  The truncated singular value decomposition may be used to find the solution
of linear discrete ill-posed problems in conjunction with Tikhonov
regularization and requires the estimation of a regularization parameter that
balances between the sizes of the fit to data function and the regularization
term. The unbiased predictive risk estimator is one suggested method for
finding the regularization parameter when the noise in the measurements is
normally distributed with known variance. In this paper we provide an algorithm
using the unbiased predictive risk estimator that automatically finds both the
regularization parameter and the number of terms to use from the singular value
decomposition. Underlying the algorithm is a new result that proves that the
regularization parameter converges with the number of terms from the singular
value decomposition. For the analysis it is sufficient to assume that the
discrete Picard condition is satisfied for exact data and that noise completely
contaminates the measured data coefficients for a sufficiently large number of
terms, dependent on both the noise level and the degree of ill-posedness of the
system. A lower bound for the regularization parameter is provided leading to a
computationally efficient algorithm. Supporting results are compared with those
obtained using the method of generalized cross validation. Simulations for
two-dimensional examples verify the theoretical analysis and the effectiveness
of the algorithm for increasing noise levels, and demonstrate that the relative
reconstruction errors obtained using the truncated singular value decomposition
are less than those obtained using the singular value decomposition. This is a
pre-print of an article published in BIT Numerical Mathematics. The final
authenticated version is available online at:
https://doi.org/10.1007%2Fs10543-019-00762-7.
