CONVERGENCE RATES FOR GREEDY ALGORITHMS IN REDUCED BASIS METHODS

The reduced basis method was introduced for the accurate online evaluation of solutions to a parameter dependent family of elliptic PDEs. Abstractly, it can be viewed as determining a good n-dimensional space H(n) to be used in approximating the elements of a compact set F in a Hilbert space H. One by now popular computational approach is to find H(n) through a greedy strategy. It is natural to compare the approximation performance of the H(n) generated by this strategy with that of the Kolmogorov widths d(n)(F) since the latter gives the smallest error that can be achieved by subspaces of fixed dimension n. The first such comparisons, given in [A. Buffa et al., ESAIM Math. Model. Numer. Anal., 2011, to appear], show that the approximation error, sigma(n)(F) := dist(F, H(n)), obtained by the greedy strategy satisfies sigma(n)(F) <= Cn2(n)d(n)(F). In this paper, various improvements of this result will be given. Among these, it is shown that whenever d(n)(F) <= Mn(-alpha) for all n > 0 and some M, alpha > 0, we also have sigma(n)(F) <= C(alpha)Mn(-alpha) for all n > 0, where C(alpha) depends only on alpha. Similar results are derived for generalized exponential rates of the form Me(-an alpha). The exact greedy algorithm is not always computationally feasible, and a commonly used computationally friendly variant can be formulated as a weak greedy algorithm. The results of this paper are established for this version as well.