##  [Coarse-to-Fine Reconstruction](/coarse-fine-reconstruction-0) 

 Definition

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

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

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

Misapplication

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

 

 

 

 

 





## Consequence

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

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

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

Semantic Tension

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

 

 

 

 

 





## Synthesis

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.