On the convergence of a heuristic parameter choice rule for Tikhonov regularization

Multiplicative regularization solves a linear inverse problem by minimizing the product of the norm of the data misfit and the norm of the solution. This technique is related to Tikhonov regularization with the parameter chosen to make the data misfit and regularization terms (of the Tikhonov objective function) equal. This suggests a heuristic parameter choice method, equivalent to the rule previously proposed by Reginska. Reginska\u27s rule is well defined provided the data is sufficiently close to exact data and does not lie in the range of the operator. If a sufficiently large portion of the data error lies outside the range of the operator, then the solution defined by Reginska\u27s rule converges weakly to the exact solution as the data error converges to zero. The regularization parameter converges to zero like the square of the norm of the data noise, leading to under-regularization for small noise levels. Nevertheless, the method performs well on a suite of test problems, as shown by comparison with the L-curve, generalized cross-validation, quasi-optimality, and Hanke--Raus parameter choice methods. A modification of the approach yields a heuristic parameter choice rule that is provably convergent (in the norm topology) under the restrictions on the data error described above, as long as the exact solution has a small amount of additional smoothness. On the test problems considered here, the modified rule outperforms all of the above heuristic methods, although it is only slightly better than the quasi-optimality rule