Residual periodograms for choosing regularization parameters for ill-posed problems

Consider an ill-posed problem transformed if necessary so that the errors in the data are independent identically normally distributed with mean zero and variance 1. We survey regularization and parameter selection from a linear algebra and statistics viewpoint and compare the statistical distributions of regularized estimates of the solution and the residual. We discuss methods for choosing a regularization parameter in order to assure that the residual for the model is statistically plausible. Ideally, as proposed by Rust (1998 Tech. Rep. NISTIR 6131, 2000 Comput. Sci. Stat. 32 333–47 ), the results of candidate parameter choices should be evaluated by plotting the resulting residual along with its periodogram and its cumulative periodogram, but sometimes an automated choice is needed. We evaluate a method for choosing the regularization parameter that makes the residuals as close as possible to white noise, using a diagnostic test based on the periodogram. We compare this method with standard techniques such as the discrepancy principle, the L-curve and generalized cross validation, showing that it performs better on two new test problems as well as a variety of standard problems.