On wavelet-based algorithms for solving differential equations

We describe an order N method for computing the Green’s function of the two-point boundary value problem for elliptic differential operators in the wavelet “system of coordinates.” For simplicity, we consider the ordinary O(h²) finite-difference scheme, and use wavelets only to perform the “linear algebra.” Our main tool is the diagonal preconditioning available for the periodized differential operators in the wavelet bases.