Most algorithms for choosing the regularization parameter in a discrete ill-posed problem are based on the norm of the residual vector. In this work we propose a different approach, where we seek to use all the information available in the residual vector. We present important relations between the residual components and the amount of information that is available in the noisy data, and we show how to use statistical tools and fast Fourier transforms to extract this information efficiently. This approach leads to a computationally inexpensive parameter-choice rule based on the normalized cumulative periodogram, which is particularly suited for large-scale problems.
Hansen, P. C., Kilmer, M. E., & Kjeldsen, R. H. (2006). Exploiting residual information in the parameter choice for discrete ill-posed problems. BIT Numerical Mathematics, 46(1), 41-59. https://doi.org/10.1007/s10543-006-0042-7