Publications

2015

• A remark on accelerated block coordinate descent for computing the proximity operators of a sum of convex functions.
  
  • Chambolle Antonin
  • Pock Thomas
  SMAI Journal of Computational Mathematics, Société de Mathématiques Appliquées et Industrielles (SMAI), 2015, 1, pp.29 - 54. We analyze alternating descent algorithms for minimizing the sum of a quadratic function and block separable non-smooth functions. In case the quadratic interactions between the blocks are pairwise, we show that the schemes can be accelerated, leading to improved convergence rates with respect to related accelerated parallel proximal descent. As an application we obtain very fast algorithms for computing the proximity operator of the 2D and 3D total variation. (10.5802/smai-jcm.3)
  DOI : 10.5802/smai-jcm.3
• Unveiling the Diversification Dynamics of Australasian Predaceous Diving Beetles in the Cenozoic
  
  • Toussaint Emmanuel F.A.
  • Condamine Fabien L.
  • Hawlitschek Oliver
  • Watts Chris H.
  • Porch Nick
  • Hendrich Lars
  • Balke Michael
  Systematic Biology, Oxford University Press (OUP), 2015, 64 (1), pp.3-24. During the Cenozoic, Australia experienced major climatic shifts that have had dramatic ecological consequences for the modern biota. Mesic tropical ecosystems were progressively restricted to the coasts and replaced by arid-adapted floral and faunal communities. Whilst the role of aridification has been investigated in a wide range of terrestrial lineages, the response of freshwater clades remains poorly investigated. To gain insights into the diversification processes underlying a freshwater radiation, we studied the evolutionary history of the Australasian predaceous diving beetles of the tribe Hydroporini (147 described species). We used an integrative approach including the latest methods in phylogenetics, divergence time estimation, ancestral character state reconstruction, and likelihood-based methods of diversification rate estimation. Phylogenies and dating analyses were reconstructed with molecular data from seven genes (mitochondrial and nuclear) for 117 species (plus 12 outgroups). Robust and well-resolved phylogenies indicate a late Oligocene origin of Australasian Hydroporini. Biogeographic analyses suggest an origin in the East Coast region of Australia, and a dynamic biogeographic scenario implying dispersal events. The group successfully colonized the tropical coastal regions carved by a rampant desertification, and also colonized groundwater ecosystems in Central Australia. Diversification rate analyses suggest that the ongoing aridification of Australia initiated in the Miocene contributed to a major wave of extinctions since the late Pliocene probably attributable to an increasing aridity, range contractions and seasonally disruptions resulting from Quaternary climatic changes. When comparing subterranean and epigean genera, our results show that contrasting mechanisms drove their diversification and therefore current diversity pattern. The Australasian Hydroporini radiation reflects a combination of processes that promoted both diversification, resulting from new ecological opportunities driven by initial aridification, and a subsequent loss of mesic adapted diversity due to increasing aridity. (10.1093/sysbio/syu067)
  DOI : 10.1093/sysbio/syu067
• Definable Zero-Sum Stochastic Games
  
  • Bolte Jérôme
  • Gaubert Stéphane
  • Vigeral Guillaume
  Mathematics of Operations Research, INFORMS, 2015, 40 (1), pp.171-191. Definable zero-sum stochastic games involve a finite number of states and action sets, reward and transition functions that are definable in an o-minimal structure. Prominent examples of such games are finite, semi-algebraic or globally subanalytic stochastic games. We prove that the Shapley operator of any definable stochastic game with separable transition and reward functions is definable in the same structure. Definability in the same structure does not hold systematically: we provide a counterexample of a stochastic game with semi-algebraic data yielding a non semi-algebraic but globally subanalytic Shapley operator. %Showing the definability of the Shapley operator in full generality appears thus as a complex and challenging issue. } Our definability results on Shapley operators are used to prove that any separable definable game has a uniform value; in the case of polynomially bounded structures we also provide convergence rates. Using an approximation procedure, we actually establish that general zero-sum games with separable definable transition functions have a uniform value. These results highlight the key role played by the tame structure of transition functions. As particular cases of our main results, we obtain that stochastic games with polynomial transitions, definable games with finite actions on one side, definable games with perfect information or switching controls have a uniform value. Applications to nonlinear maps arising in risk sensitive control and Perron-Frobenius theory are also given. (10.1287/moor.2014.0666)
  DOI : 10.1287/moor.2014.0666
• Mathematical Methods in Elasticity Imaging
  
