About the Balancing Principle for Choice of the Regularization Parameter

We discuss the balancing principle (BP) for the choice of the regularization parameter in linear ill-posed problems. As shown, the BP is closely related to Rule R1, thus known results about weak quasioptimality of Rule R1 are conveyable to BP. For BP we propose a smaller coefficient than used in original works and prove that in (iterated) Tikhonov method it leads to a smaller error of approximate solution. A general rule for the choice of the regularization parameter is introduced, including BP and many other rules as special cases. In BP a sequence of approximations is computed, we propose alternative uses of this sequence.