The reduced basis method for incompressible viscous flow calculations
SIAM Journal on Scientific and Statistical Computing
A reduced-order method for simulation and control of fluid flows
Journal of Computational Physics
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Journal of Scientific Computing
ISUMA '03 Proceedings of the 4th International Symposium on Uncertainty Modelling and Analysis
SIAM Journal on Scientific Computing
Inverse Problem Theory and Methods for Model Parameter Estimation
Inverse Problem Theory and Methods for Model Parameter Estimation
Journal of Computational Physics
Journal of Computational Physics
SIAM Journal on Scientific Computing
Output-based space-time mesh adaptation for the compressible Navier-Stokes equations
Journal of Computational Physics
Journal of Scientific Computing
Output-based mesh adaptation for high order Navier-Stokes simulations on deformable domains
Journal of Computational Physics
Hi-index | 0.02 |
We present a technique for the rapid and reliable prediction of linear-functional outputs of coercive and noncoercive linear elliptic partial differential equations with affine parameter dependence. The essential components are: (i) rapidly convergent global reduced basis approximations - (Galerkin) projection onto a space WN spanned by solutions of the governing partial differential equation at N judiciously selected points in parameter space; (ii) a posteriori error estimation - relaxations of the error-residual equation that provide inexpensive yet sharp bounds for the error in the outputs of interest; and (iii) offiine/online computational procedures - methods which decouple the generation and projection stages of the approximation process. The operation count for the online stage - in which, given a new parameter value, we calculate the output of interest and associated error bound - depends only on N (typically very small) and the parametric complexity of the problem.In this paper we propose a new "natural norm" formulation for our reduced basis error estimation framework that: (a) greatly simplifies and improves our inf-sup lower bound construction (offline) and evaluation (online) - a critical ingredient of our a posteriori error estimators; and (b) much better controls - significantly sharpens - our output error bounds, in particular (through deflation) for parameter values corresponding to nearly singular solution behavior. We apply the method to two illustrative problems: a coercive Laplacian heat conduction problem which becomes singular as the heat transfer coefficient tends to zero; and a non-coercive Helmholtz acoustics problem - which becomes singular as we approach resonance. In both cases, we observe very economical and sharp construction of the requisite natural-norm inf-sup lower bound; rapid convergence of the reduced basis approximation; reasonable effectivities (even for near-singular behavior) for our deflated output error estimators; and significant - several order of magnitude - (online) computational savings relative to standard finite element procedures.