A Multiresolution Strategy for Numerical HomogenizationM

The term homogenization refers to a collection of methods for the description of the relations between the equations of \microstructure" and those of \macrostructure". It is a diverse eld since there is usually more than one way to formulate the problem. We refer to 3] , 8] and references therein for examples of various formulations and solutions of problems of homogenization. Ordinarily, one considers at most two \scales" of variation of the coeecients of the equations governing the microscopic behavior; the goal is to extract the quantities describing the behavior at a coarse scale (maybe as a limit). Thus, the behaviour at possible intermediate scales has been ignored basically due to the lack of tools for its description. Recently the notion of Multiresolution Analysis (MRA) was introduced by Meyer 9] and Mallat 6] as a general framework for construction of the wavelet bases. Using MRA, the notion of the non-standard representation of operators was introduced in 1]. For a wide class of operators (e.g. Calderr on-Zygmund or pseudo-diierential operators), the non-standard form is sparse and permits fast algorithms for evaluation of these operators on functions. The non-standard form also permits an explicit description of the interaction between the scales and, thus, appear to be an appropriate tool for the problems of homogenization. This paper is the rst of a series where we use MRA to develop a multiresolution strategy for the numerical solution and homogenization of equations. We consider linear systems of integral equations in one variable, including those equivalent to ODE's and semi-discrete versions of PDE's. In other papers of this series, we plan to consider nonlinear integral equations and equations in more than one variable. The linear homogenization procedure is exact in that it yields a linear system of equations whose solutions are projections on the coarse scale of the solutions of the