Publications

2022

• Optimizing Noisy Complex Systems Liable to Failure
  
  • Lunz Davin
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2022, 82 (1), pp.25-48. Inspired by complex systems in social and industrial contexts, we consider a family of coupled diffusion processes modeling system components, and an associated system objective. Each process is inherently noisy, driven by a controllable drift, and fails upon reaching a critical state. Interdependence is captured via the global objective and the governing dynamics (correlated noise, cascading failures). Analytical and numerical calculations reveal that the optimal strategies to steer such systems so as to maximise the objective are highly coupled, depending strongly on the state of the entire system. Strikingly, they exhibit a rich set of bifurcations, describing qualitatively different strategies throughout the parameter space. (10.1137/21M1416126)
  DOI : 10.1137/21M1416126
• 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
• Entropic turnpike estimates for the kinetic Schrödinger problem
  
  • Chiarini Alberto
  • Conforti Giovanni
  • Greco Giacomo
  • Ren Zhenjie
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2022, 27 (none). (10.1214/22-EJP850)
  DOI : 10.1214/22-EJP850
• 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
• 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.
• 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.
• A class of short-term models for the oil industry addressing speculative storage
  
  • Achdou Yves
  • Bertucci Charles
  • Lasry Jean-Michel
  • Lions Pierre Louis
  • Rostand Antoine
  • Scheinkman Jose
  Finance and Stochastics, Springer Verlag (Germany), 2022, 26 (3), pp.631-669. This is a work in progress. The aim is to propose a plausible mechanism for the short term dynamics of the oil market based on the interaction of economic agents. This is a theoretical research which by no means aim at describing all the aspects of the oil market. In particular, we use the tools and terminology of game theory, but we do not claim that this game actually exists in the real world. In parallel, we are currently studying and calibrating a long term model for the oil industry, which addresses the interactions of a monopolists with a competitive fringe of small producers. It is the object of another paper that will be available soon. The present premiminary version does not contain all the economic arguments and all the connections with our long term model. It mostly addresses the description of the model, the equations and numerical simulations focused on the oil industry short term dynamics. A more complete version will be available soon. (10.1007/s00780-022-00481-y)
  DOI : 10.1007/s00780-022-00481-y
• Reconstruction from the Fourier transform on the ball via prolate spheroidal wave functions
  
  • Isaev Mikhail
  • Novikov Roman
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2022, 163 (July), pp.318-333. We give new formulas for finding a compactly supported function v on R^d, d≥1, from its Fourier transform Fv given within the ball B_r. For the one-dimensional case, these formulas are based on the theory of prolate spheroidal wave functions (PSWF's). In multidimensions, well-known results of the Radon transform theory reduce the problem to the one-dimensional case. Related results on stability and convergence rates are also given. (10.1016/j.matpur.2022.05.008)
  DOI : 10.1016/j.matpur.2022.05.008
• Convergence of a finite-volume scheme for a heat equation with a multiplicative Lipschitz noise
  
  • Bauzet Caroline
  • Nabet Flore
  • Schmitz Kerstin
  • Zimmermann Aleksandra
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2022, 57 (2), pp.745-783. We study here the approximation by a finite-volume scheme of a heat equation forced by a Lipschitz continuous multiplicative noise in the sense of Itô. More precisely, we consider a discretization which is semi-implicit in time and a two-point flux approximation scheme (TPFA) in space. We adapt the method based on the theorem of Prokhorov to obtain a convergence in distribution result, then Skorokhod's representation theorem yields the convergence of the scheme towards a martingale solution and the Gyöngy-Krylov argument is used to prove convergence in probability of the scheme towards the unique variational solution of our parabolic problem. (10.1051/m2an/2022087)
  DOI : 10.1051/m2an/2022087
• A moment closure based on a projection on the boundary of the realizability domain: Extension and analysis
  
  • Pichard Teddy
  Kinetic and Related Models, AIMS, 2022, 15 (5), pp.793. A closure relation for moments equation in kinetic theory was recently introduced in [38], based on the study of the geometry of the set of moments. This relation was constructed from a projection of a moment vector toward the boundary of the set of moments and corresponds to approximating the underlying kinetic distribution as a sum of a chosen equilibrium distribution plus a sum of purely anisotropic Dirac distributions. The present work generalizes this construction for kinetic equations involving unbounded velocities, i.e. to the Hamburger problem, and provides a deeper analysis of the resulting moment system. Especially, we provide representation results for moment vectors along the boundary of the moment set that implies the well-definition of the model. And the resulting moment model is shown to be weakly hyperbolic with peculiar properties of hyperbolicity and entropy of two subsystems, corresponding respectively to the equilibrium and to the purely anisotropic parts of the underlying kinetic distribution. (10.3934/krm.2022014)
  DOI : 10.3934/krm.2022014
• Differentially Private Federated Learning on Heterogeneous Data
  