  • Ammari Habib
  • Bretin Elie
  • Garnier Josselin
  • Kang Hyeonbae
  • Lee Hyundae
  • Wahab Abdul
  , 2015, pp.240 p..
• A Priori Error Estimate of a Multiscale Finite Element Method for Transport Modeling
  
  • Ouaki Franck
  • Allaire Grégoire
  • Desroziers Sylvain
  • Enchéry Guillaume
  SeMA Journal: Boletin de la Sociedad Española de Matemática Aplicada, Springer, 2015, 67 (1), pp.1-37. This work proposes an \textit{a priori} error estimate of a multiscale finite element method to solve convection-diffusion problems where both velocity and diffusion coefficient exhibit strong variations at a scale which is much smaller than the domain of resolution. In that case, classical discretization methods, used at the scale of the heterogeneities, turn out to be too costly. Our method, introduced in~\cite{ECCOMAS}, aims at solving this kind of problems on coarser grids with respect to the size of the heterogeneities by means of particular basis functions. These basis functions are defined using cell problems and are designed to reproduce the variations of the solution on an underlying fine grid. Since all cell problems are independent from each other, these problems can be solved in parallel, which makes the method very efficient when used on parallel architectures. This article focuses on the proof of an \textit{a priori} error estimate of this method.
• Configuration Tracking for Mechanical Systems by Kinematic Reduction and Fast Oscillating Controls
  
  • Barbero-Liñán M.
  • Sigalotti Mario
  , 2015.
• Controllability of spin-boson systems
  
  • Boscain Ugo
  • Mason Paolo
  • Panati Gianluca
  • Sigalotti Mario
  Journal of Mathematical Physics, American Institute of Physics (AIP), 2015, 56. In this paper we study the so-called spin-boson system, namely {a two-level system} in interaction with a distinguished mode of a quantized bosonic field. We give a brief description of the controlled Rabi and Jaynes--Cummings models and we discuss their appearance in the mathematics and physics literature. We then study the controllability of the Rabi model when the control is an external field acting on the bosonic part. Applying geometric control techniques to the Galerkin approximation and using perturbation theory to guarantee non-resonance of the spectrum of the drift operator, we prove approximate controllability of the system, for almost every value of the interaction parameter.
• Second order mean field games with degenerate diffusion and local coupling
  
  • Cardaliaguet Pierre
  • Graber J.
  • Porretta Alessio
  • Tonon Daniela
  Nonlinear Differential Equations and Applications, Springer Verlag, 2015, 22 (5), pp.1287-1317. We analyze a (possibly degenerate) second order mean field games system of partial differential equations. The distinguishing features of the model considered are (1) that it is not uniformly parabolic, including the first order case as a possibility, and (2) the coupling is a local operator on the density. As a result we look for weak, not smooth, solutions. Our main result is the existence and uniqueness of suitably defined weak solutions, which are characterized as minimizers of two optimal control problems. We also show that such solutions are stable with respect to the data, so that in particular the degenerate case can be approximated by a uniformly parabolic (viscous) perturbation.
• Hawkes Processes in Finance
  
  • Bacry Emmanuel
  • Mastromatteo Iacopo
  • Muzy Jean-François
  Market microstructure and liquidity, World scientific publishing company, 2015, 01 (01), pp.1550005. no abstract (10.1142/S2382626615500057)
  DOI : 10.1142/S2382626615500057
• Dispersal is a major driver of the latitudinal diversity gradient of Carnivora
  
