We present an efficient approach for reducing the statistical uncertaintyassociated with direct Monte Carlo simulations of the Boltzmann equation.As with previous variance-reduction approaches, the resulting relativestatistical uncertainty in hydrodynamic quantities (statistical uncertainty normalized by thecharacteristic value of quantity of interest) is smalland independent of the magnitude of the deviation from equilibrium,making the simulation of arbitrarily small deviations from equilibriumpossible. In contrast to previous variance-reduction methods, themethod presented here is able to substantially reduce variance withvery little modification to the standard DSMC algorithm. This is achievedby introducing an auxiliary equilibrium simulation which, via an importanceweight formulation, uses the same particle data as the non-equilibrium(DSMC) calculation; subtracting the equilibrium from the non-equilibriumhydrodynamic fields drastically reduces the statistical uncertaintyof the latter because the two fields are correlated. The resulting formulation is simple to code and provides considerable computational savings for a wide range of problems of practical interest. It is validated by comparing our results with DSMC solutions for steadyand unsteady, isothermal and non-isothermal problems; in all casesvery good agreement between the two methods is found.

In this paper we study different algorithms for backwardstochastic differential equations (BSDE in short) basing on randomwalk framework for 1-dimensional Brownian motion. Implicit andexplicit schemes for both BSDE and reflected BSDE are introduced.Then we prove the convergence of different algorithms and presentsimulation results for different types of BSDEs.

This work studies the null-controllability of a class of abstract parabolic equations. The main contribution in the general caseconsists in giving a short proof of an abstract version of a sufficient condition for null-controllability which has been proposed by Lebeau and Robbiano. We do not assume that the control operator is admissible. Moreover, we give estimates of the control cost.In the special case of the heat equation in rectangular domains, we provide an alternative way to check the Lebeau-Robbiano spectral condition. We then show that the sophisticated Carleman and interpolation inequalities used in previous literature may be replaced by a simple result of Turán. In this case, we provide explicit values for the constants involved in the above mentioned spectral condition. As far as we are aware, this is the first proof of the null-controllability of the heat equation with arbitrary control domain in a n-dimensional open set which avoids Carleman estimates.

An optimal control problem forsemilinear parabolic partial differential equations is considered.The control variable appears in the leading term of the equation.Necessary conditions for optimal controls are established by themethod of homogenizing spike variation. Results for problems withstate constraints are also stated.

The Bidomain model is nowadays one of the most accurate mathematical descriptions of the action potential propagation in the heart.However, its numerical approximation is in general fairly expensive as a consequence of the mathematical featuresof this system. For this reason, a simplification of this model, called Monodomain problem is quite oftenadopted in order to reduce computational costs. Reliability of this model is however questionable, in particular in the presence of applied currents and in the regions where the upstroke or the late recovery of the action potential is occurring.In this paper we investigate a domain decomposition approach for this problem, where the entire computationaldomain is suitably split and the two models are solved in different subdomains. Since the mathematical features of the two problems are rather different, the heterogeneous coupling is non trivial. Here we investigate appropriate interface matching conditionsfor the coupling on non overlapping domains. Moreover, we pursue an Optimized Schwarz approach for the numerical solution of the heterogeneous problem. Convergence of the iterative method is analyzed by means of a Fourier analysis. We investigate the parameters to be selected in the matching radiation-type conditions to accelerate the convergence. Numerical results both in two and three dimensions illustrate the effectiveness of the coupling strategy.

Data assimilation refers to any methodology that uses partialobservational data and the dynamics of a system for estimating themodel state or its parameters. We consider here a non classicalapproach to data assimilation based in null controllabilityintroduced in [Puel, C. R. Math. Acad. Sci. Paris 335 (2002) 161–166] and [Puel, SIAM J. Control Optim.48 (2009) 1089–1111] and we apply it to oceanography.More precisely, we are interested in developing this methodologyto recover the unknown final state value (state value at the end of the measurement period) in a quasi-geostrophicocean model from satellite altimeter data, which allows in fact tomake better predictions of the ocean circulation. The main idea ofthe method is to solve several null controllability problems for the adjoint system inorder to obtain projections of the final state on a reduced basis.Theoretically, we have to prove the well posedness of theinvolved systems associated to the method and we also need anobservability property to show the existence of null controls for the adjoint system. Tothis aim, we use a global Carleman inequality for the associatedvelocity-pressure formulation of the problem which was previouslyproved in [Fernández-Cara et al., J. Math. Pures Appl.83(2004) 1501–1542]. We present numerical simulations using a regularizedversion of this data assimilation methodology based on nullcontrollability for elements of a reduced spectral basis.After proving the convergence of the regularized solutions, weanalyze the incidence of the observatory size and noisy data inthe recovery of the initial value for a quality prediction.

