A Greedy Method for Solving Classes of PDE Problems

Abstract

Motivated by the successful use of greedy algorithms for Reduced Basis Methods, a greedy method is proposed that selects N input data in an asymptotically optimal way to solve well-posed operator equations using these N data. The operator equations are defined as infinitely many equations given via a compact set of functionals in the dual of an underlying Hilbert space, and then the greedy algorithm, defined directly in the dual Hilbert space, selects N functionals step by step. When N functionals are selected, the operator equation is numerically solved by projection onto the span of the Riesz representers of the functionals. Orthonormalizing these yields useful Reduced Basis functions. By recent results on greedy methods in Hilbert spaces, the convergence rate is asymptotically given by Kolmogoroff N-widths and therefore optimal in that sense. However, these N-widths seem to be unknown in PDE applications. Numerical experiments show that for solving elliptic second-order Dirichlet problems, the greedy method of this paper behaves like the known P-greedy method for interpolation, applied to second derivatives. Since the latter technique is known to realize Kolmogoroff N-widths for interpolation, it is hypothesized that the Kolmogoroff N-widths for solving second-order PDEs behave like the Kolmogoroff N-widths for second derivatives, but this is an open theoretical problem