A kind of a posteriori parameter choices for the iterated Tikhonov regularization method (Q2367860)

Let \(X\) and \(Y\) be real Hilbert spaces and \(A: X\to Y\) be a bounded linear operator with nonclosed range \(R(A)\). If \(y\in D(A^ +)=R(A)+R(A)^ \perp\), there exists a unique Moore-Penrose generalized solution to the equation \(Ax=y\). In practice, however, the given equation is usually of an approximate form (1) \(Ax=y_ \delta\), where \(y_ \delta\in B_ \delta(y)=\{z\in Y/\| Q(z-y)\|\leq \delta\}\) with \(\delta>0\) and \(Q\) being the orthogonal projection operator from \(Y\) onto \(R(A)\). Finding an approximation of the generalized solution \(x^ +\) from (1) is an ill-posed problem. The author is concerned with an important method of the solution (the iterated Tikhonov regularization method of order \(n\)) which can be written as \(x^ \delta_{\alpha,n}=R_{\alpha,n} y_ \delta=U_ n(\alpha,A^* A)A^* y_ \delta\), where \(\alpha>0\) is called the regularization parameter and \(U_ n(\alpha,\lambda)=\bigl[1- (\alpha/(\lambda+\alpha))^ n\bigr]/\lambda\), \(\lambda\in (0,\infty)\). The author analyzes a kind of a posteriori parameter choice for the general iterated Tikhonov regularization which includes Morozov's and Gfrerer's principles as special cases.