Multiscale computation: from fast solvers to systematic upscaling

Most fundamental problems in physics, chemistry and engineering involve computation too hard even for future supercomputers, if conventional mathematical approaches are used. The reason is always a product of several complexity factors associated with the wide range of space and time scales characteristic to such problems. Each of these complexity factors can in principle be removed by various multiscale algorithms, i.e., employing separate processing at each scale of the problem, combined with interscale iterative interactions. Starting from multigrid fast solvers for discretized partial differential equations and from renormalization group methods in theoretical physics, the multiscale computational methodology has recently been extended to many other areas and new types of problems: linear and highly nonlinear, deterministic and stochastic, discrete and continuous, with particles and macromolecules, graphs and images.