| Coefficients of PDE's are often changing across many spatial or temporal scales, whereas we might be interested in the behavior of the solution only on some relatively coarse scale. The multiresolution strategy for reduction and homogenization provides a method for finding an equation for the projection of the solution to a coarse scale. This equation explicitly incorporates the fine-scale behavior of the coefficients.; We present the multiresolution strategy for reduction and homogenization of differential equations, and apply it to linear wave equations in which the coefficients describe a layered medium (the problem reduces to a system of ordinary differential equations in this case) and to elliptic partial differential equations. For the layered-medium wave equations, we discuss and compare the multiresolution approach with classical techniques. For elliptic operators, it is known that the non-standard form has fast off-diagonal decay and the rate of decay is controlled by the number of vanishing moments of the wavelet basis. We prove that if an appropriate (e.g. high order) basis is used, 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 eigevenvalues of strictly elliptic operators. We also introduce a modified reduction procedure which preserves the small eigenvalues with greater accuracy than the usual reduction procedure and obtain estimates for the perturbation of those eigenvalues. |
