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

In this chapter 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 Boyaval et al. (2008), multi-scale Boyaval (2008); Nguyen (2008b)) contexts. In this paper we first present reduced basis approximation and a posteriori error estimation Prud’homme et al. (2002); Rozza et al. (to appear 2008) for general linear parabolic equations — building on Grepl and Patera (2005); Haasdonk and Ohlberger (2008) — and subsequently for a nonlinear parabolic equation, the incompressible Navier–Stokes equations — building on Nguyen et al. (2008). 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 Grepl (2005); Starnes (2002).