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Motivated by the successful use of greedy algorithms for Reduced BasisMethods, a greedy method is proposed that selects N input data in anasymptotically optimal way to solve well-posed operator equations using these Ndata. The operator equations are defined as infinitely many equations given viaa compact set of functionals in the dual of an underlying Hilbert space, andthen the greedy algorithm, defined directly in the dual Hilbert space, selectsN functionals step by step. When N functionals are selected, the operatorequation is numerically solved by projection onto the span of the Rieszrepresenters of the functionals. Orthonormalizing these yields useful ReducedBasis functions. By recent results on greedy methods in Hilbert spaces, theconvergence rate is asymptotically given by Kolmogoroff N-widths and thereforeoptimal in that sense. However, these N-widths seem to be unknown in PDEapplications. Numerical experiments show that for solving elliptic second-orderDirichlet problems, the greedy method of this paper behaves like the knownP-greedy method for interpolation, applied to second derivatives. Since thelatter technique is known to realize Kolmogoroff N-widths for interpolation, itis hypothesized that the Kolmogoroff N-widths for solving second-order PDEsbehave like the Kolmogoroff N-widths for second derivatives, but this is anopen theoretical problem.

A Greedy Method for Solving Classes of PDE Problems