  • Rolland Jonathan
  • Condamine Fabien L.
  • Beeravolu Reddy Champak
  • Jiguet Frédéric
  • Morlon Hélène
  Global Ecology and Biogeography, Wiley, 2015, 24 (9), pp.1059 - 1071. <strong>Aim</strong> Understanding the relative contribution of diversification rates (speciation and extinction) and dispersal in the formation of the latitudinal diversity gradient - the decrease in species richness with increasing latitude - is a main goal of biogeography. The mammalian order Carnivora, which comprises 286 species, displays the traditional latitudinal diversity gradient seen in almost all mammalian orders. Yet the processes driving high species richness in the tropics may be fundamentally different in this group from that in other mammalian groups. Indeed, a recent study suggested that in Carnivora, unlike in all other major mammalian orders, net diversification rates are not higher in the tropics than in temperate regions. Our goal was thus to understand the reasons why there are more species of Carnivora in the tropics. <strong>Location</strong> World-wide. <strong>Methods</strong> We reconstructed the biogeographical history of Carnivora using a time-calibrated phylogeny of the clade comprising all terrestrial species and dispersal-extinction-cladogenesis models. We also analysed a fossil dataset of carnivoran genera to examine how the latitudinal distribution of Carnivora varied through time. <strong>Results</strong> Our biogeographical analyses suggest that Carnivora originated in the East Palaearctic (i.e. Central Asia, China) in the early Palaeogene. Multiple independent lineages dispersed to low latitudes following three main paths: toward Africa, toward India/Southeast Asia and toward South America via the Bering Strait. These dispersal events were probably associated with local extinctions at high latitudes. Fossil data corroborate a high-latitude origin of the group, followed by late dispersal events toward lower latitudes in the Neogene. <strong>Main conclusions</strong> Unlike most other mammalian orders, which originated and diversified at low latitudes and dispersed out of the tropics', Carnivora originated at high latitudes, and subsequently dispersed southward. Our study provides an example of combining phylogenetic and fossil data to understand the generation and maintenance of global-scale geographical variations in species richness. (10.1111/geb.12354)
  DOI : 10.1111/geb.12354
• Phaseless inverse scattering in the one-dimensional case
  
  • Novikov Roman
  Eurasian Journal of Mathematical and Computer Applications, Eurasian National University, Kazakhstan (Nur-Sultan), 2015, 3 (1), pp.64-70. We consider the one-dimensional Schrödinger equation with a potential satisfying the standard assumptions of the inverse scattering theory and supported on the half-line x ≥ 0. For this equation at fixed positive energy we give explicit formulas for finding the full complex valued reflection coefficient to the left from appropriate phaseless scattering data measured on the left, i.e. for x < 0. Using these formulas and known inverse scattering results we obtain global uniqueness and reconstruction results for phaseless inverse scattering in dimension d = 1.
• Artificial boundary conditions for axisymmetric eddy current probe problems
  
  • Haddar Houssem
  • Jiang Zixian
  • Lechleiter Armin
  Computers & Mathematics with Applications, Elsevier, 2015, 68 (12, Part A,), pp.1844–1870. We study different strategies for the truncation of computational domains in the simulation of eddy current probes of elongated axisymmetric tubes. For axial fictitious boundaries, an exact Dirichlet-to-Neumann map is proposed and mathematically analyzed via a non-selfadjoint spectral problem: under general assumptions we show convergence of the solution to an eddy current problem involving a truncated Dirichlet-to-Neumann map to the solution on the entire, unbounded axisymmetric domain as the truncation parameter tends to infinity. Under stronger assumptions on the physical parameters of the eddy current problem, convergence rates are shown. We further validate our theoretical results through numerical experiments for a realistic physical setting inspired by eddy current probes of nuclear reactor core tubes. (10.1016/j.camwa.2014.10.008)
  DOI : 10.1016/j.camwa.2014.10.008
• Lyapunov and Minimum-Time Path Planning for Drones
  
