Spacevarying

Spacevarying, often written space-varying or spatially varying, is a descriptive term used across science, engineering, and mathematics to denote quantities whose values depend on location in space. It contrasts with space-invariant or homogeneous conditions, where properties are uniform throughout the domain.

In mathematics and engineering, space-varying coefficients appear in differential operators. For example, a second-order differential operator with coefficients that depend on the spatial variable x, such as Lx u = -d/dx (a(x) du/dx) + b(x) u, has behavior that changes with position. Space-varying systems challenge analysis and often require numerical methods that accommodate locality.

In signal and image processing, space-variant systems possess impulse responses or kernels that depend on the input location. This includes adaptive filters, space-variant blur in imaging, and nonuniform deconvolution. Modeling these systems typically uses local parameters or piecewise definitions.

In physics and materials science, many phenomena are space-varying. Graded or functionally graded materials have properties that vary with position. Optical media with a refractive index that changes in space produce nonuniform light propagation; quantum systems with position-dependent potentials yield spatially varying behavior.

In statistics and geographic sciences, space-varying relationships are captured by models such as geographically weighted regression, where regression coefficients are allowed to change over space. This aligns with broader concepts of spatial heterogeneity and non-stationarity.

Modeling space-varying phenomena requires methods that respect locality, such as finite element or finite difference schemes with spatially variable coefficients, kernel-based local modeling, or adaptive mesh refinement. Challenges include ensuring stability, identifiability, and computational efficiency.

See also spatial variation, spatial heterogeneity, non-stationarity, and geographically weighted regression.