multiplescale

Multiplescale (or multiscale) refers to an approach or phenomenon that involves processes acting at two or more spatial, temporal, or organizational scales. It can describe a system where coarse-grained dynamics at a large scale interact with fine-scale details, requiring representation and analysis across multiple levels rather than a single scale.

In mathematics and engineering, multiscale analysis includes homogenization, asymptotic methods, and multiscale finite element methods. In signal processing, scale-space theory and wavelet-based multiresolution analysis decompose signals into components at different resolutions. In physics and materials science, renormalization and coarse-graining capture how microscopic rules give rise to macroscopic behavior.

Applications span climate and earth sciences, where models couple processes from clouds to global circulation; materials science, where heterogeneous composites are described by effective properties; biology, where models connect molecular, cellular, and organ scales; and computer vision, where features are extracted at multiple resolutions.

Challenges include choosing appropriate scales, dealing with scale interactions and nonlinear coupling, and computational cost. The concept dates to developments in statistical physics and applied mathematics, with formal multiscale methods emerging in the late 20th century and gaining prominence with wavelet theory and multiscale finite element methods.

See also: multiscale modeling, homogenization, wavelets, scale-space, renormalization group.