  • Noble Maxence
  • Bellet Aurélien
  • Dieuleveut Aymeric
  , 2022. Federated Learning (FL) is a paradigm for large-scale distributed learning which faces two key challenges: (i) training efficiently from highly heterogeneous user data, and (ii) protecting the privacy of participating users. In this work, we propose a novel FL approach (DP-SCAFFOLD) to tackle these two challenges together by incorporating Differential Privacy (DP) constraints into the popular SCAFFOLD algorithm. We focus on the challenging setting where users communicate with a "honest-but-curious" server without any trusted intermediary, which requires to ensure privacy not only towards a third party observing the final model but also towards the server itself. Using advanced results from DP theory and optimization, we establish the convergence of our algorithm for convex and non-convex objectives. Our paper clearly highlights the trade-off between utility and privacy and demonstrates the superiority of DP-SCAFFOLD over the state-ofthe-art algorithm DP-FedAvg when the number of local updates and the level of heterogeneity grows. Our numerical results confirm our analysis and show that DP-SCAFFOLD provides significant gains in practice.
• Scintillation of partially coherent light in time-varying complex media
  
  • Garnier Josselin
  • Sølna Knut
  Journal of the Optical Society of America. A Optics, Image Science, and Vision, Optical Society of America, 2022, 39 (8), pp.1309. We present a theory for wave scintillation in the situation of a time-dependent partially coherent source and a time-dependent randomly heterogeneous medium. Our objective is to understand how the scintillation index of the measured intensity depends on the source and medium parameters. We deduce from an asymptotic analysis of the random wave equation a general form of the scintillation index, and we evaluate this in various scaling regimes. The scintillation index is a fundamental quantity that is used to analyze and optimize imaging and communication schemes. Our results are useful to quantify the scintillation index under realistic propagation scenarios and to address such optimization challenges. (10.1364/JOSAA.453358)
  DOI : 10.1364/JOSAA.453358
• An ODE Method to Prove the Geometric Convergence of Adaptive Stochastic Algorithms
  
  • Akimoto Youhei
  • Auger Anne
  • Hansen Nikolaus
  Stochastic Processes and their Applications, Elsevier, 2022, 145, pp.269-307. We consider stochastic algorithms derived from methods for solving deterministic optimization problems, especially comparison-based algorithms derived from stochastic approximation algorithms with a constant step-size. We develop a methodology for proving geometric convergence of the parameter sequence {θn}n≥0 of such algorithms. We employ the ordinary differential equation (ODE) method, which relates a stochastic algorithm to its mean ODE, along with a Lyapunov-like function Ψ such that the geometric convergence of Ψ(θn) implies -- in the case of an optimization algorithm -- the geometric convergence of the expected distance between the optimum and the search point generated by the algorithm. We provide two sufficient conditions for Ψ(θn) to decrease at a geometric rate: Ψ should decrease "exponentially" along the solution to the mean ODE, and the deviation between the stochastic algorithm and the ODE solution (measured by Ψ) should be bounded by Ψ(θn) times a constant. We also provide practical conditions under which the two sufficient conditions may be verified easily without knowing the solution of the mean ODE. Our results are any-time bounds on Ψ(θn), so we can deduce not only the asymptotic upper bound on the convergence rate, but also the first hitting time of the algorithm. The main results are applied to a comparison-based stochastic algorithm with a constant step-size for optimization on continuous domains. (10.1016/j.spa.2021.12.005)
  DOI : 10.1016/j.spa.2021.12.005
• Gaussian Agency problems with memory and Linear Contracts
  
  • Abi Jaber Eduardo
  • Villeneuve Stéphane
  Finance and Stochastics, Springer Verlag (Germany), 2022. Can a principal still offer optimal dynamic contracts that are linear in end-of-period outcomes when the agent controls a process that exhibits memory? We provide a positive answer by considering a general Gaussian setting where the output dynamics are not necessarily semi-martingales or Markov processes. We introduce a rich class of principal-agent models that encompasses dynamic agency models with memory. From the mathematical point of view, we develop a methodology to deal with the possible non-Markovianity and non-semimartingality of the control problem, which can no longer be directly solved by means of the usual Hamilton-Jacobi-Bellman equation. Our main contribution is to show that, for one-dimensional models, this setting always allows for optimal linear contracts in end-of-period observable outcomes with a deterministic optimal level of effort. In higher dimension, we show that linear contracts are still optimal when the effort cost function is radial and we quantify the gap between linear contracts and optimal contracts for more general quadratic costs of efforts.
• 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
• Anisotropic and crystalline mean curvature flow of mean-convex sets
  
  • Chambolle Antonin
  • Novaga Matteo
  Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Scuola Normale Superiore, 2022, 23 (2), pp.623-643. We consider a variational scheme for the anisotropic (including crystalline) mean curvature flow of sets with strictly positive anisotropic mean curvature. We show that such condition is preserved by the scheme, and we prove the strict convergence in BV of the time-integrated perimeters of the approximating evolutions, extending a recent result of De Philippis and Laux to the anisotropic setting. We also prove uniqueness of the flat flow obtained in the limit. (10.2422/2036-2145.202005_009)
  DOI : 10.2422/2036-2145.202005_009
• 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
• 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
• 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
• 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
• 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, 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
• 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
• 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.