A discontinuous Galerkin finite element method for an optimalcontrol problem related to semilinear parabolic PDE's is examined.The schemes under consideration are discontinuous in time butconforming in space. Convergence of discrete schemes of arbitraryorder is proven. In addition, the convergence of discontinuousGalerkin approximations of the associated optimality system to thesolutions of the continuous optimality system is shown. The proofis based on stability estimates at arbitrary time points underminimal regularity assumptions, and a discrete compactnessargument for discontinuous Galerkin schemes (see Walkington[SINUM (June 2008) (submitted), preprint available at http://www.math.cmu.edu/~noelw], Sects. 3, 4).

We design a particle interpretation of Feynman-Kac measures on path spacesbased on a backward Markovian representation combined with a traditionalmean field particle interpretation of the flow of their final timemarginals. In contrast to traditional genealogical tree based models, thesenew particle algorithms can be used to compute normalized additivefunctionals “on-the-fly” as well as theirlimiting occupation measures with a given precision degree that does notdepend on the final time horizon.We provide uniform convergence results w.r.t. the time horizon parameter aswell as functional central limit theorems and exponential concentrationestimates, yielding what seems to be the first results of this type for thisclass of models. We also illustrate these results in the context offiltering of hidden Markov models, as well as in computational physics andimaginary time Schroedinger type partial differential equations, with aspecial interest in the numerical approximation of the invariant measureassociated to h-processes.

This paper gives an error analysis of themulti-configuration time-dependent Hartree (MCTDH)method for the approximation of multi-particle time-dependent Schrödinger equations. The MCTDH method approximates the multivariate wave function by a linear combination of products of univariate functions and replaces the high-dimensional linear Schrödinger equation by a coupled system of ordinary differential equations and low-dimensional nonlinear partial differential equations. The main result of this paper yields an L 2 error bound of the MCTDH approximation in terms of a best-approximation error bound in a stronger norm and of lower bounds of singular values of matrix unfoldings of the coefficient tensor. This result permits us to establish convergence of the MCTDH method to the exact wave function under appropriate conditions on the approximability of the wave function, and it points to reasons for possible failure in other cases.

This paper considers Schrödinger operators, and presents a probabilistic interpretation of the variation (or shape derivative) of the Dirichlet groundstate energy when the associated domain is perturbed. This interpretation relies on the distribution on the boundary of a stopped random process with Feynman-Kac weights. Practical computations require in addition the explicit approximation of the normal derivative of the groundstate on the boundary. We then propose to use this formulation in the case of the so-called fixed node approximation of Fermion groundstates, defined by the bottom eigenelements of the Schrödinger operator of a Fermionic system with Dirichlet conditions on the nodes (the set of zeros) of an initially guessed skew-symmetric function. We show that shape derivatives of the fixed node energy vanishes if and only if either (i) the distribution on the nodes of the stopped random process is symmetric; or (ii) the nodes are exactly the zeros of a skew-symmetric eigenfunction of the operator. We propose an approximation of the shape derivative of the fixed node energy that can be computed with a Monte-Carlo algorithm, which can be referred to as Nodal Monte-Carlo (NMC). The latter approximation of the shape derivative also vanishes if and only if either (i) or (ii) holds.

The paper is devoted to the computation of two-phase flows in a porous mediumwhen applying the two-fluid approach. The basic formulation is presented first, together with the main properties of the model. A few basic analytic solutions are then provided, some of them correspondingto solutions of the one-dimensional Riemann problem. Three distinct Finite-Volume schemes are then introduced. The first two schemes, which rely on the Rusanov scheme,are shown to give wrong approximations in some cases involving sharp porous profiles.The third one, which is an extension of a scheme proposed by Kröner and Thanh [SIAM J. Numer. Anal. 43 (2006) 796–824]for the computation of single phase flows in varying cross section ducts,provides fair results in all situations. Properties of schemes and numerical results are presented. Analytic tests enable to compute the L 1 norm of the error.

