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