Abstract

Multiscale Computational Methods

Achi Brandt
Weizmann Institute of Science
Applied Mathematics and Computer Science

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.