TItle: Multiscale Computation: From Multigrid solvers to systematic
upscaling

Speaker: Achi Brandt

    Department of Applied Mathematics & Computer Science
      The Weizmann Institute of Science, Israel

Abstract:

Most numerical methods for solving large-scale systems tend to be
extremely costly, for several general reasons, each of which can in
principle be removed by multiscale algorithms. Algorithms to be
briefly surveyed: fast multigrid solvers for discretized
partial-differential equations (PDEs) and for most other systems of
local equations; fast summation of long-range (e.g., electrostatic)
interactions and fast solvers of integral and inverse PDE problems;
collective computation of many eigenfunctions; slowdown-free Monte
Carlo simulations; multilevel methods of global optimization; and
general procedures for "systematic upscaling".

SYSTEMATIC UPSCALING is a methodical approach for deriving, scale
after scale, collective variables and governing numerical equations
(or transition probabilities rules) at increasingly larger scales,
starting from a microscopic scale where first-principle laws are
known. Iterating back and forth between all levels allows the
computation at each scale to be short and confined to small "windows".

The multiscale methods are key to removing computational bottlenecks
in many areas of science and engineering, such as: QCD (elementary
particle) computation; ab-initio quantum chemistry real-time path
integrals; density-functional calculation of electronic structures;
molecular dynamics of fluids, materials and macromolecules; turbulent
flows; tomography (medical-imaging reconstruction); image segmentation
and picture recognition; and various large-scale graph optimization,
clustering and classification problems.  Future directions will be
outlined.