title: Multilevel Upscaling through Variational Coarsening

abstract.
In many applications, homogenization (or upscaling) techniques are
necessary to develop computationally feasible models on scales coarser
than the variation of the coefficients of the continuum model. The
accuracy of such techniques depends dramatically on assumptions that
underlie the particular upscaling methodology used. For example,
decoupling of fine- and coarse-scale effects in the underlying medium may
utilize artificial internal boundary conditions on sub-cell
problems. Such assumptions, however, may be at odds with the true,
fine-scale solution, yielding coarse-scale errors that may be
unbounded.

The development of robust efficient multilevel solvers, such as
multigrid, has naturally led to the development of general multiscale
concepts, such as operator-induced variational coarsening. Such
approaches implicitly treat multiscale aspects of the fine-scale model as
they generate a sequence of coarser representations and, thus,
implicitly create multiscale basis functions and coarse-scale
closures.  I will discuss a multilevel upscaling (MLUPS) algorithm based
on Dendy's Black Box Multigrid method and discuss its
relationship to the multiscale finite element (MSFEM) algorithm.  This
research is in collaboration with J. David Moulton from Los Alamos
National Laboratory.