Publications

2011

• Optimal structure of gas transmission trunklines
  
  • Bonnans J. Frederic
  • André Jean
  Optimization and Engineering, Springer Verlag, 2011, 12 (1), pp.175-198. In this paper, we consider the optimal design of a straight pipeline system. Suppose a gas pipeline is to be designed to transport a specified flowrate from the entry point to the gas demand point. Physical and contractual requirements at supply and delivery nodes are known as well as the costs to buy and lay a pipeline or build a compressor station. In order to minimize the overall cost of creation of this mainline, the following design variables need to be determined: the number of compressor stations, the lengths of pipeline segments between compressor stations, the diameters of the pipeline segments, the suction and discharge pressures at each compressor station. To facilitate the calculation of the design of a pipeline, gas engineers proposed, in several handbooks, to base their cost-assessments on some optimal properties from previous experiences and usual engineering practices: the distance between compressors is constant, all diameters are equal, and all inlet (resp. outlet) pressures are equal. The goals of this paper are (1) to state on which assumptions we can consider that the optimal properties are valid and (2) to propose a rigorous proof of the optimal properties (based on nonlinear programming optimality conditions) within a more general framework than before.
• Energy contracts management by stochastic programming techniques
  
  • Cen Zhihao
  • Bonnans J. Frederic
  • Christel Thibault
  Annals of Operations Research, Springer Verlag, 2011, 200 (1), pp.199-222. We consider the problem of optimal management of energy contracts, with bounds on the local (time step) amounts and global (whole period) amounts to be traded, integer constraint on the decision variables and uncertainty on prices only. After building a finite state Markov chain by using vectorial quantization tree method, we rely on the stochastic dual dynamic programming (SDDP) method to solve the continuous relaxation of this stochastic optimization problem. An heuristic for computing sub optimal solutions to the integer optimization problem, based on the Bellman values of the continuous relaxation, is provided. Combining the previous techniques, we are able to deal with high-dimension state variables problems. Numerical tests applied to realistic energy markets problems have been performed. (10.1007/s10479-011-0973-5)
  DOI : 10.1007/s10479-011-0973-5
• Minimal Time Problems with Moving Targets and Obstacles
  
  • Bokanowski Olivier
  • Zidani Hasnaa
  , 2011, 18, Part 1, pp.2589-2593. We consider minimal time problems governed by nonlinear systems under general time dependant state constraints and in the two-player games setting. In general, it is known that the characterization of the minimal time function, as well as the study of its regularity properties, is a difficult task in particular when no controlability assumption is made. In addition to these difficulties, we are interested here to the case when the target, the state constraints and the dynamics are allowed to be time-dependent. We introduce a particular "reachability" control problem, which has a supremum cost function but is free of state constraints. This auxiliary control problem allows to characterize easily the backward reachable sets, and then, the minimal time function, without assuming any controllability assumption. These techniques are linked to the well known level-set approachs. Partial results of the study have been published recently by the authors in SICON. Here, we generalize the method to more complex problems of moving target and obstacle problems. Our results can be used to deal with motion planning problems with obstacle avoidance. (10.3182/20110828-6-IT-1002.02261)
  DOI : 10.3182/20110828-6-IT-1002.02261
• Accelerated Boundary Element Method for Diffuse Optical Imaging
  
  • Elisee Josias
  • Bonnet Marc
  • Arridge Simon
  Optics Letters, Optical Society of America - OSA Publishing, 2011, 36, pp.4101-4103. The boundary element method (BEM) is a useful tool in Diffuse Optical Imaging (DOI) when modelling large optical regions whose parameters are piecewise constant, but is computationally expensive. We present here an acceleration technique, the single-level Fast Multipole Method, for a highly lossy medium. The enhanced practicability of the BEM in DOI is demonstrated through test examples on single-layer problems, where order of magnitude reduction factors on solution time are achieved, and on a realistic three-layer model of the neonatal head. Our experimental results agree very closely with theoretical predictions of computational complexity. (10.1364/OL.36.004101)
  DOI : 10.1364/OL.36.004101
• Coupling discontinuous Galerkin methods and retarded potentials for transient wave propagation on unbounded domains
  
  • Abboud Toufic
  • Joly Patrick
  • Rodríguez Jerónimo
  • Terrasse Isabelle
  Journal of Computational Physics, Elsevier, 2011, 230 (15), pp.5877-5907. This work deals with the numerical simulation of wave propagation on unbounded domains with localized heterogeneities. To do so, we propose to combine a discretization based on a discontinuous Galerkin method in space and explicit finite differences in time on the regions containing heterogeneities with the retarded potential method to account the unbounded nature of the computational domain. The coupling formula enforces a discrete energy identity ensuring the stability under the usual CFL condition in the interior. Moreover, the scheme allows to use a smaller time step in the interior domain yielding to quasi-optimal discretization parameters for both methods. The aliasing phenomena introduced by the local time stepping are reduced by a post-processing by averaging in time obtaining a stable and second order consistent (in time) coupling algorithm. We compute the numerical rate of convergence of the method for an academic problem. The numerical results show the feasibility of the whole discretization procedure. © 2011 Elsevier Inc. (10.1016/j.jcp.2011.03.062)
  DOI : 10.1016/j.jcp.2011.03.062
• An Error Estimate for the Finite Difference Scheme for One-Phase Obstacle Problem
  
