Multiscale solvers and systematic upscaling in computational physics

doi:10.1016/j.cpc.2005.03.097

Computer Physics Communications

Volume 169, Issues 1–3, 1 July 2005, Pages 438-441

https://doi.org/10.1016/j.cpc.2005.03.097 Get rights and content

Abstract

Multiscale algorithms can overcome the scale-born bottlenecks that plague most computations in physics. 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 multigrid solvers for partial differential equations, highly efficient multiscale techniques have more recently been developed 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 (as needed in QCD); fast integral transforms; integral equations; astrophysics; molecular dynamics of macromolecules and fluids; many-atom electronic structures; global and discrete-state optimization; practical graph problems; image segmentation and recognition; tomography (medical imaging); fast Monte-Carlo sampling in statistical physics; and general, systematic methods of upscaling (accurate numerical derivation of large-scale equations from microscopic laws).

Section snippets

The scale gap

Despite their dizzying speed, modern supercomputers are still incapable of handling many most vital scientific problems. This is primarily due to the scale gap, which exists between the microscopic scale at which physical laws are given and the much larger scale of phenomena we wish to understand.

This gap implies, first of all, a huge number of variables (e.g., atoms or gridpoints or picture elements), and possibly even a much larger number of interactions. Moreover, computers simulate physical

Multigrid and renormalization

Past studies have demonstrated that scale-born slowness can be overcome by multiscale algorithms. Such algorithms have first been developed in the form of fast multigrid solvers for discretized PDEs [1], [2], [10], [11], [12]. These solvers are based on two processes: (1) classical relaxation schemes, which are generally slow to converge but fast to smooth the error function; (2) approximating the smooth error on a coarser grid (typically having twice the meshsize), by solving there equations

Systematic upscaling (SU): An outline

Local equations and interactions. Computationally we deal only with discrete systems; their n variables $u_{1}, u_{2}, \dots, u_{n}$ will typically be either values of discretized functions (grid values, or finite elements, etc.), or locations of particles. An equation or interaction is called local if it involves only $O (1)$ neighboring variables. For simplicity of discussion we describe SU first for systems of local equations (including energy minimization with local interactions) or local interactions at

Acknowledgment

The research is supported by the Israel Science Foundation grant 295/01 and by the US Air Force Office of Scientific Research, contract F33615-03-D5408.

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Multiscale solvers and systematic upscaling in computational physics

Abstract

Section snippets

The scale gap

Multigrid and renormalization

Systematic upscaling (SU): An outline

Acknowledgment

Appl. Math. Comp.

Nucl. Phys. B (Proc. Suppl.)

Comp. Phys. Comm.

Multi-level adaptive solutions to boundary value problems

Math. Comp.

Guide to multigrid development

Wave-ray multigrid method for standing wave equations

Electronic Trans. Num. Annal.

Renormalization multigrid (RMG): Statistically optimal renormalization group flow and coarse-to-fine Monte Carlo acceleration

J. Stat. Phys.

Multi-level approaches to discrete-state and stochastic problems