as well as in the presence of noise and constraints. We also discuss the choice of the target values, budget-based targets, and the aggregation of runtimes by using simulated restarts, averages, and empirical cumulative distributions which generalize convergence graphs of single runs. The presented performance assessment is to a large extent implemented in the comparing continuous optimizers (COCO) platform freely available at https://github.com/numbbo/coco. (10.1109/TEVC.2022.3210897)
  DOI : 10.1109/TEVC.2022.3210897
• Surrogate-Assisted Bounding-Box approach applied to constrained multi-objective optimisation under uncertainty
  
  • Rivier Mickael
  • Congedo Pietro Marco
  Reliability Engineering and System Safety, Elsevier, 2022, 217, pp.108039. This paper is devoted to tackling constrained multi-objective optimisation under uncertainty problems. A Surrogate-Assisted Bounding-Box approach (SABBa) is formulated here to deal with approximated robustness and reliability measures, which can be adaptively refined. A Bounding-Box is defined as a multi-dimensional product of intervals, centred on the estimated objectives and constraints, that contains the true underlying values. The accuracy of these estimations can be tuned throughout the optimisation so as to reach high levels only on promising designs, which allows quick convergence toward the optimal area. In SABBa, this approach is supplemented with a Surrogate-Assisting (SA) strategy, which permits to further reduce the overall computational cost. The adaptive refinement within the Bounding-Box approach is guided by the computation of the Pareto Optimal Probability (POP) of each box. We first assess the proposed method on several analytical uncertainty-based optimisation test-cases with respect to an a priori metamodel approach in terms of a probabilistic modified Hausdorff distance to the true Pareto optimal set. The method is then applied to three engineering applications: the design of two-bar truss in structural mechanics, the shape optimisation of an Organic Rankine Cycle turbine blade and the design of a thermal protection system for atmospheric reentry. (10.1016/j.ress.2021.108039)
  DOI : 10.1016/j.ress.2021.108039
• Assessment of a non-conservative Residual Distribution scheme for solving a four-equation two-phase system with phase transition
  
  • Bacigaluppi Paola
  • Carlier Julien
  • Pelanti Marica
  • Congedo Pietro Marco
  • Abgrall Rémi
  Journal of Scientific Computing, Springer Verlag, 2022, 90 (1). This work focuses on a four-equation model for simulating two-phase mixtures with phase transition. The main assumption consists in a homogeneous temperature, pressure and velocity fields between the two phases. In particular, we tackle the study of time dependent problems with strong discontinuities and phase transition. This work presents the extension of a non-conservative residual distribution scheme to solve a four-equation two-phase system with phase transition. This non-conservative formulation allows avoiding the classical oscillations obtained by many approaches, that might appear for the pressure profile across contact discontinuities. The proposed method relies on a Finite Volume based Residual Distribution scheme which is designed for an explicit second-order time stepping. We test the non-conservative Residual Distribution scheme on several benchmark problems and assess the results via a cross-validation with the approximated solution obtained via a conservative approach, based on an HLLC solver. Furthermore, we check both methods for mesh convergence and show the effective robustness on very severe test cases, that involve both problems with and without phase transition. (10.1007/s10915-021-01706-6)
  DOI : 10.1007/s10915-021-01706-6
• Risk-Averse Stochastic Programming vs. Adaptive Robust Optimization: A Virtual Power Plant Application
  
