Abstract
The Reduced Basis Method (RBM) is a popular certified model reduction
approach for solving parametrized partial differential equations. One critical
stage of the \textit{offline} portion of the algorithm is a greedy algorithm,
requiring maximization of an error estimate over parameter space. In practice
this maximization is usually performed by replacing the parameter domain
continuum with a discrete "training" set. When the dimension of parameter space
is large, it is necessary to significantly increase the size of this training
set in order to effectively search parameter space. Large training sets
diminish the attractiveness of RBM algorithms since this proportionally
increases the cost of the offline {phase}.
In this work we propose novel strategies for offline RBM algorithms that
mitigate the computational difficulty of maximizing error estimates over a
training set. The main idea is to identify a subset of the training set, a
"surrogate parameter domain" (SPD), on which to perform greedy algorithms. The
SPD's we construct are much smaller in size than the full training set, yet our
examples suggest that they are accurate enough to represent the solution
manifold of interest at the current offline RBM iteration. We propose two
algorithms to construct the SPD: Our first algorithm, the Successive
Maximization Method (SMM) method, is inspired by inverse transform sampling for
non-standard univariate probability distributions. The second constructs an SPD
by identifying pivots in the Cholesky Decomposition of an approximate error
correlation matrix. We demonstrate the algorithm through numerical experiments,
showing that the algorithm is capable of accelerating offline RBM procedures
without degrading accuracy, assuming that the solution manifold has low
Kolmogorov width.