A Progressive Reduced Basis/Empirical Interpolation Method for Nonlinear Parabolic Problems

Abstract

We investigate new developments of the combined Reduced-Basis and Empirical Interpolation Methods (RB-EIM) for parametrized nonlinear parabolic problems. In many situations, the cost of the EIM in the offline stage turns out to be prohibitive since a significant number of nonlinear timedependent problems need to be solved using the high-fidelity (or full-order) model. In the present work, we develop a new methodology, the Progressive RB-EIM (PREIM) method for nonlinear parabolic problems. The purpose is to reduce the offline cost while maintaining the accuracy of the RB approximation in the online stage. The key idea is a progressive enrichment of both the EIM approximation and the RB space, in contrast to the standard approach where the EIM approximation and the RB space are built separately. PREIM uses high-fidelity computations whenever available and RB computations otherwise. Another key feature of each PREIM iteration is to select twice the parameter in a greedy fashion, the second selection being made after computing the high-fidelity solution for the firstly selected value of the parameter. Numerical examples are presented on nonlinear heat transfer problems. * This work is partially supported by Electricité De France (EDF) and a CIFRE PhD fellowship from ANRT † University Paris-Est, CERMICS (ENPC), 77455 Marne la Vallée Cedex 2 and INRIA Paris, parabolic problems). The integer M is called the rank of the EIM and controls the accuracy of the approximation. Although the EIM is performed during the offline stage of the RB method, its cost can become a critical issue since the EIM can require an important number of HF computations for an accurate approximation of the nonlinearity. The cost of the EIM typically scales with the size of the training set P tr .The goal of the present work is to overcome this issue. To this purpose, we devise a new methodology, the Progressive RB-EIM (PREIM) method, which aims at reducing the computational cost of the offline stage while maintaining the accuracy of the RB approximation in the online stage. The key idea is a progressive enrichment of both the EIM approximation and the RB space, in contrast to the standard approach where the EIM approximation and the RB space are built separately. In PREIM, the number of HF computations is at most M , and it is in general much lower than M in a time-dependent context where the greedy selection of the pair (µ, k) to build the EIM approximation (where µ is the parameter and k refers to the discrete time node) can lead to repeated values of µ for many different values of k. In other words, PREIM can select multiple space fields within the same HF trajectory to build the EIM space functions. In this context, only a modest number of HF trajectories needs to be computed, yielding significant computational savings with respect to the standard offline stage. PREIM is driven by convergence criteria on the quality of both the EIM and the RB approximation, as in the standard RB-EIM procedure. Moreover, as the PREIM iteration progresses, more a...

PREIM for nonlinear parabolic problems

“…• Q m : Set of interpolation functions (q j ) 1≤ j≤m three main steps (the detailed algorithm appears in [1]):…”

“…The purpose of PREIM (Progressive RB-EIM) [1] is to reduce the offline costs of nonlinear parabolic reduced order models with accurate RB approximations in the online stage. The key idea is a progressive enrichment of both the EIM approximation and the RB space, in contrast to the standard approach where the EIM approximation and the RB space are built separately.…”

Adaptive basis construction and improved error estimation for parametric nonlinear dynamical systems

“…In addition, the standard EIM needs FOM simulations at all samples in a training set, which is time consuming. This issue is also pointed out in References 14 and 15. In Reference 16, RBM‐EIM is implemented such that both the RB and the interpolation basis are updated simultaneously.…”

“…Recently, adaptive schemes have been proposed in References 14 and 15 for RB and EIM bases construction. The authors in Reference 15 propose the simultaneous EIM‐RB (SER) method of simultaneously enriching the RB and EIM bases for nonlinear but stationary problems.…”

“…A progressive EIM (PREIM) method for nonlinear, dynamical (time‐dependent) systems is proposed in Reference 14. A special case of the method is a natural extension of the SER method in Reference 15.…”