  It is shown that the computational efficiency of the discrete least-squares
(DLS) approximation of solutions of stochastic elliptic PDEs is improved by
incorporating a reduced-basis method into the DLS framework. The goal is to
recover the entire solution map from the parameter space to the finite element
space. To this end, first, a reduced-basis solution using a weak greedy
algorithm is constructed, then a DLS approximation is determined by evaluating
the reduced-basis approximation instead of the full finite element
approximation. The main advantage of the new approach is that one only need
apply the DLS operator to the coefficients of the reduced-basis expansion,
resulting in huge savings in both the storage of the DLS coefficients and the
online cost of evaluating the DLS approximation. In addition, the recently
developed quasi-optimal polynomial space is also adopted in the new approach,
resulting in superior convergence rates for a wider class of problems than
previous analyzed. Numerical experiments are provided that illustrate the
theoretical results.
