To see just how ill-conditioned the transfer matrix in the previous
equation, calculate the condition number of, A_transfer.dat, using the
singular value decomposition (A_transfer.dat was constructed by using
the finite element method, which will be described in detail later in the
chapter). First solve equation (14) using all the singular values
using the data files, Phi_H.dat and Phi_T.dat. Then solve the system by
discarding the ones near zero (i.e. by using the TSVD approach). Next
solve (14) using the Tikhonov regularization technique by varying the
regularization parameter 
.
As you continue solving the inverse problem for different regularization parameters, you might realize that you don't really know what you are looking for in terms of an answer (except to compare it to the SVD solution). Assuming we don't have a ``true'' solution to compare with, how is it then, that we know which regularization to choose a priori? This is a very good question and one that has intrigued (plagued) scientists and mathematicians working on inverse problems for years. While there has been much research into methods for choosing the best regularization parameter in an a priori manner, most schemes that have been put forth depend in some way on the data and/or constraints placed on the solution, and, thus, are problem dependent.
For an excellent
overview of the problem of finding optimal a priori estimates of the
regularization parameter and some solutions, see [19].
