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Exercise 4: Effect of the regularization parameter.

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].