Homogenization

A multiscale analytical procedure that derives effective macroscopic equations and parameters from microscopically heterogeneous models by averaging or asymptotic expansion under a separation of scales or statistical homogeneity.

A set of mathematical techniques for deriving effective macroscopic equations and coefficients that approximate the behaviour of media or operators with fine-scale spatial heterogeneity, typically by averaging or limit processes as the microstructure scale tends to zero.

The operation of adjoining a new variable (commonly denoted z or t) and multiplying terms so that a polynomial becomes homogeneous of a fixed total degree, producing a homogeneous polynomial in an extended set of variables.

The process of adjoining an auxiliary variable (often denoted t) and multiplying monomials by appropriate powers of that variable so that an affine polynomial becomes homogeneous of a fixed total degree; this produces a homogeneous polynomial whose projective zero set compactifies the affine variety and facilitates degree bookkeeping and projective methods.