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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• Spontaneous Periodic Orbits in the Navier–Stokes Flow
  
  • van den Berg Jan Bouwe
  • Breden Maxime
  • Lessard Jean-Philippe
  • van Veen Lennaert
  Journal of Nonlinear Science, Springer Verlag, 2021, 31 (2), pp.41. In this paper, a general method to obtain constructive proofs of existence of periodicorbits in the forced autonomous Navier–Stokes equations on the three-torus isproposed. After introducing a zero finding problem posed on a Banach space of geometricallydecaying Fourier coefficients, a Newton–Kantorovich theorem is applied toobtain the (computer-assisted) proofs of existence. The required analytic estimates toverify the contractibility of the operator are presented in full generality and symmetriesfrom themodel are used to reduce the size of the problem to be solved. As applications,we present proofs of existence of spontaneous periodic orbits in the Navier–Stokesequations with Taylor–Green forcing. (10.1007/s00332-021-09695-4)
  DOI : 10.1007/s00332-021-09695-4
• Importance of mass and enthalpy conservation in the modeling of titania nanoparticles flame synthesis
  
  • Orlac'Ch Jean-Maxime
  • Darabiha Nasser
  • Giovangigli Vincent
  • Franzelli Benedetta
  Combustion Theory and Modelling, Taylor & Francis, 2021, 25, pp.389-412. In most simulations of fine particles in reacting flows, including sooting flames, en-thalpy exchanges between gas and particle phases and differential diffusion between the two phases are most often neglected, since the particle mass fraction is generally very small. However, when the nanoparticles mass fraction is very large representing up to 50 % of the mixture mass, the conservation of the total enthalpy and/or the total mass becomes critical. In the present paper, we investigate the impact of mass and enthalpy conservation in the modeling of titania nanoparticles synthesis in flames, classically characterized by a high conversation rate and consequently a high nanoparticles concentration. It is shown that when the nanoparticles concentration is high, neglecting the enthalpy of the particle phase may lead to almost 70 % relative error on the temperature profile and to relative errors on the main titania species mass fractions and combustion products ranging from 20 % to 100 %. It is also established that neglecting the differential diffusion of the gas phase with respect to the particle phase is also significant, with almost 15 % relative error on the TiO2 mole fraction, although the effect on combustion products is minor. (10.1080/13647830.2021.1886330)
  DOI : 10.1080/13647830.2021.1886330
• Convergence and a posteriori error analysis for energy-stable finite element approximations of degenerate parabolic equations
  
  • Cancès Clément
  • Nabet Flore
  • Vohralík Martin
  Mathematics of Computation, American Mathematical Society, 2021, 90 (328), pp.517-563. We propose a finite element scheme for numerical approximation of degenerate parabolic problems in the form of a nonlinear anisotropic Fokker-Planck equation. The scheme is energy-stable, only involves physically motivated quantities in its definition, and is able to handle general unstructured grids. Its convergence is rigorously proven thanks to compactness arguments, under very general assumptions. Although the scheme is based on Lagrange finite elements of degree 1, it is locally conservative after a local postprocess giving rise to an equilibrated flux. This also allows to derive a guaranteed a posteriori error estimate for the approximate solution. Numerical experiments are presented in order to give evidence of a very good behavior of the proposed scheme in various situations involving strong anisotropy and drift terms. (10.1090/mcom/3577)
  DOI : 10.1090/mcom/3577
• Coherent Soliton States Hidden in Phase Space and Stabilized by Gravitational Incoherent Structures
  
  • Garnier Josselin
  • Baudin Kilian
  • Fusaro Adrien
  • Picozzi Antonio
  Physical Review Letters, American Physical Society, 2021, 127 (1), pp.014101. We consider the problem of the formation of soliton states from a modulationally unstable initial condition in the framework of the Schrödinger-Poisson (or Newton-Schrödinger) equation accounting for gravitational interactions. We unveil a previously unrecognized regime: By increasing the nonlinearity, the system self-organizes into an incoherent localized structure that contains “hidden” coherent soliton states. The solitons are hidden in the sense that they are fully immersed in random wave fluctuations: The radius of the soliton is much larger than the correlation radius of the incoherent fluctuations, while its peak amplitude is of the same order of such fluctuations. Accordingly, the solitons can hardly be identified in the usual spatial or spectral domains, while their existence is clearly unveiled in the phase-space representation. Our multiscale theory based on coupled coherent-incoherent wave turbulence formalisms reveals that the hidden solitons are stabilized and trapped by the incoherent localized structure. Furthermore, hidden binary soliton systems are identified numerically and described theoretically. The regime of hidden solitons is of potential interest for self-gravitating Boson models of “fuzzy" dark matter. It also sheds new light on the quantum-to-classical correspondence with gravitational interactions. The hidden solitons can be observed in nonlocal nonlinear optics experiments through the measurement of the spatial spectrogram. (10.1103/PhysRevLett.127.014101)
  DOI : 10.1103/PhysRevLett.127.014101
• Non-convex functionals penalizing simultaneous oscillations along independent directions: rigidity estimates
  
