Compact discrepancy and chi-squared principles for over-determined inverse problems

We discuss and analyze the classical discrepancy principle and the recently proposed and closely related chi-squared principle for selecting the regularization parameter of an inverse problem. Some properties that deteriorate the performance of these methods for over-determined inverse problems are highlighted. We propose a so-called compact version of the discrepancy and chi-squared principles and demonstrate that for over-determined inverse problems the propagation of measurement uncertainty into the determination of the regularization parameter can be partly suppressed. This is achieved by expanding the cost function in terms of the singular vectors of the system matrix and by replacing the solution independent stochastic term by its statistical expectation. Although the motivation of the method requires a singular value decomposition, we show that in practice this step can be avoided. The performance of the regular and compact discrepancy and chi-squared principles is demonstrated on two benchmark problems. The compact versions of the discrepancy and chi-squared principles (i.e. with suppression of uncertainty) are shown to always improve the quality of the regularized solutions.