rbm

In this paper we present reduced basis (RB) approximations and associated rigorous a posteriori error bounds for the parametrized unsteady Boussinesq equations. The essential ingredients are Galerkin projection onto a low-dimensional space associated with a smooth parametric manifold ? to provide dimension reduction; an efficient proper orthogonal decomposition?Greedy sampling method for identification of optimal and numerically stable approximations ? to yield rapid convergence; accurate (online) calculation of the solution-dependent stability factor by the successive constraint method ? to quantify the growth of perturbations/residuals in time; rigorous a posteriori bounds for the errors in the RB approximation and associated outputs ? to provide certainty in our predictions; and an offline?online computational decomposition strategy for our RB approximation and associated error bound ? to minimize marginal cost and hence achieve high performance in the real-time and many-query contexts. The method is applied to a transient natural convection problem in a two-dimensional “complex” enclosure ? a square with a small rectangle cutout ? parametrized by Grashof number and orientation with respect to gravity. Numerical results indicate that the RB approximation converges rapidly and that furthermore the (inexpensive) rigorous a posteriori error bounds remain practicable for parameter domains and final times of physical interest.

In this paper we present rigorous a posteriori L 2 error bounds for reduced basis approximations of the unsteady viscous Burgers? equation in one space dimension. The a posteriori error estimator, derived from standard analysis of the error-residual equation, comprises two key ingredients?both of which admit efficient Offline-Online treatment. The first is a sum over timesteps of the square of the dual norm of the residual; the second is an accurate upper bound (computed by the Successive Constraint Method) for the exponential-in-time stability factor. These error bounds serve both Offline for construction of the reduced basis space by a new POD-Greedy procedure and Online for verification of fidelity. The a posteriori error bounds are practicable for final times (measured in convective units) T?O(1) and Reynolds numbers ? ?1?1; we present numerical results for a (stationary) steepening front for T=2 and 1?? ?1?200.

We report here on the recent application of a now classical general reduction technique, the Reduced-Basis (RB) approach initiated by C. Prud?homme et al. in J. Fluids Eng. 124(1), 70?80, 2002, to the specific context of differential equations with random coefficients. After an elementary presentation of the approach, we review two contributions of the authors in Comput. Methods Appl. Mech. Eng. 198(41?44), 3187?3206, 2009, which presents the application of the RB approach for the discretization of a simple second order elliptic equation supplied with a random boundary condition, and in Commun. Math. Sci., 2009, which uses a RB type approach to reduce the variance in the Monte-Carlo simulation of a stochastic differential equation. We conclude the review with some general comments and also discuss possible tracks for further research in the direction.