  • Goldman Michael
  • Merlet Benoît
  Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Scuola Normale Superiore, 2021, 22 (3), pp.1473--1509. We study a family of non-convex functionals {E} on the space of measurable functions u : Ω 1 × Ω 2 ⊂ R n 1 × R n 2 → R. These functionals vanish on the non-convex subset S(Ω 1 × Ω 2) formed by functions of the form u(x 1 , x 2) = u 1 (x 1) or u(x 1 , x 2) = u 2 (x 2). We investigate under which conditions the converse implication "E(u) = 0 ⇒ u ∈ S(Ω 1 × Ω 2)" holds. In particular, we show that the answer depends strongly on the smoothness of u. We also obtain quantitative versions of this implication by proving that (at least for some parameters) E(u) controls in a strong sense the distance of u to S(Ω 1 × Ω 2). (10.2422/2036-2145.201906_006)
  DOI : 10.2422/2036-2145.201906_006
• Coupling techniques for nonlinear hyperbolic equations. II. Resonant interfaces with internal structure
  
  • Boutin Benjamin
  • Coquel Frédéric
  • Lefloch Philippe G.
  Networks and Heterogeneous Media, American Institute of Mathematical Sciences, 2021, 16 (2), pp.283-315. In the first part of this series, an augmented PDE system was introduced in order to couple two nonlinear hyperbolic equations together. This formulation allowed the authors, based on Dafermos's self-similar viscosity method, to establish the existence of self-similar solutions to the coupled Riemann problem. We continue here this analysis in the restricted case of one-dimensional scalar equations and investigate the internal structure of the interface in order to derive a selection criterion associated with the underlying regularization mechanism and, in turn, to characterize the nonconservative interface layer. In addition, we identify a new criterion that selects double-waved solutions that are also continuous at the interface. We conclude by providing some evidence that such solutions can be non-unique when dealing with non-convex flux-functions. (10.3934/nhm.2021007)
  DOI : 10.3934/nhm.2021007
• Concurrent shape optimization of the part and scanning path for additive manufacturing
  
  • Boissier Mathilde
  • Allaire Grégoire
  • Tournier Christophe
  , 2021.
• A weak solution theory for stochastic Volterra equations of convolution type
  
  • Jaber Eduardo Abi
  • Cuchiero Christa
  • Larsson Martin
  • Pulido Sergio
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2021, 31 (6), pp.2924-2952. We obtain general weak existence and stability results for stochastic convolution equations with jumps under mild regularity assumptions, allowing for non-Lipschitz coefficients and singular kernels. Our approach relies on weak convergence in $L^p$ spaces. The main tools are new a priori estimates on Sobolev--Slobodeckij norms of the solution, as well as a novel martingale problem that is equivalent to the original equation. This leads to generic approximation and stability theorems in the spirit of classical martingale problem theory. We also prove uniqueness and path regularity of solutions under additional hypotheses. To illustrate the applicability of our results, we consider scaling limits of nonlinear Hawkes processes and approximations of stochastic Volterra processes by Markovian semimartingales.
• Log-Sobolev Inequality for the Continuum Sine-Gordon Model
  
  • Bauerschmidt Roland
  • Bodineau Thierry
  Commun.Pure Appl.Math., 2021, 74 (10), pp.2064-2113. We derive a multiscale generalisation of the Bakry-Émery criterion for a measure to satisfy a log-Sobolev inequality. Our criterion relies on the control of an associated PDE well-known in renormalisation theory: the Polchinski equation. It implies the usual Bakry-Émery criterion, but we show that it remains effective for measures that are far from log-concave. Indeed, using our criterion, we prove that the massive continuum sine-Gordon model with β < 6π satisfies asymptotically optimal log-Sobolev inequalities for Glauber and Kawasaki dynamics. These dynamics can be seen as singular SPDEs recently constructed via regularity structures, but our results are independent of this theory. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. (10.1002/cpa.21926)
  DOI : 10.1002/cpa.21926