  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 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 computationsotherwise. 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.
