Fast Macroscopic Forcing: Exploiting locality for operator recovery

The macroscopic forcing method (MFM), introduced by Mani and Park (2021), is an approach for recovering lower-dimensional scaled-up operators by successively forcing a direct simulation. The MFM has already successfully recovered eddy diffusivities for RANS-like turbulence models. However, standard algorithms for MFM apply forcings to each coarse-scale degree of freedom and conduct a fine-scale simulation, which is expensive, or exploit rather strict assumptions about the coarse operator. We present an MFM algorithm that is cheaper and more general. It applies sparse reconstruction, which exposes local features in the differential operator and reconstructs the coarse one in only a few matrix-vector products. For non-local operators, we prepend this approach by peeling long-range effects with dense matrix-vector products to expose a more local operator. We demonstrate the algorithm's performance on scalar transport and channel flow problems.