Multiscale Scientific Computation: Six Year Research Summary

Achi Brandt
Department of Applied Math
The Weizmann Inst. of Science
Rehovot, 76100, Israel


The Gauss Center research on multiscale computational methods is reported, emphasizing main ideas and inter-relations between various fields, and listing the relevant bibliography. The reported areas include: top-efflciency multigrid methods in fluid dynamics; atmospheric flows and data assimilation; feedback optimal control; PDE solvers on unbounded domains; wave/ray methods for highly indefinite equations; rigorous quantitative analysis of multigrid; many-eigenfunction problems and ab-initio quantum chemistry; fast evaluation of integral transforms on adaptive grids; multigrid Dirac solvers; fast inverse-matrix and determinant updates;multiscale Monte-Carlo methods in statistical physics; molecular mechanics (including fast force summation, fast macromolecular energy minimization, Monte-Carlo methods at equilibrium, both for macromolecules and for large ensembles of small molecules, and the combination of small-scale equilibrium with large-scale dynamics); image processing (edge detection and segmentation); and tomography (medical imaging and radar reconstruction).

Key words. Scientific computation, multiscale, multi-resolution, multigrid, fluid dynamics, atmospheric flows, data assimilation, optimal control, wave problems, Dirac equations, inverse matrix, Schrodinger operator, Monte-Carlo algorithms, critical slowing down, molecular mechanics, fast force summation, energy minimization, integro-differential equations, tomography , medical imaging, radar, image processing, edge detection, segmentation, algebraic multigrid.

AMS subject classification. 34A50, 35A40, 44-04, 45{04, 65C05, 65F10, 65F15, 65F40, 65K10, 65M30, 65M50, 65M55, 65N22, 65N25, 65N38, 65N55, 65R10, 65R20, 65Y05, 68U10, 70-08, 76-04, 76M20, 81-08, 81T80, 82-08, 82B80, 82C80, 92E99

Contributed June 7, 1999.