The construction of a well-conditioned integral equation for iterative solution of scattering problems with a variable Leontovitch boundary condition is proposed. A suitable parametrix is obtained by using a new unknown and an approximation of the transparency condition. We prove the well-posedness of the equation for any wavenumber. Finally, some numerical comparisons with well-tried method prove the efficiency of the new formulation.

Motivated by the development of efficient Monte Carlo methodsfor PDE models in molecular dynamics,we establish a new probabilistic interpretation of a family of divergence formoperators with discontinuous coefficients at the interfaceof two open subsets of $\mathbb{R}^d$ . This family of operators includes the case of thelinearized Poisson-Boltzmann equation used tocompute the electrostatic free energy of a molecule.More precisely, we explicitly construct a Markov process whoseinfinitesimal generator belongs to this family, as the solution of a SDEincluding a non standard local time term related to the interfaceof discontinuity. We then prove an extendedFeynman-Kac formula for the Poisson-Boltzmann equation.This formula allows us to justifyvarious probabilistic numerical methods toapproximate the free energy of a molecule.We analyse the convergence rate of these simulation procedures andnumerically compare them on idealized molecules models.

The paper studies the convergence behavior ofMonte Carlo schemes for semiconductors.A detailed analysis of the systematic error with respect to numerical parameters is performed.Different sources of systematic error are pointed out andillustrated in a spatially one-dimensional test case.The error with respect to the number of simulation particlesoccurs during the calculation of the internal electric field.The time step error, which is related to the splitting of transport andelectric field calculations, vanishes sufficiently fast.The error due to the approximation of the trajectories ofparticles depends on the ODE solver used in the algorithm.It is negligible compared to the other sources of time steperror, when a second order Runge-Kutta solver is used. The error related to the approximate scattering mechanismis the most significant source of error with respect to the time step.

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.

This paper describes the extension of arecently developed numerical solver for the Landau-LifshitzNavier-Stokes (LLNS) equations to binary mixtures in threedimensions. The LLNS equations incorporate thermal fluctuations intomacroscopic hydrodynamics by using white-noise fluxes. Thesestochastic PDEs are more complicated in three dimensions due to thetensorial form of the correlations for the stochastic fluxes and inmixtures due to couplings of energy and concentration fluxes (e.g.,Soret effect). We present various numerical tests of systems in andout of equilibrium, including time-dependent systems, anddemonstrate good agreement with theoretical results and molecularsimulation.

With the pioneering work of [Pardoux and Peng, Syst. Contr. Lett.14 (1990) 55–61; Pardoux and Peng,Lecture Notes in Control and Information Sciences176 (1992) 200–217]. We have at our disposalstochastic processes which solve the so-called backward stochasticdifferential equations. These processes provide us with a Feynman-Kacrepresentation for the solutions of a class of nonlinear partial differential equations (PDEs) which appearin many applications in the field of Mathematical Finance. Therefore thereis a great interest among both practitioners and theoreticians to developreliable numerical methods for their numerical resolution. In this survey, we present a number of probabilistic methods forapproximating solutions of semilinear PDEs all based on the correspondingFeynman-Kac representation. We also include a general introduction tobackward stochastic differential equations and their connection with PDEsand provide a generic framework that accommodates existing probabilisticalgorithms and facilitates the construction of new ones.

This paper discusses analytical and numerical issues related toelliptic equations with random coefficients which are generallynonlinear functions of white noise. Singularity issues are avoidedby using the Itô-Skorohod calculus to interpret the interactionsbetween the coefficients and the solution. The solution is constructedby means of the Wiener Chaos (Cameron-Martin) expansions. Theexistence and uniqueness of the solutions are established underrather weak assumptions, the main of which requires only that theexpectation of the highest order (differential) operator is anon-degenerate elliptic operator. The deterministic coefficientsof the Wiener Chaos expansion of the solution solve a lower-triangularsystem of linear elliptic equations (the propagator). This structureof the propagator insures linear complexity of the related numericalalgorithms. Using the lower triangular structure and linearity of thepropagator, the rate of convergence is derived for a spectral/hp finiteelement approximation. The results of related numerical experiments arepresented.