Computation of differential operators in wavelet coordinates

In [Found. Comput. Math., 2 (2002), pp. 203-245], Cohen, Dahmen, and DeVore proposed an adaptive wavelet algorithm for solving general operator equations. Assuming that the operator defines a boundedly invertible mapping between a Hilbert space and its dual, and that a Riesz basis of wavelet type for this Hilbert space is available, the operator equation is transformed into an equivalent well-posed infinite matrix-vector system. This system is solved by an iterative method, where each application of the infinite stiffness matrix is replaced by an adaptive approximation. It was shown that if the errors of the best linear combinations from the wavelet basis with $N$ terms are $\mathcal{O}(N^{-s})$ for some $s>0$, which is determined by the Besov regularity of the solution and the order of the wavelet basis, then approximations yielded by the adaptive method with $N$ terms also have errors of $\mathcal{O}(N^{-s})$. Moreover, their computation takes only $\mathcal{O}(N)$ operations, provided $s<s^*$, with $s^*$ being a measure of how well the infinite stiffness matrix with respect to the wavelet basis can be approximated by computable sparse matrices. Under appropriate conditions on the wavelet basis, for both differential and singular integral operators and for the relevant range of $s$, in [SIAM J. Math. Anal., 35(5) (2004), pp. 1110-1132] we showed that $s^{\ast}>s$, assuming that each entry of the stiffness matrix is exactly available at unit cost.

Generally these entries have to be approximated using numerical quadrature. In this paper, restricting ourselves to differential operators, we develop a numerical integration scheme that computes these entries giving an additional error that is consistent with the approximation error, whereas in each column the average computational cost per entry is ${\mathcal O}(1)$. As a consequence, we can conclude that the adaptive wavelet algorithm has optimal computational complexity.