Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Parabolic PDEs: Application to Real‐Time Bayesian Parameter Estimation

By C. Nguyen, Gianluigi Rozza, Dinh Bao Phuong Huynh, Anthony T. Patera. 2010

96 citations recorded by the source corpus.

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.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/eacaa9789e0daa112a83bd361d3f41c7db08d4fd

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Citations in the corpus

Stabilized reduced basis method for parametrized advection-diffusion PDEs (2014, methodology): “2 Sampling strategy and a posteriori error estimates To construct the reduced basis in the time-dependent case, we will follow the so called PODgreedy approach [19,40,45].”
Sparse-grid, reduced-basis Bayesian inversion (2015, result): “Based on previous results on sparsity of solution families of affine-parametric operator equations from [20, 28, 52, 31, 29, 30, 29, 30], Reduced Basis Methods (RBM), or more general model reduction techniques, have been applied to solve Bayesian inverse problems recently in [39, 35, 25, 33, 21].”
Stochastic Collocation Algorithms Using l1-Minimization for Bayesian Solution of Inverse Problems (2015, methodology): “This is then used for posterior evaluation or sampling [1, 16, 17, 20, 23, 30, 39].”
Certified reduced basis approximation for parametrized partial differential equations and applications (2011, methodology): “An efficient combination of the two techniques greedy-POD in parameter-time has been proposed [22,23] and is currently used for the treatment of parabolic problems [24]; see Sect.”
A Stochastic Newton MCMC Method for Large-Scale Statistical Inverse Problems with Application to Seismic Inversion (2012, methodology): “This is then used for posterior evaluation or sampling [3, 10, 16, 35, 46].”