An adaptive and efficient greedy procedure for the optimal training of parametric reduced‐order models

Abstract

SUMMARYAn adaptive and efficient approach for constructing reduced-order models (ROMs) that are robust to changes in parameters is developed. The approach is based on a greedy sampling of the underlying high-dimensional model (HDM) together with an efficient procedure for exploring the configuration space and identifying parameters for which the error is likely to be high. Because this exploration is based on a surrogate model for an error indicator, it is amenable to a fast training phase. Furthermore, a model for the exact error based on the error indicator is constructed and used to determine when the greedy procedure reaches a desired error tolerance. An efficient procedure for updating the reduced-order basis constructed by proper orthogonal decomposition is also introduced in this paper, avoiding the cost associated with computing largescale singular value decompositions. The proposed procedure is then shown to require less evaluations of a posteriori error estimators than the classical procedure in order to identify locations of the parameter space to be sampled. It is illustrated on the training of parametric ROMs for three linear and nonlinear mechanical systems, including the realistic prediction of the response of a V-hull vehicle to underbody blasts. Copyright

Somea posteriorierror bounds for reduced-order modelling of (non-)parametrized linear systems

“…However, one can use sparse grids or other adaptive sampling techniques [15,27,47] to avoid the exponential growth. Nevertheless, suffering from curse of dimensionality is a common problem with almost all PMOR methods and algorithms.…”

PEBL-ROM: Projection-error based local reduced-order models

“…In addition to this single deterministic system, parameters can also enter the system and hence the sampling of snapshots does typically involve both the choice of time instances and parameter values. The choice of sampling parameters is crucial to the definition of an accurate parametric ROM but is not the focus of the present paper and the reader is referred to the following references [14][15][16]. The dynamical system, suitable solver and sampling procedure are assumed to be given in the present work.…”

Accurate Solution of Bayesian Inverse Uncertainty Quantification Problems Combining Reduced Basis Methods and Reduction Error Models

“…ROM uncertainty is indeed quite similar to the so-called model form uncertainty, arising whenever a limited understanding of the modeled process is available [18,17]. Concerning the modeling of the reduction error itself, different approaches to approximation have been considered very recently: the so-called approximation error model [1,26], Gaussian process (GP)-based calibration (or GP machine learning) [39], interpolant-based error indicators [33], and regression-based error indicators [13]. In all of these cases, a statistical representation of the ROM error through calibration experiments is used to model the difference between the full-order and the lower-order model.…”

“…In contrast to the approaches taken in [13,33], where the objective was to construct reduction error models that could be used to train or adapt the ROM accordingly in the case that no other error estimator was readily available, our proposed approach is instead aimed at the accurate solution of inverse problems using ROMs, and may use existing ROM error bounds as additional information, if available. In previous work [21] we have shown that using low-fidelity ROMs as surrogates for solving inverse problems can lead to biased and overly optimistic posterior distributions.…”