The design of most adaptive wavelet methods for elliptic partial differential equations follows a general concept proposed by A. Cohen, W. Dahmen and R. DeVore in [3, 4]. The essential steps are: transformation of the variational formulation into the well-conditioned infinite-dimensional l 2 problem, finding of the convergent iteration process for the l 2 problem and finally derivation of its finite dimensional version which works with an inexact right hand side and approximate matrix-vector multiplications. In our contribution, we shortly review all these parts and wemainly pay attention to approximate matrix-vector multiplications. Effective approximation of matrix-vector multiplications is enabled by an off-diagonal decay of entries of the wavelet stiffness matrix. We propose here a new approach which better utilize actual decay of matrix entries.