Discretization-invariant operator family

Let $A$ and $U$ be real Banach spaces of functions on bounded domains $D$ and $D^{\mathrm{\prime }}$, with the input point evaluations used below defined. For every $n\ge 1$, choose finite-dimensional normed spaces $E_n,F_n$, an exactly defined sampling map $S_n:A\to E_n$ (for example, $S_n(a)=(a({\xi }_{n,1}),\dots ,a({\xi }_{n,N_n}))$ at specified input points), a finite-dimensional map ${\tilde{G}}_n:E_n\to F_n$, and an output reconstruction map $R_n:F_n\to U$. The reconstruction supplies a function with a defined value at every point of $D^{\mathrm{\prime }}$. The corresponding continuum-valued discrete map is $G_n(a):=R_n({\tilde{G}}_n(S_n(a)))\in U\mathrm{.}$ A family $(G_n{)}_{n\ge 1}$ is a discretization-invariant approximation of an operator $G:A\to U$ if its sampling maps and output reconstructions are part of the family and it approximates this single continuum operator as the input discretization is refined. In the compact-uniform version, for every compact $K\subset A$, ${\sup }_{a\in K}\mathrm{\parallel }{G}_n(a)-G(a){\mathrm{\parallel }}_U⟶0.$ For $K=\mathrm{\varnothing }$, the supremum of this nonnegative error is defined to be $0$. When $A$ consists of almost-everywhere equivalence classes, the sampling map must be explicitly defined on those classes or use a specified representative convention; point evaluation is not inferred from the equivalence class alone. This definition does not identify unrelated grid-specific maps as one operator family.

metric compactness

Kovachki et al., Definition 4 and §3.1: discretization-invariance conditions