Continuous and Discrete Adjoints to the Euler Equations for Fluids

Adjoints are used in optimization to speed-up computations, simplify optimality conditions or compute sensitivities. Because time is reversed in adjoint equations with first order time derivatives, boundary conditions and transmission conditions through shocks can be difficult to understand. In this article we analyze the adjoint equations that arise in the context of compressible flows governed by the Euler equations of fluid dynamics. We show that the continuous adjoints and the discrete adjoints computed by automatic differentiation agree numerically; in particular the adjoint is found to be continuous at the shocks and usually discontinuous at contact discontinuities by both.Comment: 30 pages 16 figure

Computational Issues for Optimal Shape Design in Hemodynamics

A Fluid-Structure Interaction model is studied for aortic flow, based on Koiter's shell model for the structure, Navier-Stokes equation for the fluid and transpiration for the coupling. It accounts for wall deformation while yet working on a fixed geometry. The model is established first. Then a numerical approximation is proposed and some tests are given. The model is also used for optimal design of a stent and possible recovery of the arterial wall elastic coefficients by inverse methods

A reduced basis for option pricing

We introduce a reduced basis method for the efficient numerical solution of partial integro-differential equations which arise in option pricing theory. Our method uses a basis of functions constructed from a sequence of Black-Scholes solutions with different volatilities. We show that this choice of basis leads to a sparse representation of option pricing functions, yielding an approximation whose precision is exponential in the number of basis functions. A Galerkin method using this basis for solving the pricing PDE is presented. Numerical tests based on the CEV diffusion model and the Merton jump diffusion model show that the method has better numerical performance relative to commonly used finite-difference and finite-element methods. We also compare our method with a numerical Proper Orthogonal Decomposition (POD). Finally, we show that this approach may be used advantageously for the calibration of local volatility functions.

Dynamic Programming for Mean-field type Control

International audienceFor mean-field type control problems, stochastic dynamic programming requires adaptation. We propose to reformulate the problem as a distributed control problem by assuming that the PDF $\rho$ of the stochastic process exists. Then we show that Bellman's principle applies to the dynamic programming value function $V(\tau,\rho_\tau)$ where the dependency on $\rho_\tau$ is functional as in P.L. Lions' analysis of mean-filed games (2007). We derive HJB equations and apply them to two examples, a portfolio optimization and a systemic risk model

A total linearization method for solving viscous free boundary flow problems by the finite element method

In this paper a total linearization method is derived for solving steady viscous free boundary flow problems (including capillary effects) by the finite element method. It is shown that the influence of the geometrical unknown in the totally linearized weak formulation can be expressed in terms of boundary integrals. This means that the implementation of the method is simple. Numerical experiments show that the iterative method gives accurate results and converges very fast

La dynamique de la Technologie Intranet au service d'un système d'information avec Bases de données interactives.

International audienceInternet s'accroît de manière exponentielle depuis le début des années 90. Ce phénomène est dû en partie aux entreprises qui ont vu dans cette technologie l'outil par excellence pour élargir leur marché et étudier leur environnement. L'information est aujourd'hui au coeur de la vie de l'entreprise. Mais elle est aussi une matière première indispensable qui est souvent désorganisée aussi bien dans sa collecte que dans son traitement et sa diffusion. La solution pour délivrer l'information " just-in-time " est l'implémentation d'un système de communication universel. Un réseau qui relierait tous les intervenants de l'entreprise s'apparente manifestement à l'Internet. Et, utiliser Internet au sein des entreprises, c'est le principe des Intranets. En effet, l'Intranet permet une communication sécurisée et transversale des "intranautes", ainsi qu'un apport en intelligence de l'entreprise. La gestion optimale de cette plus-value passe par une interactivité de l'Intranet. Pour ce faire, un système de Bases de Données doit être construit. Ces bases peuvent concerner l'information interne à l'entreprise et sa mémoire (compte-rendu des congrès/colloque, annuaires, ...) ainsi que l'information externe enrichie par les experts (benchmarking : sites des concurrents, brevets, ...) Cette technologie peut se baser sur plusieurs types d'outils qui se déclinent en suites logicielles ou logiciels intégrés. C'est à l'entreprise de choisir la technologie adéquate en fonction de ses besoins, moyens, et compétences. Pour illustrer cet état de fait, FAURECIA développe un système d'information qui repose sur un Intranet interactif. Pour être de plus en plus réactive sur ses marchés, l'entreprise doit être de plus en plus communicante, en son sein et avec ses partenaires (clientèle et fournisseurs). Pour s'en rapprocher, elle pourra leur ouvrir son Intranet. C'est le concept de l'Extranet. Cette technologie intègre la notion de "l'entreprise étendue", ou "l'entreprise virtuelle". Et parmi les fondements de ce nouveau concept managérial, l'extension du système d'information en amont et aval de l'entreprise. De plus en plus, les relations " business-to-business " utilisent ces nouvelles technologies

Vibrato and Automatic Differentiation for High Order Derivatives and Sensitivities of Financial Options

International audienceThis paper deals with the computation of second or higher order greeks of financial securities. It combines two methods, Vibrato and automatic differentiation and compares with other methods. We show that this combined technique is faster than standard finite difference, more stable than automatic differentiation of second order derivatives and more general than Malliavin Calculus. We present a generic framework to compute any greeks and present several applications on different types of financial contracts: European and American options, multidimensional Basket Call and stochastic volatility models such as Heston's model. We give also an algorithm to compute derivatives for the Longstaff-Schwartz Monte Carlo method for American options. We also extend automatic differentiation for second order derivatives of options with non-twice differentiable payoff

Analysis of a coupled fluid-structure model with applications to hemodynamics

We propose and analyze a simplified fluid-structure coupled model for flows with compliant walls. As in [F. Nobile and C. Vergara, SIAM J. Sci. Comput., 30 (2008), pp. 731-763], the wall reaction to the fluid is modeled by a small displacement viscoelastic shell where the tangential stress components and displacements are neglected. We show that within this small displacement approximation a transpiration condition can be used which does not require an update of the geometry at each time step, for pipe flow at least. Such simplifications lead to a model which is well posed and for which a semi-implicit time discretization can be shown to converge. We present some numerical results and a comparison with a standard test case taken from hemodynamics. The model is more stable and less computer demanding than full models with moving mesh. We apply the model to a three-dimensional arterial flow with a stent.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo Regiona