Most numerical methods for solving physical problems tend to be extremely costly, for several general reasons that will be explained. Model studies have shown that each of these reasons can in principle be removed by multiscale (e.g., multigrid) algorithms. These algorithms
employ separate processing at each scale of the physical space, combined with interscale iterative interactions, in ways which use finer scales very sparingly. Having been developed first and well known as solvers for elliptic PDEs, highly efficient multiscale techniques have more recently been developed for non elliptic and time-dependent problems, and for many other types of computational tasks, including: inverse PDE
problems; highly indefinite (e.g., standing wave) equations; Dirac equations in disordered gauge fields; fast computation and updating of
large determinants; general fast integral transforms; integral equations; many body interactions; molecular dynamics of macromolecules
and fluids; many-atom electronic structures; global and discrete-state optimization; practical network and graph problems; image segmentation
and recognition; clusteringin in high dimension and low-dimensional embedding; tomography (medical imaging); fast Monte-Carlo sampling
in statistical physics; real-time path-integral; and general, systematic methods of upscaling (accurate numerical derivation of large-scale equations from microscopic laws).
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