General linear scaling, independent of physical
localization, and even much better scalings, can be
obtained by multiscale (e.g., multigrid) algorithms,
based on systematic transitions to increasingly
coarser-grained equations. Multigrid methods provide
one-shot non-linear solvers, eliminating the need
for self-consistency iterations. Local grid
refinements are conveniently incorporated for little
extra computations. The mathematical localization of
eigenfunctions will be discussed. Systematic
upscaling from electronic structures to molecular
dynamics will be described.
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