  • Arakelyan Avetik
  • Barkhudaryan Rafayel
  • Poghosyan Michael
  Journal of Contemporary Mathematical Analysis, 2011, 46 (3), pp.131-141.
• On the use of Lamb modes in the linear sampling method for elastic waveguides
  
  • Bourgeois Laurent
  • Le Louër Frédérique
  • Lunéville Éric
  Inverse Problems, IOP Publishing, 2011, 27 (5), pp.055001. This paper is devoted to a modal formulation of the linear sampling method in elastic 2D or 3D waveguide, that is we use the guided modes (called Lamb modes in 2D) as incident waves and the corresponding far fields in order to retrieve some obstacles. We provide the mathematical background to tackle the problem of identifiability and the justification of the linear sampling method for such a case. The elastic waveguide raises a specific issue: it concerns the projection of the scattered field on a transverse basis, which requires the introduction of new variables that mix displacement and stress components and satisfy the so-called Fraser's biorthogonality condition. Some numerical experiments in 2D show the feasibility of the reconstruction in the case of a finite number of incident waves formed by Lamb modes. © 2011 IOP Publishing Ltd. (10.1088/0266-5611/27/5/055001)
  DOI : 10.1088/0266-5611/27/5/055001
• Optimal Control of the Atmospheric Reentry of a Space Shuttle by an Homotopy Method
  
  • Hermant Audrey
  Optimal Control Applications and Methods, Wiley, 2011, 32 (6), pp.627-646. This paper deals with the optimal control problem of the atmospheric reentry of a space shuttle with a second-order state constraint on the thermal flux. We solve the problem using the shooting algorithm combined with an homotopy method which automatically determines the structure of the optimal trajectory (composed of one boundary arc and one touch point). (10.1002/oca.961)
  DOI : 10.1002/oca.961
• About identification of defects in an elastic-plastic medium from boundary measurements in the antiplane case
  
  • Bourgeois Laurent
  • Dardé Jérémi
  Applicable Analysis, Taylor & Francis, 2011, 90 (10), pp.1481-1497. This article is devoted to a new method to find defects in some elasticplastic structure in the antiplane case from overdetermined data on a subpart of its boundary. This iterative method merges the method of quasireversibility and a simple level set technique. The foundations of the method are justified from a theoretical point of view and its efficiency is shown with the help of numerical experiments. © 2011 Taylor & Francis. (10.1080/00036811.2010.549481)
  DOI : 10.1080/00036811.2010.549481
• Probabilistic representation for solutions of an irregular porous media type equation: the degenerate case
  
  • Barbu Viorel
  • Roeckner Michael
  • Russo Francesco
  Probability Theory and Related Fields, Springer Verlag, 2011. We consider a possibly degenerate porous media type equation over all of $\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. This equation is motivated by some singular behaviour arising in complex self-organized critical systems. The main idea consists in approximating the equation by equations with monotone non-degenerate coefficients and deriving some new analytical properties of the solution. (10.1007/s00440-010-0291-x)
  DOI : 10.1007/s00440-010-0291-x
• Stable reconstruction of generalized impedance boundary conditions
  
  • Bourgeois Laurent
  • Chaulet Nicolas
  • Haddar Houssem
  Inverse Problems, IOP Publishing, 2011, 27 (9), pp.095002. (10.1088/0266-5611/27/9/095002)
  DOI : 10.1088/0266-5611/27/9/095002
• 10e colloque national en calcul des structures (CSMA 2011)
  
  • Bonnet Marc
  • Cornuault Christian
  • Pagano Stéphane
  , 2011, pp.273 articles. Ce dixième colloque a vocation d'aborder tous les thèmes en rapport avec le calcul des structures : Modélisation et comportement : Structures minces, élasto-visco-plasticité, rupture, fatigue, endommagement, matériaux composites, mise en forme, procédés de fabrication et d'assemblage. Méthodes et techniques de résolution : Parallélisme, sous-structuration, contact, frottement, approches multiéchelles, approches stochastiques, prise en compte des incertitudes, méthodes alternatives aux éléments finis. Contrôle et optimisation : Estimation d'erreur et applications, maillage adaptatif, problèmes inverses, identification, recalage, optimisation de forme et de processus, structures adaptatives, contrôle actif, structures intelligentes. Dynamique, interactions et couplages : Dynamique rapide et vibrations non linéaires, crash, vibroacoustique, interactions multiphysiques, sol-structure, fluide-structure. Applications : Industries mécaniques, biens d'équipement, aéronautique et spatial, véhicules et transport, construction navale, génie civil, énergie, nanotechnologies, biomécanique.
• Stability of acoustic propagation in 2D duct flows: A low frequency approach
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Duruflé Marc
  • Joly Patrick
  • Joubert Lauris
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2011, 21 (5), pp.1121-1151. In this paper, we derive a simplified "quasi-1D" limit model for the propagation of low frequency acoustic waves in a laminar flow filling a 2D duct. We analyze the well-posedness of this model in function of the Mach profile of the flow. This can be reduced to the study of the spectrum of a bounded non-normal operator. As a by-product of this analysis, we establish new results for hydrodynamic instabilities of KelvinHelmholtz type in compressible fluids. © 2011 World Scientific Publishing Company. (10.1142/s0218202511005271)
  DOI : 10.1142/s0218202511005271
• A discontinuous Galerkin solver for front propagation
  
