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Reduced Basis Greedy Selection Using Random Training Sets

Reduced bases have been introduced for the approximation of parametrized PDEs in applications where many online queries are required. Their numerical efficiency for such problems has been theoretically confirmed in \cite{BCDDPW,DPW}, where it is shown that the reduced basis space $V_n$ of dimension $n$, constructed by a certain greedy strategy, has approximation error similar to that of the optimal space associated to the Kolmogorov $n$-width of the solution manifold. The greedy construction of the reduced basis space is performed in an offline stage which requires at each step a maximization of the current error over the parameter space. For the purpose of numerical computation, this maximization is performed over a finite {\em training set} obtained through a discretization. of the parameter domain. To guarantee a final approximation error $\varepsilon$ for the space generated by the greedy algorithm requires in principle that the snapshots associated to this training set constitute an approximation net for the solution manifold with accuracy or order $\varepsilon$. Hence, the size of the training set is the $\varepsilon$ covering number for $\mathcal{M}$ and this covering number typically behaves like $\exp(C\varepsilon^{-1/s})$ for some $C>0$ when the solution manifold has $n$-width decay $O(n^{-s})$. Thus, the shear size of the training set prohibits implementation of the algorithm when $\varepsilon$ is small. The main result of this paper shows that, if one is willing to accept results which hold with high probability, rather than with certainty, then for a large class of relevant problems one may replace the fine discretization by a random training set of size polynomial in $\varepsilon^{-1}$. Our proof of this fact is established by using inverse inequalities for polynomials in high dimensions.