Upscaling Interpretation of Nonlocal Fields, Gradients, and Divergences

The interrelations between weight-function upscaling (measurement) and the definition of various nonlocal operators is explored. Let < f > = f∗g where f∗g is the convolution product which represents the effect of upscaling via an instrument (defined by g) on a field variable f and its localized counterpart. Nonlocal field variables are defined and employed for upscaling. In this talk it will be shown via Fourier transform, for judicious choice of the arbitrary function Gp, that Gpf(x) = < grad f > (x), where Gpf (x) is the nonlocal gradient of f and grad f is the classical gradient. Upscaled representations for the adjoint of Gp and the nonlocal divergence are obtained. A nonlocal self-diffusion equation is upscaled and written in terms of nonlocal operators.