Multiscale

Multiscale refers to systems, models, and analyses that involve processes occurring at more than one spatial or temporal scale. In such contexts, phenomena at a small scale can influence larger-scale behavior, and large-scale conditions can constrain or shape finer details. A multiscale approach seeks to describe or simulate a system by linking descriptions across scales rather than relying on a single, uniform resolution.

Mathematics and computational science provide several techniques. Homogenization theory derives effective macroscopic properties of heterogeneous media by averaging or upscaling microstructure. Multiscale modeling couples models at different scales, such as a detailed microscale model embedded within a coarser macroscale simulation. Numerical methods include the multiscale finite element method (MsFEM), the heterogeneous multiscale method (HMM), multigrid and multiresolution approaches, and wavelet-based analyses that decompose data across scales. Renormalization group ideas from statistical physics provide a conceptual framework for understanding how behavior changes with scale.

Applications span materials science, geophysics, and environmental science, where porous media, composites, or fractured rocks exhibit scale-dependent properties; climate and ocean dynamics involve interactions across global and regional scales; biology and neuroscience use multiscale models to connect genes and cells to tissues and organs.

Challenges include ensuring consistency between scales, error estimation, and computational cost, as well as developing robust coupling strategies between models operating at different resolutions or assumptions.

Historically, multiscale concepts emerged in applied mathematics and engineering with homogenization and asymptotic methods, later expanded by wavelet theory and multiscale algorithms in computational physics and applied sciences.