Metric based upscaling

Heterogeneous multi scale structures can be found everywhere in nature. The number of operations needed to simulate the physics of these structures blows up exponentially with the number scales. Numerous upscaling techniques have been developed to describe these systems at a macroscopic level, these techniques are based on transfer of averaged quantities from a microscopic level to a macroscopic level.

We show that half of the information is lost in the process: an upscaled metric and an upscaled measure of volumes. We address this issue by proposing a new method preserving that information. It allows us to solve numerically massively multi scale partial differential equations (characterized by broad and continuous spectrum of heterogeneous scales, with no assumption of periodicity or egodicity at a given scale)

As the spectrum of scales involved in a PDE broadens, its solutions lose their regularity. However we can define a highly irregular (multi-fractal in the limit "infinite number of scales") metric such that the solutions remain differentiable with respect to that metric. This allows the upscaling of the PDE (or SDE) with commutative operators and the downscaling of solutions by Taylor expansion with respect to the new metric.