## Reduced basis approximation and a posteriori error estimation for parametrized parabolic PDEs; Application to real-time Bayesian parameter estimation

In this paper we consider reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized linear and non-linear parabolic partial differential equations. The essential ingredients are Galerkin projection onto a low-dimensional space associated with a smooth parametric manifold'' --- dimension reduction; efficient and effective Greedy and POD-Greedy sampling methods for identification of optimal and numerically stable approximations --- rapid convergence; rigorous and sharp a posteriori error bounds (and associated stability factors) for the linear-functional outputs of interest --- certainty; and Offline-Online computational decomposition strategies --- minimum marginal cost for high performance in the real-time/embedded (e.g., parameter estimation, control) and many-query (e.g., design optimization, uncertainty quantification, multi- scale) contexts. In this paper we first present reduced basis approximation and a posteriori error estimation for general linear parabolic equations and subsequently for a nonlinear parabolic equation, the incompressible Navier-- Stokes equations. We then present results for the application of our (parabolic) reduced basis methods to Bayesian parameter estimation: detection and characterization of a delamination crack by transient thermal analysis.

Editor(s):
Tenorio, L.
van Bloemen Waanders, B.
Mallick, B.
Willcox, K.
Biegler, L.
Biros, G.
Ghattas, O.
Heinkenschloss, M.
Keyes, D.
Marzouk, Y.
Published in:
Large Scale Inverse Problems and Quantification of Uncertainty, Chapter 8, 151-178
Year:
2010
Publisher:
UK, John Wiley & Sons
Keywords:
Note:
EPFL-IACS report 11.2008
Laboratories: