In linear systems, ill-conditioning often becomes more severe when adaptive refinement is applied. To reduce solver time, preconditioners are usually introduced. Conceptually, ill-conditioning reminds me of turbulence modeling, where we deal with both large and small eddies, and apply filters to suppress or enhance them to make the problem tractable.
My question is: Is it possible to design new preconditioners inspired by this turbulence-filtering analogy? For example, could one develop a preconditioner that selectively dampens or amplifies certain components of the system (analogous to eddies), potentially requiring access to eigenvalue information in the early stages of development?
Such an idea was the basis of the multigrid methods. The different layers have different characteristic lenghts, the wider is the lenght the higher the frequencies of the error are suppressed.
I think multigrid is not exactly what I mean. Multigrid already uses a kind of filtering, but it works in an indirect way, with different levels, interpolation, and extra cost. What I am looking for is a more direct idea: a matrix-based preconditioner that works like a spectral filter. It should reduce the condition number by directly damping or amplifying certain parts of the spectrum of A, without going to coarse grids. The goal is a preconditioner that acts like an explicit filter, built from the structure of the matrix itself.
Precondizioner strategies are largely adopted in solving low-Mach flows since they are ill-conditioned by solving the set of compressible flow model equations.
You can find inspiration from that field.
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