Efficient adaptive operator application on wavelet expansions

One key step in solving partial differential equations using adaptive wavelet methods is the ability to efficiently apply an operator to a wavelet expansion. Whereas this problem has been generally solved in theory, the known solution is still a little slow and hard to implement. Here, we propose a more practical algorithm for a useful set of linear operators containing in particular all linear differential operators. Our algorithm is general as it works for many wavelet systems. It is fast as it is linear with a small constant factor. It is exact as coefficients are computed without approximation. It is simple since the matrix entries of the operator need to be known only for wavelets at the same scale.