A Training Set Subsampling Strategy for the Reduced Basis Method

Abstract

We present a subsampling strategy for the offline stage of the Reduced Basis Method. The approach is aimed at bringing down the considerable offline costs associated with using a finely-sampled training set. The proposed algorithm exploits the potential of the pivoted QR decomposition and the discrete empirical interpolation method to identify important parameter samples. It consists of two stages. In the first stage, we construct a low-fidelity approximation to the solution manifold over a fine training set. Then, for the available low-fidelity snapshots of the output variable, we apply the pivoted QR decomposition or the discrete empirical interpolation method to identify a set of sparse sampling locations in the parameter domain. These points reveal the structure of the parametric dependence of the output variable. The second stage proceeds with a subsampled training set containing a by far smaller number of parameters than the initial training set. Different subsampling strategies inspired from recent variants of the empirical interpolation method are also considered. Tests on benchmark examples justify the new approach and show its potential to substantially speed up the offline stage of the Reduced Basis Method, while generating reliable reduced-order models.

Adaptive Interpolatory MOR by Learning the Error Estimator in the Parameter Domain

“…11: end while solution to the dual system by considering the right hand side as the i-th row vector of C(µ), namely, C T (: , i) in Eq. (9). Correspondingly, the residual r pr ( μ) is obtained by solving Eq.…”

“…This is due to the fact that at each iteration of the greedy algorithm, an error estimator needs to be repeatedly computed for all the samples in the training set. Many adaptive training techniques have been proposed recently for RBM [9,18,19,23]. In contrast, no efficient training techniques are proposed for the interpolatory MOR methods, though similar greedy algorithms using fixed training sets are proposed in [11,13].…”

“…In contrast, no efficient training techniques are proposed for the interpolatory MOR methods, though similar greedy algorithms using fixed training sets are proposed in [11,13]. In this work, we extend the adaptive training technique in [9] for RBM to an adaptive training technique for the interpolatory MOR methods in [11,13].…”

“…The idea is similar to the one in [9] for the RBM. However, in [9], an error estimator in time domain is used, where the inf-sup constant needs to be computed for each parameter in the training set, which is computationally inefficient for largescale systems. In this work, we use an inf-sup-constant-free error estimator newly proposed in [13].…”

“…The error estimator is evaluated only on a coarse training set including a few parameter samples. The algorithm is an extension of the work in [9] to interpolatory model order reduction (MOR) in frequency domain. Beyond the work in [9], we use a newly proposed inf-sup-constant-free error estimator in the frequency domain [13], which is often much tighter than the error estimator using the inf-sup constant.…”