  • Lima Ricardo
  • Conejo Antonio
  • Giraldi Loïc
  • Le Maitre Olivier
  • Hoteit Ibrahim
  • Knio Omar
  INFORMS Journal on Computing, Institute for Operations Research and the Management Sciences (INFORMS), 2022. This paper compares risk-averse optimization methods to address the self-scheduling and market involvement of a virtual power plant (VPP). The decision-making problem of the VPP involves uncertainty in the wind speed and electricity price forecast. We focus on two methods: risk-averse two-stage stochastic programming (SP) and two-stage adaptive robust optimization (ARO). We investigate both methods concerning formulations, uncertainty and risk, decomposition algorithms, and their computational performance. To quantify the risk in SP, we use the conditional value at risk (CVaR) because it can resemble a worst-case measure, which naturally links to ARO. We use two efficient implementations of the decomposition algorithms for SP and ARO; we assess (1) the operational results regarding first-stage decision variables, estimate of expected profit, and estimate of the CVaR of the profit and (2) their performance taking into consideration different sample sizes and risk management parameters. The results show that similar first-stage solutions are obtained depending on the risk parameterizations used in each formulation. Computationally, we identified three cases: (1) SP with a sample of 500 elements is competitive with ARO; (2) SP performance degrades comparing to the first case and ARO fails to converge in four out of five risk parameters; (3) SP fails to converge, whereas ARO converges in three out of five risk parameters. Overall, these performance cases depend on the combined effect of deterministic and uncertain data and risk parameters. Summary of Contribution: The work presented in this manuscript is at the intersection of operations research and computer science, which are intrinsically related with the scope and mission of IJOC. From the operations research perspective, two methodologies for optimization under uncertainty are studied: risk-averse stochastic programming and adaptive robust optimization. These methodologies are illustrated using an energy scheduling problem. The study includes a comparison from the point of view of uncertainty modeling, formulations, decomposition methods, and analysis of solutions. From the computer science perspective, a careful implementation of decomposition methods using parallelization techniques and a sample average approximation methodology was done . A detailed comparison of the computational performance of both methods is performed. Finally, the conclusions allow establishing links between two alternative methodologies in operations research: stochastic programming and robust optimization. (10.1287/ijoc.2022.1157)
  DOI : 10.1287/ijoc.2022.1157
• Quantitative particle approximation of nonlinear Fokker-Planck equations with singular kernel
  
  • Olivera Christian
  • Richard Alexandre
  • Tomasevic Milica
  Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, Scuola Normale Superiore [1974-....], 2022. In this work, we study the convergence of the empirical measure of moderately interacting particle systems with singular interaction kernels. First, we prove quantitative convergence of the time marginals of the empirical measure of particle positions towards the solution of the limiting nonlinear Fokker-Planck equation. Second, we prove the well-posedness for the McKean-Vlasov SDE involving such singular kernels and the convergence of the empirical measure towards it (propagation of chaos). Our results only require very weak regularity on the interaction kernel, which permits to treat models for which the mean field particle system is not known to be well-defined. For instance, this includes attractive kernels such as Riesz and Keller-Segel kernels in arbitrary dimension. For some of these important examples, this is the first time that a quantitative approximation of the PDE is obtained by means of a stochastic particle system. In particular, this convergence still holds (locally in time) for PDEs exhibiting a blow-up in finite time. The proofs are based on a semigroup approach combined with a fine analysis of the regularity of infinite-dimensional stochastic convolution integrals. (10.2422/2036-2145.202105_087)
  DOI : 10.2422/2036-2145.202105_087
• Design of a mode converter using thin resonant ligaments
  
  • Chesnel Lucas
  • Heleine Jérémy
  • Nazarov Sergei A
  Communications in Mathematical Sciences, International Press, 2022, 20 (2), pp.425–445. The goal of this work is to design an acoustic mode converter. More precisely, the wave number is chosen so that two modes can propagate. We explain how to construct geometries such that the energy of the modes is completely transmitted and additionally the mode 1 is converted into the mode 2 and conversely. To proceed, we work in a symmetric waveguide made of two branches connected by two thin ligaments whose lengths and positions are carefully tuned. The approach is based on asymptotic analysis for thin ligaments around resonance lengths. We also provide numerical results to illustrate the theory. (10.4310/CMS.2022.v20.n2.a6)
  DOI : 10.4310/CMS.2022.v20.n2.a6
• Optimal Electricity Demand Response Contracting with Responsiveness Incentives
  
