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It is shown that the computational efficiency of the discrete least-squares(DLS) approximation of solutions of stochastic elliptic PDEs is improved byincorporating a reduced-basis method into the DLS framework. The goal is torecover the entire solution map from the parameter space to the finite elementspace. To this end, first, a reduced-basis solution using a weak greedyalgorithm is constructed, then a DLS approximation is determined by evaluatingthe reduced-basis approximation instead of the full finite elementapproximation. The main advantage of the new approach is that one only needapply the DLS operator to the coefficients of the reduced-basis expansion,resulting in huge savings in both the storage of the DLS coefficients and theonline cost of evaluating the DLS approximation. In addition, the recentlydeveloped quasi-optimal polynomial space is also adopted in the new approach,resulting in superior convergence rates for a wider class of problems thanprevious analyzed. Numerical experiments are provided that illustrate thetheoretical results.

An improved discrete least-squares/reduced-basis method for parameterized elliptic PDEs