  The solution, $x$, of the linear system of equations $A x\approx b$ arising
from the discretization of an ill-posed integral equation with a square
integrable kernel $H(s,t)$ is considered. The Tikhonov regularized solution $
x(\lambda)$ is found as the minimizer of $J( x)=\{ \|A x - b\|_2^2 + \lambda^2
\|L x\|_2^2\}$. $ x(\lambda)$ depends on regularization parameter $\lambda$
that trades off the data fidelity, and on the smoothing norm determined by $L$.
Here we consider the case where $L$ is diagonal and invertible, and employ the
Galerkin method to provide the relationship between the singular value
expansion and the singular value decomposition for square integrable kernels.
The resulting approximation of the integral equation permits examination of the
properties of the regularized solution $ x(\lambda)$ independent of the sample
size of the data. We prove that estimation of the regularization parameter can
be obtained by consistently down sampling the data and the system matrix,
leading to solutions of coarse to fine grained resolution. Hence, the estimate
of $\lambda$ for a large problem may be found by downsampling to a smaller
problem, or to a set of smaller problems, effectively moving the costly
estimate of the regularization parameter to the coarse representation of the
problem. Moreover, the full singular value decomposition for the fine scale
system is replaced by a number of dominant terms which is determined from the
coarse resolution system, again reducing the computational cost. Numerical
results illustrate the theory and demonstrate the practicality of the approach
for regularization parameter estimation using generalized cross validation,
unbiased predictive risk estimation and the discrepancy principle applied for
both the system of equations, and the augmented system of equations.