  • Aïd René
  • Possamaï Dylan
  • Touzi Nizar
  Mathematics of Operations Research, INFORMS, 2022. Demand response programs in retail electricity markets are very popular. However, despite their success in reducing average consumption, the random responsiveness of consumers to price events makes their efficiency questionable to achieve the flexibility needed for electric systems with a large share of renewable energy. This paper aims at designing demand response contracts that allow to act on both the average consumption and its variance. The interaction between a risk-averse producer and a risk-averse consumer is modelled as a principal–agent problem, thus accounting for the moral hazard underlying demand response contracts. The producer, facing the limited flexibility of production, pays an appropriate incentive compensation to encourage the consumer to reduce his average consumption and to enhance his responsiveness. We provide a closed-form solution for the optimal contract in the linear case. We show that the optimal contract has a rebate form where the initial condition of the consumption serves as a baseline and where the consumer is charged a price for energy and a price for volatility. The first-best price for energy is a convex combination of the marginal cost and the marginal value of energy, where the weights are given by the risk-aversion ratios, and the first-best price for volatility is the risk-aversion ratio times the marginal cost of volatility. The second-best price, for energy and volatility, is a decreasing nonlinear function of time inducing decreasing effort. The price for energy is lower (respectively, higher) than the marginal cost of energy during peak-load (respectively, off-peak) periods. We illustrate the potential benefits issued from the implementation of an incentive mechanism on the responsiveness of the consumer by calibrating our model with publicly available data.
• Multidimensional fully adaptive lattice Boltzmann methods with error control based on multiresolution analysis
  
  • Bellotti Thomas
  • Gouarin Loïc
  • Graille Benjamin
  • Massot Marc
  Journal of Computational Physics, Elsevier, 2022, 471, pp.111670. Lattice-Boltzmann methods are known for their simplicity, efficiency and ease of parallelization, usually relying on uniform Cartesian meshes with a strong bond between spatial and temporal discretization. This fact complicates the crucial issue of reducing the computational cost and the memory impact by automatically coarsening the grid where a fine mesh is unnecessary, still ensuring the overall quality of the numerical solution through error control. This work provides a possible answer to this interesting question, by connecting, for the first time, the field of lattice-Boltzmann Methods (LBM) to the adaptive multiresolution (MR) approach based on wavelets. To this end, we employ a MR multi-scale transform to adapt the mesh as the solution evolves in time according to its local regularity. The collision phase is not affected due to its inherent local nature and because we do not modify the speed of the sound, contrarily to most of the LBM/Adaptive Mesh Refinement (AMR) strategies proposed in the literature, thus preserving the original structure of any LBM scheme. Besides, an original use of the MR allows the scheme to resolve the proper physics by efficiently controlling the accuracy of the transport phase. We carefully test our method to conclude on its adaptability to a wide family of existing lattice Boltzmann schemes, treating both hyperbolic and parabolic systems of equations, thus being less problem-dependent than the AMR approaches, which have a hard time guaranteeing an effective control on the error. The ability of the method to yield a very efficient compression rate and thus a computational cost reduction for solutions involving localized structures with loss of regularity is also shown, while guaranteeing a precise control on the approximation error introduced by the spatial adaptation of the grid. The numerical strategy is implemented on a specific open-source platform called SAMURAI with a dedicated data-structure relying on set algebra. (10.1016/j.jcp.2022.111670)
  DOI : 10.1016/j.jcp.2022.111670
• Non-linear boundary condition for non-ideal electrokinetic equations in porous media
  
  • Allaire Grégoire
  • Brizzi Robert
  • Labbez Christophe
  • Mikelić Andro
  Applicable Analysis, Taylor & Francis, 2022, 101 (12), pp.4203-4234. This paper studies the partial differential equation describing the charge distribution of an electrolyte in a porous medium. Realistic non-ideal effects are incorporated through the mean spherical approximation (MSA) model which takes into account finite size ions and screening effects. The main novelty is the consideration of a non-constant surface charge density on the pore walls. Indeed, a chemical equilibrium reaction is considered on the boundary to represent the dissociation of ionizable sites on the solid walls. The surface charge density is thus given as a non-linear function of the electrostatic potential. Even in the ideal case, the resulting system is a new variant of the famous Poisson-Boltzmann equation, which still has a monotone structure under quantitative assumptions on the physical parameters. In the non-ideal case, the MSA model brings in additional non-linearities which break down the monotone structure of the system. We prove existence, and sometimes uniqueness, of the solution. Some numerical experiments are performed in 2-d to compare this model with that for a constant surface charge. (10.1080/00036811.2022.2080672)
  DOI : 10.1080/00036811.2022.2080672