  • Bokanowski Olivier
  • Cheng Yingda
  • Shu Chi-Wang
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2011, 33 (2), pp.923-938. We propose a new discontinuous Galerkin (DG) method based on [Cheng and Shu, JCP, 2007] to solve a class of Hamilton-Jacobi equations that arises from optimal control problems. These equations are connected to front propagation problems or minimal time problems with non isotropic dynamics. Several numerical experiments show the relevance of our method, in particular for front propagation.
• Characterization of the value function of final state constrained control problems with BV trajectories
  
  • Briani Ariela
  • Zidani Hasnaa
  Communications on Pure and Applied Mathematics, Wiley, 2011, 10 (6), pp.1567-1587. This paper aims to investigate a control problem governed by differential equations with Radon measure as data and with final state constraints. By using a known reparametrization method (by Dal Maso and Rampazzo [18]), we obtain that the value function can be characterized by means of an auxiliary control problem of absolutely continuous trajectories, involving time-measurable Hamiltonian. We study the characterization of the value function of this auxiliary problem and discuss its numerical approximations. (10.3934/cpaa.2011.10.1567)
  DOI : 10.3934/cpaa.2011.10.1567
• Dynamic consistency for Stochastic Optimal Control problems
  
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • Cohen Guy
  • de Lara Michel
  • Girardeau Pierre
  Annals of Operations Research, Springer Verlag, 2011, pp.17 p.. For a sequence of dynamic optimization problems, we aim at discussing a notion of consistency over time. This notion can be informally introduced as follows. At the very first time step $t_0$, the decision maker formulates an optimization problem that yields optimal decision rules for all the forthcoming time step $t_0, t_1, \dots, T$; at the next time step $t_1$, he is able to formulate a new optimization problem starting at time $t_1$ that yields a new sequence of optimal decision rules. This process can be continued until final time $T$ is reached. A family of optimization problems formulated in this way is said to be time consistent if the optimal strategies obtained when solving the original problem remain optimal for all subsequent problems. The notion of time consistency, well-known in the field of Economics, has been recently introduced in the context of risk measures, notably by Artzner et al. (2007) and studied in the Stochastic Programming framework by Shapiro (2009) and for Markov Decision Processes (MDP) by Ruszczynski (2009). We here link this notion with the concept of ''state variable'' in MDP, and show that a significant class of dynamic optimization problems are dynamically consistent, provided that an adequate state variable is chosen. (10.1007/s10479-011-1027-8)
  DOI : 10.1007/s10479-011-1027-8
• On stochastic calculus related to financial assets without semimartingales
  
  • Coviello Rosanna
  • Di Girolami Cristina
  • Russo Francesco
  Bulletin des Sciences Mathématiques, Elsevier, 2011, 135, pp.733-774. This paper does not suppose a priori that the evolution of the price of a financial asset is a semimartingale. Since possible strategies of investors are self-financing, previous prices are forced to be finite quadratic variation processes. The non-arbitrage property is not excluded if the class $\mathcal{A}$ of admissible strategies is restricted. The classical notion of martingale is replaced with the notion of $\mathcal{A}$-martingale. A calculus related to $\mathcal{A}$-martingales with some examples is developed. Some applications to no-arbitrage, viability, hedging and the maximization of the utility of an insider are expanded. We finally revisit some no arbitrage conditions of Bender-Sottinen-Valkeila type. (10.1016/j.bulsci.2011.06.008)
  DOI : 10.1016/j.bulsci.2011.06.008
• Mathematical and numerical modelling of wave propagation in fractal trees
  
  • Joly Patrick
  • Semin Adrien
  Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2011.
• Application of convex lexicographical optimization to the balance of GRTgaz gas grid
  
  • Bonnans Joseph Frederic
  • Paraisy Ruben
  • Veyrat Sébastien
  • Adam Soizic
  Journal of Global Optimization, Springer Verlag, 2011, 49 (3), pp.415--423.
• An alternative to Dirichlet-to-Neumann maps for waveguides
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Legendre Guillaume
  Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2011, 349 (17-18), pp.1005-1009. We are interested by the treatment of the radiation condition at infinity for the numerical solution of a problem set in an unbounded waveguide. We propose an alternative to the classical approach involving a modal expression of Dirichlet-to-Neumann (DtN) operators. This method is particularly simple to implement since it only requires the solution of boundary value problems with local boundary conditions. The corresponding approximate solution is comparable in accuracy to the one obtained by truncating the infinite series in the DtN maps. (10.1016/j.crma.2011.08.006)
  DOI : 10.1016/j.crma.2011.08.006