  • Maillot Thibault
  • Boscain Ugo
  • Gauthier Jean-Paul
  • Serres Ulysse
  Journal of Dynamical and Control Systems, Springer Verlag, 2015, 21 (1), pp.1-34. (10.1007/s10883-014-9222-y)
  DOI : 10.1007/s10883-014-9222-y
• The topological derivative of stress-based cost functionals in anisotropic elasticity
  
  • Delgado Gabriel
  • Bonnet Marc
  Computers & Mathematics with Applications, Elsevier, 2015, 69, pp.1144-1166. The topological derivative of cost functionals J that depend on the stress (through the displacement gradient, assuming a linearly elastic material behavior) is considered in a quite general 3D setting where both the background and the inhomogeneity may have arbitrary anisotropic elastic properties. The topological derivative dJ(z) of J quantifies the asymptotic behavior of J under the nucleation in the background elastic medium of a small anisotropic inhomogeneity of characteristic radius a at a specified location z. The fact that the strain perturbation inside an elastic inhomogeneity remains finite for arbitrarily small a makes the small-inhomogeneity asymptotics of stress-based cost functionals quite different than that of the more usual displacement-based functionals. The asymptotic perturbation of J is shown to be of order O(a^3) for a wide class of stress-based cost functionals having smooth densities. The topological derivative of J, i.e. the coefficient of the O(a^3) perturbation, is established, and computational procedures then discussed. The resulting small-inhomogeneity expansion of J is mathematically justified (i.e. its remainder is proved to be of order o(a^3)). Several 2D and 3D numerical examples are presented, in particular demonstrating the proposed formulation of \dJ on cases involving anisotropic elasticity and non-quadratic cost functionals. (10.1016/j.camwa.2015.03.010)
  DOI : 10.1016/j.camwa.2015.03.010
• Developmental Partial Differential Equations
  
  • Pouradier Duteil Nastassia
  • Rossi Francesco
  • Boscain Ugo
  • Piccoli Benedetto
  , 2015. In this paper, we introduce the concept of Developmental Partial Differential Equation (DPDE), which consists of a Partial Differential Equation (PDE) on a time-varying manifold with complete coupling between the PDE and the manifold’s evolution. In other words, the manifold’s evolution depends on the solution to the PDE, and vice versa the differential operator of the PDE depends on the manifold’s geometry. DPDE is used to study a diffusion equation with source on a growing surface whose growth depends on the intensity of the diffused quantity. The surface may, for instance, represent the membrane of an egg chamber and the diffused quantity a protein activating a signaling pathway leading to growth. Our main objective is to show controllability of the surface shape using a fixed source with variable intensity for the diffusion. More specifically, we look for a control driving a symmetric manifold shape to any other symmetric shape in a given time interval. For the diffusion we take directly the Laplace-Beltrami operator of the surface, while the surface growth is assumed to be equal to the value of the diffused quantity. We introduce a theoretical framework, provide approximate controllability and show numerical results. Future applications include a specific model for the oogenesis of Drosophila melanogaster.
• MatVPC: A User-Friendly MATLAB-Based Tool for the Simulation and Evaluation of Systems Pharmacology Models
  
  • Biliouris Kostas
  • Lavielle Marc
  • Trame Mirjam
  CPT: Pharmacometrics and Systems Pharmacology, American Society for Clinical Pharmacology and Therapeutics ; International Society of Pharmacometrics, 2015. Quantitative systems pharmacology (QSP) models are progressively entering the arena of contemporary pharmacology. The efficient implementation and evaluation of complex QSP models necessitates the development of flexible computational tools that are built into QSP mainstream software. To this end, we present MatVPC, a versatile MATLAB-based tool that accommodates QSP models of any complexity level. MatVPC executes Monte Carlo simulations as well as automatic construction of visual predictive checks (VPCs) and quantified VPCs (QVPCs). VPC is a model diagnostic tool that facilitates the evaluation of both the structural and the stochastic part of a model. It is constructed by superimposing the observations over the model simulations while accounting for both the interindivid-ual variability as well as the residual variability. 1 Once underutilized, 2 the VPC now is recognized as one of the most valuable model diagnostics in pharmacological model evaluation. 3–5 Its superiority over comparable diagnostic tools has been established 6 and reflected by the fact that regulatory agencies recommend it as one of the central model diagnostics. 7 (10.1002/psp4.12011)
  DOI : 10.1002/psp4.12011
• Inverse scattering without phase information
  
