This decomposition/reassembling procedure can be described mathematically as
a windowing procedure which is a specific implementation of generalized frames. By
applying frame theory, we show that a collection of local wavefield propagators, combined
via a suitable partition of unity, remains a stable propagator. This is a highly desirable
property in numerical simulations and key to accurate solutions. These results apply more generally to combinations of
linear operators that are useful for many nonstationary filtering operations.
One of the main ingredients of multigrid are scaling of misfit
function from grid to grid and solving a coarse problem. Local
smoothing during the scaling is especially efficient in case of
limited computational resources. On the other hand, in this situation
the convergence of gradient method even more depends on the smoothness
of the exact solution of the synthetic data.