The course will present the basics of the multiscale computational methodology,
with further lectures and/or group interactions in more specific directions of
interest to participants, possibly leading to future collaborations.
The multiscale computational methodology is a systematic approach,
based on multigrid and renormalizationgroup ideas, to achieve efficient calculations
of physical systems that include very many degrees of freedom (particle
locations, discretefunction values, etc.). It includes fast multigrid solvers for
discretized partialdifferential equations (as well as other large systems of local
equations); collective computation of many
eigenfunctions; slowdownfree 'Monte
Carlo (MC) simulators; multilevel methods of global optimization and general
procedures for "systematic upscaling".
Systematic upscaling means the methodical derivation, scale after scale,
of increasinglylargerscale numerical "laws" (discrete equations or statistical "actions",
generally in the form of numerical tables), starting at a microscopic scale
where firstprinciple laws are known and leading to processing rules of collective
variables at much larger scales. Using a small (e.g., 2 or 3) coarsetofine scale
ratio at each coarsening step serves to avoid severe computational slowdowns.
The multiscale computational methods are applicable to many heavyweight
nano problems,such as: densityfunctional calculation of electronic structures and
derivation of force fields for moleculardynamics (MD) and molecular static (MS)
simulations; acceleration and upscaling of various MC, kinetic MC, MD and MS
simulations of fluids, condensed matter and macromolecules; fast summation of
longrange (e.g., electrostatic) interactions; and fast solvers for steadystate and
timeimplicit equations of various continuum and mixed continuumatomistic models
of materials, species concentrations, moving interfaces, etc. Systematic upscaling
is particularly important, since typical nano structures are complicated and
lack constitutive equations, and the associated problems involve a wide range of
scales.
Some of these topics, as well as multiscale algorithms in related areas such
as global optimization, medical imaging and image processing, may be chosen for
indepth lectures and interactions during or following the course.