I have a linear system of equations $Ax = b$ where the number of unknowns $N$ is intractably large but the right-hand side has only small support and the unknown $x$ is known to decay exponentially. I hence want to find a sequence of reduced linear systems $A_n x_n = b_n$ with only $n \leq N$ unknowns such that $x_n \to x$ as $n \to N$. Is there any theory available on how to find suitable $A_n, b_n$ such that I can guarantee well-posedness of the reduced problems and convergence, possibly under additional assumptions on $A$ and $b$?I have a fairly limited understanding of what wavelets are and how they work, but as far as I see we should face the same problem there: we have an in principle infinite expansion of the solution in terms of the wavelet basis functions, but the smoothness of the solution implies that the corresponding coefficients decay and hence we can truncate. This makes me wonder how the above problems of well-posedness and convergence for truncated systems are addressed there.
Reduction of linear system with decaying unknown
wavelet