A progressive reduced basis/empirical interpolation method for nonlinear parabolic problems

We investigate new developments of the combined reduced-basis and empirical interpolation methods (RB-EIMs) 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 time-dependent 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 trajectory for the first selected value of the parameter. Numerical examples are presented on nonlinear heat transfer problems.