  • Novikov Roman
  Séminaire Laurent Schwartz - EDP et applications, Centre de mathématiques Laurent Schwartz, 2015, 2014-2015, pp.Exposé no. 16, 13 pp. We report on nonuniqueness, uniqueness and reconstruction results in quantum mechanical and acoustic inverse scattering without phase information. We are motivated by recent and very essential progress in this domain. This paper is an extended version of the talk given at Séminaire Laurent Schwartz on March 31, 2015. (10.5802/slsedp.74)
  DOI : 10.5802/slsedp.74
• Ensuring robustness of domain decomposition methods by block strategies
  
  • Gosselet Pierre
  • Rixen Daniel
  • Spillane Nicole
  • Roux François-Xavier
  , 2015. no abstract
• Training Schr\"odinger's cat: quantum optimal control
  
  • Glaser Stefffen J.
  • Boscain Ugo
  • Calarco Tommaso
  • Koch Christiane P.
  • Köckenberger Walter
  • Kosloff Ronnie
  • Kuprov Ilya
  • Luy Burkard
  • Schirmer Sophie
  • Schulte-Herbrüggen Thomas
  • Sugny Dominique
  • Wilhelm Frank K.
  , 2015. It is control that turns scientific knowledge into useful technology: in physics and engineering it provides a systematic way for driving a system from a given initial state into a desired target state with minimized expenditure of energy and resources -- as famously applied in the Apollo programme. As one of the cornerstones for enabling quantum technologies, optimal quantum control keeps evolving and expanding into areas as diverse as quantum-enhanced sensing, manipulation of single spins, photons, or atoms, optical spectroscopy, photochemistry, magnetic resonance (spectroscopy as well as medical imaging), quantum information processing and quantum simulation. --- Here state-of-the-art quantum control techniques are reviewed and put into perspective by a consortium uniting expertise in optimal control theory and applications to spectroscopy, imaging, quantum dynamics of closed and open systems. We address key challenges and sketch a roadmap to future developments.
• Energy release rate for non smooth cracks in planar elasticity
  
  • Babadjian Jean-François
  • Chambolle Antonin
  • Lemenant Antoine
  Journal de l'École polytechnique — Mathématiques, École polytechnique, 2015, 2, pp.117-152. This paper is devoted to the characterization of the energy release rate of a crack which is merely closed, connected, and with density $1/2$ at the tip. First, the blow-up limit of the displacement is analyzed, and the convergence to the corresponding positively $1/2$-homogenous function in the cracked plane is established. Then, the energy release rate is obtained as the derivative of the elastic energy with respect to an infinitesimal additional crack increment.
• Coupling techniques for nonlinear hyperbolic equations. IV. Well-balanced schemes for scalar multi-dimensional and multi-component laws
  
