  In this work we consider the problem of finding optimal regularization
parameters for general-form Tikhonov regularization using training data. We
formulate the general-form Tikhonov solution as a spectral filtered solution
using the generalized singular value decomposition of the matrix of the forward
model and a given regularization matrix. Then, we find the optimal
regularization parameter by minimizing the average of the errors between the
filtered solutions and the true data. We extend the approach to the
multi-parameter Tikhonov problem for the case where all the matrices involved
are simultaneously diagonalizable. For problems where this is not the case, we
describe an approach to compute optimal or near-optimal regularization
parameters by using operator approximations for the original problem. Several
tests are performed for 1D and 2D examples using different norms on the errors,
showing the effectiveness of this approach.
