  Sparse polynomial chaos expansions (PCE) are a popular surrogate modelling
method that takes advantage of the properties of PCE, the sparsity-of-effects
principle, and powerful sparse regression solvers to approximate computer
models with many input parameters, relying on only few model evaluations.
Within the last decade, a large number of algorithms for the computation of
sparse PCE have been published in the applied math and engineering literature.
We present an extensive review of the existing methods and develop a framework
for classifying the algorithms. Furthermore, we conduct a unique benchmark on a
selection of methods to identify which approaches work best in practical
applications. Comparing their accuracy on several benchmark models of varying
dimensionality and complexity, we find that the choice of sparse regression
solver and sampling scheme for the computation of a sparse PCE surrogate can
make a significant difference, of up to several orders of magnitude in the
resulting mean-squared error. Different methods seem to be superior in
different regimes of model dimensionality and experimental design size.