The Cahn-Hilliard variational inequality is a non-standard parabolic variational inequality of fourth order for which straightforward numerical approaches cannot be applied. We propose a primal-dual active set method which can be interpreted as a semi-smooth Newton method as solution technique for the discretized Cahn-Hilliard variational inequality. A (semi-)implicit Euler discretization is used in time and a piecewise linear finite element discretization of splitting type is used in space leading to a discrete variational inequality of saddle point type in each time step. In each iteration of the primal-dual active set method a linearized system resulting from the discretization of two coupled elliptic equations which are defined on different sets has to be solved. We show local convergence of the primal-dual active set method and demonstrate its efficiency with several numerical simulations.

This paper deals with a model describing damage processes in a (nonlinear) elastic body which is in contact with adhesion with a rigid support. On the basis of phase transitions theory, we detailthe derivation of the model written in terms of a PDE system, combined with suitable initial and boundary conditions. Some internal constraints on the variables are introduced in the equations and on the boundary, to get physical consistency. We prove theexistence of global in time solutions (to a suitable variational formulation) of therelated Cauchy problem by means of a Schauder fixed point argument, combinedwith monotonicity and compactness tools. We also perform an asymptotic analysis of the solutions as the interfacial damage energy (between the body and the contact surface) goes to +∞.

Optimization problems with convex but non-smooth cost functional subject to an elliptic partial differential equation are considered.The non-smoothness arises from a L1-norm in the objective functional. The problem is regularized to permit the use of the semi-smooth Newtonmethod. Error estimates with respect to the regularization parameter are provided. Moreover, finite element approximations are studied. A-priori as well as a-posteriori error estimates are developed and confirmed by numerical experiments.

In the present paper, we consider nonlinear optimal control problems with constraints on the state of the system. We are interested in the characterization of the value function without any controllability assumption. In the unconstrained case, it is possible to derive a characterization of the value function by means of a Hamilton-Jacobi-Bellman (HJB) equation. This equation expresses the behavior of the value function along the trajectories arriving or starting from any position x. In the constrained case, when no controllability assumption is made, the HJB equation may have several solutions. Our first result aims to give the precise information that should be added to the HJB equation in order to obtain a characterization of the value function. This result is very general and holds even when the dynamics is not continuous and the state constraints set is not smooth. On the other hand we study also some stability results for relaxed or penalized control problems.

We consider the Dirichlet Laplacian in a thin curvedthree-dimensional rod. The rod is finite. Its cross-section isconstant and small, and rotates along the reference curve in anarbitrary way. We find a two-parametric set of the eigenvalues ofsuch operator and construct their complete asymptotic expansions. Weshow that this two-parametric set contains any prescribed number ofthe first eigenvalues of the considered operator. We obtain thecomplete asymptotic expansions for the eigenfunctions associatedwith these first eigenvalues.

In this paper, we carry out the numerical analysis of adistributed optimal control problem governed by a quasilinearelliptic equation of non-monotone type. The goal is to prove thestrong convergence of the discretization of the problem by finiteelements. The main issue is to get error estimates for thediscretization of the state equation. One of the difficulties inthis analysis is that, in spite of the partial differentialequation has a unique solution for any given control, theuniqueness of a solution for the discrete equation is an openproblem.

This paper is devoted to the study of a coupled system which consists ofa wave equation and a heat equation coupled through a transmission conditionalong a steady interface. This system is a linearized model forfluid-structure interaction introduced by Rauch, Zhang and Zuazuafor a simple transmission condition and by Zhang and Zuazua for anatural transmission condition. Using an abstract theorem of Burq and a new Carleman estimate proved near the interface, wecomplete the results obtained by Zhang and Zuazua and by Duyckaerts.We prove, without a Geometric Control Condition, a logarithmic decayof the energy.

