Certified Reduced Basis Methods for Parametrized Partial Differential Equations
We discuss reduced basis approximation and associated a posteriori error estimation for rapid and reliable solution of parametrized partial differential equations in the many-query and real-time contexts. Applications include parameter estimation, model validation, uncertainty quantification, design and control, and multiscale analysis. We illustrate our methods with examples drawn from heat transfer, solid mechanics, acoustics, and fluid dynamics.
The reduced basis approach can be pursued first at the component level and then at the system level.
At the component level, the crucial ingredients are rapidly convergent Galerkin approximations over a space spanned by “snapshots” on the parametrically induced solution manifold; Empirical Interpolation treatment of non-affine parameter dependence; Successive Constraint constructions for (coercivity, inf-sup) stability-constant lower bounds; rigorous and sharp a posteriori error bounds for the field and outputs/quantities of interest; efficient (h-p) POD (in time)/Greedy (in parameter) selection of quasi-optimal samples; and construction-evaluation and Offline-Online computational procedures.
We also describe and demonstrate recent implementations on hierarchical architectures: the Offline stage is associated to a large parallel supercomputer; the Online stage is associated to a “slim” deployed or embedded platform in the field — for real-time input-output prediction as well as visualization. In our current implementations the Offine computations are performed on the Ranger TeraGrid supercomputer and the Online computations are performed on Android Smartphones.
At the system level we appeal to a static condensation formulation of the reduced basis element method. We develop (Offline) reduced basis models for components in a library which are then (Online) connected via ports to synthesize a full system: computationally, the Offline stage provides reduced basis bubble approximations which in the Online stage are then incorporated into a Schur complement defined over (relatively few) global ports. The treatment of interfaces ensures complete “Lego” interchangeability of components in the Online stage and hence considerable flexibility in geometric and parametric variation. Furthermore, rigorous a posteriori error bounds are provided: standard reduced basis error bounds at the component level are aggregated by matrix perturbation approaches into error bounds at the system level. The method shares many features of earlier proposals within the model order reduction and domain decomposition communities which we shall highlight in the presentation.
Finally we consider the application of reduced basis methods in a new frequentistic mathematical model validation framework. The composite hypothesis testing formulation exploits the rapid many-query evaluation of the reduced basis approximation to yield real-time results; the a posteriori error bounds serve to rigorously distinguish between reduced basis and mathematical model contributions to bias. The formulation also includes a new representation of prediction-experiment misfit which permits, through the Hotelling T 2 statistic, estimation of experimental noise — assumed Gaussign but not necessarily homoscedastic or uncorrelated — with relatively few data/realizations. Connections to other frequentistic validation approaches will be described.
Work in collaboration with Phuong Huynh, David Knezevic, and Jens Eftang.