In many practical problems coefficients of PDEs are changing across many spatial or temporal scales, whereas we might be interested in the behavior of the solution only on some relatively coarse scale. We approach the problem of capturing the influence of fine scales on the behavior of the solution on a coarse scale using the multiresolution strategy. Considering adjacent scales of a multiresolution analysis, we explicitly eliminate variables associated with the finer scale, which leaves us with a coarse-scale equation. We use the term reduction to designate a recursive application of this procedure over a finite number of scales.
We present a multiresolution strategy for reduction of self-adjoint, strictly elliptic operators in one and two dimensions. It is known that the non-standard form for a wide class of operators has fast off-diagonal decay and the rate of decay is controlled by the number of vanishing moments of the wavelet. We prove that the reduction procedure preserves the rate of decay over any finite number of scales and therefore results in sparse matrices for computational purposes. Furthermore, the reduction procedure approximately preserves small eigenvalues of self-adjoint, strictly elliptic operators. We also introduce a modified reduction procedure which preserves the small eigenvalues with greater accuracy than the standard reduction procedure and obtain estimates for the perturbation of those eigenvalues. Finally, we discuss potential extensions of the reduction procedure to parabolic and hyperbolic problems.