  • Boutin Benjamin
  • Coquel Frédéric
  • LeFloch Philippe G.
  Mathematics of Computation, American Mathematical Society, 2015, 84 (294), pp.1663-1702. This series of papers is devoted to the formulation and the approximation of coupling problems for nonlinear hyperbolic equations. The coupling across an interface in the physical space is formulated in term of an augmented system of partial differential equations. In an earlier work, this strategy allowed us to develop a regularization method based on a thick interface model in one space variable for coupling scalar equations. In the present paper, we significantly extend this framework and, in addition, encompass equations in several space variables. This new formulation includes the coupling of several distinct scalar conservation laws and allows for a possible covering in space. Our main contributions are, on one hand, the design and analysis of a well–balanced finite volume method on general triangulations and, on the other hand, a proof of convergence of this method toward entropy solutions, extending Coquel, Cockburn, and LeFloch's theory (restricted to a single conservation law without coupling). The core of our analysis is, first, the derivation of entropy inequalities as well as a discrete entropy dissipation estimate and, second, a proof of convergence toward the entropy solution of the coupling problem. (10.1090/S0025-5718-2015-02933-0)
  DOI : 10.1090/S0025-5718-2015-02933-0
• Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity
  
  • Méléard Sylvie
  • Mirrahimi Sepideh
  Communications in Partial Differential Equations, Taylor & Francis, 2015, 40 (5), pp.957-993. We perform an asymptotic analysis of models of population dynamics with a fractional Laplacian and local or nonlocal reaction terms. The first part of the paper is devoted to the long time/long range rescaling of the fractional Fisher-KPP equation. This rescaling is based on the exponential speed of propagation of the population. In particular we show that the only role of the fractional Laplacian in determining this speed is at the initial layer where it determines the thickness of the tails of the solutions. Next, we show that such rescaling is also possible for models with non-local reaction terms, as selection-mutation models. However, to obtain a more relevant qualitative behavior for this second case, we introduce, in the second part of the paper, a second rescaling where we assume that the diffusion steps are small. In this way, using a WKB ansatz, we obtain a Hamilton-Jacobi equation in the limit which describes the asymptotic dynamics of the solutions, similarly to the case of selection-mutation models with a classical Laplace term or an integral kernel with thin tails. However, the rescaling introduced here is very different from the latter cases. We extend these results to the multidimensional case. (10.1080/03605302.2014.963606)
  DOI : 10.1080/03605302.2014.963606
• Rare event simulation using reversible shaking transformations
  
  • Gobet Emmanuel
  • Liu Gang
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2015, 37 (5), pp.A2295-A2316. We introduce random transformations called reversible shaking transformations which we use to design two schemes for estimating rare event probability. One is based on interacting particle systems (IPS) and the other on time-average on a single path (POP) using ergodic theorem. We discuss their convergence rates and provide numerical experiments including continuous stochastic processes and jump processes. Our examples cover rather important situations related to insurance, queueing system and random graph for instance. Both schemes have good performance, with a seemingly better one for POP. (10.1137/14098418X)
  DOI : 10.1137/14098418X
• A deterministic approximation method in shape optimization under random uncertainties
  
  • Allaire Grégoire
  • Dapogny Charles
  SMAI Journal of Computational Mathematics, Société de Mathématiques Appliquées et Industrielles (SMAI), 2015, 1, pp.83-143. This paper is concerned with the treatment of uncertainties in shape optimization. We consider uncertainties in the loadings, the material properties, the geometry and the vibration frequency, both in the parametric and geometric optimization setting. We minimize objective functions which are mean values, variances or failure probabilities of standard cost functions under random uncertainties. By assuming that the uncertainties are small and generated by a finite number N of random variables, and using first-or second-order Taylor expansions, we propose a deterministic approach to optimize approximate objective functions. The computational cost is similar to that of a multiple load problems where the number of loads is N. We demonstrate the effectiveness of our approach on various parametric and geometric optimization problems in two space dimensions. (10.5802/smai-jcm.5)
  DOI : 10.5802/smai-jcm.5
• Mean field games systems of first order
  
  • Cardaliaguet Pierre
  • Graber Philip Jameson
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2015, 21 (3), pp.690–722. We consider a system of mean field games with local coupling in the deterministic limit. Under general structure conditions on the Hamiltonian and coupling, we prove existence and uniqueness of the weak solution, characterizing this solution as the minimizer of some optimal control of Hamilton-Jacobi and continuity equations. We also prove that this solution converges in the long time average to the solution of the associated ergodic problem.