  Compressive sensing has become a powerful addition to uncertainty
quantification in recent years. This paper identifies new bases for random
variables through linear mappings such that the representation of the quantity
of interest is more sparse with new basis functions associated with the new
random variables. This sparsity increases both the efficiency and accuracy of
the compressive sensing-based uncertainty quantification method. Specifically,
we consider rotation-based linear mappings which are determined iteratively for
Hermite polynomial expansions. We demonstrate the effectiveness of the new
method with applications in solving stochastic partial differential equations
and high-dimensional ($\mathcal{O}(100)$) problems.