The aim of this paper is to provide a rigorous variational formulation forthe detection of points in 2-d biological images. To this purposewe introduce a new functional whose minimizers give the points we want to detect. Then we define an approximating sequence of functionals forwhich we prove the Γ-convergence to the initial one.

We consider the numerical solution, in two- and three-dimensionalbounded domains, of the inverse problem for identifying the locationof small-volume, conductivity imperfections in a medium with homogeneousbackground. A dynamic approach, based on the wave equation, permitsus to treat the important case of “limited-view” data. Our numericalalgorithm is based on the coupling of a finite element solution ofthe wave equation, an exact controllability method and finally a Fourierinversion for localizing the centers of the imperfections. Numericalresults, in 2- and 3-D, show the robustness and accuracy of the approachfor retrieving randomly placed imperfections from both complete andpartial boundary measurements.

In the setting ofa real Hilbert space ${\cal H}$, we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order evolutionequations

          ü(t) + γ$\dot{u}$(t) + ∇ϕ(u(t)) + A(u(t)) = 0,

where ∇ϕ is the gradient operator of a convexdifferentiable potential function ϕ : ${\cal H}\to \R$, A : ${\cal H}\to {\cal H}$ is a maximal monotone operator which is assumed to beλ-cocoercive, and γ > 0 is a damping parameter.Potential and non-potential effects are associated respectively to∇ϕ and A. Under condition λγ2 > 1, it is proved that each trajectory asymptotically weaklyconverges to a zero of ∇ϕ + A. This condition, whichonly involves the non-potential operator and the dampingparameter, is sharp and consistent with time rescaling. Passingfrom weak to strong convergence of the trajectories is obtained byintroducing an asymptotically vanishing Tikhonov-like regularizingterm. As special cases, we recover the asymptotic analysis of theheavy ball with friction dynamic attached to a convex potential, thesecond-order gradient-projection dynamic, and the second-orderdynamic governed by the Yosida approximation of a general maximalmonotone operator. The breadth and flexibility of the proposed framework is illustrated through applications in the areas of constrained optimization,dynamical approach to Nash equilibria for noncooperative games, and asymptotic stabilization in the case of a continuum of equilibria.

We develop a functional analytical framework for a linearperidynamic model of a spring network system in any space dimension.Various properties of the peridynamic operators are examined forgeneral micromodulus functions. These properties are utilized toestablish the well-posedness of both the stationary peridynamicmodel and the Cauchy problem of the time dependent peridynamicmodel. The connections to the classical elastic models are alsoprovided.

The penalty method when applied to the Stokes problem provides a very efficient algorithm for solving any discretization of this problem since it gives rise to a system of two equations where the unknowns are uncoupled. For a spectral or spectral element discretization of the Stokes problem, we prove a posteriori estimates that allow us to optimize the penalty parameter as a function of the discretization parameter. Numerical experiments confirm the interest of this technique.

In this paper we construct a new H(div)-conforming projection-basedp-interpolation operator that assumes only H r (K) $\cap$ ${\bf \tilde H}$ -1/2(div, K)-regularity(r > 0) on the reference element (either triangle or square) K.We show that this operator is stable with respect to polynomial degrees andsatisfies the commuting diagram property. We also establish an estimate for theinterpolation error in the norm of the space ${\bf \tilde H}$ -1/2(div, K),which is closely related to the energy spaces for boundary integral formulationsof time-harmonic problems of electromagnetics in three dimensions.

We propose and analyze a finite element method for approximating solutions to the Navier-Stokes-alpha model (NS-α) that utilizes approximate deconvolution and a modified grad-div stabilization and greatlyimproves accuracy in simulations. Standard finite element schemesfor NS-α suffer from two major sources of error if their solutions are considered approximationsto true fluid flow: (1) the consistency error arising from filtering; and (2) the dramatic effect of the large pressure erroron the velocity error that arises from the (necessary) use of the rotational form nonlinearity.The proposed scheme “fixes” these two numericalissues through the combined use of a modified grad-div stabilization that acts in both the momentum and filter equations, and an adapted approximate deconvolution technique designed to work with the altered filter. We provethe scheme is stable, optimally convergent, and the effect of the pressureerror on the velocity error is significantly reduced. Several numerical experiments are given that demonstrate theeffectiveness of the method.