A mapping from coarse model variables or observables to a reconstructed fine-scale field that is consistent with the coarse constraints and any chosen priors or regularization; intended to recover subgrid structure absent from the coarse representation.

Principle

Recover unresolved degrees of freedom by imposing consistency with coarse constraints plus additional physical, statistical, or geometric priors to make the inverse mapping well-posed.

Demonstration

Reconstructing a high-resolution velocity field on a fine grid from cell-averaged velocities on a coarse grid by performing interpolation followed by a solenoidal projection and small-scale spectral synthesis to match energy spectra.

Misapplication

Naïvely upsampling coarse values with pointwise interpolation without enforcing conservation laws or statistical properties, producing nonphysical subgrid oscillations or violating integral constraints.

Consequence

Enables multiscale coupling and nested modeling where fine-scale diagnostics or forcings are required; improves interpretability of coarse outputs when priors reflect true subgrid physics.

Reversal

The inverse operation is Fine-to-Coarse Aggregation, which compresses fine-scale fields into coarse observables; reversing a correct reconstruction recovers coarse inputs but not necessarily the original fine field unless reconstruction is lossless.

Boundary

Applies only when coarse constraints (e.g., cell averages, moments, low-pass coefficients) are specified; does not invent information beyond priors and is limited by uncertainty quantification of the chosen reconstruction model.

Semantic Tension

Tension between deterministic interpolation (single best estimate) and stochastic downscaling (probabilistic ensemble of plausible fine fields) over the same coarse data.

Synthesis

A disciplined inverse mapping that combines coarse constraints with physical or statistical priors to regenerate plausible fine-scale structure useful for coupling, diagnostics, or initialization while acknowledging irrecoverable uncertainty.