Abstract
Multiscale schemes for transferring information from fine to coarse scales are typically based
on homogenization techniques. Such schemes smooth the fine scale features of the underlying
fields, often resulting in the inability to accurately retain the fine scale correlations. In
addition, higher-order statistical moments (beyond mean) of the relevant field variables are
not necessarily preserved. As a superior alternative to averaging homogenization methods, a
wavelet-based scheme for the exchange of information between a reactive and diffusive field in
the context of multiscale reaction-diffusion problems is proposed and analyzed. The scheme is
shown to be efficient in passing information along scales, from fine to coarse, i.e., upscaling as
well as from coarse to fine, i.e., downscaling. It incorporates fine scale statistics (higher-order
moments beyond mean), mainly due to the capability of wavelets to represent fields hierarchically.
Critical to the success of the scheme is the identification of dominant scales containing
the majority of the useful information. The dominant scales in effect specify the coarsest resolution
possible. The scheme is applied in detail to the analysis of a diffusive system with
a chemically reacting boundary. Reactions are simulated using kinetic Monte Carlo (kMC)
and diffusion is solved by finite differences (FDs). Spatial scale differences are present at the
interface of the kMC sites and the diffusion grid. The computational efficiency of the scheme is
compared to results obtained by averaging homogenization, and to results from a benchmark
scheme that ensures spatial scale parity between kMC and FD.
