What do we learn from the discrepancy principle? (Q873747)

The problem under consideration is an operator equation \(y_\delta=Ax+\delta\xi\) with a linear, compact and injective operator \(A\) acting between Hilbert spaces \(X\) and \(Y\) and with bounded deterministic noise \(\xi\), \(\| \xi\| \leq 1\). The author analyzes the regularization of the equation's solution in the form of \(g_\alpha : (0, \| A^*A\| ] \rightarrow \mathbb R^+\), i.e., the regularized solution is \(x_{\alpha, \delta}:=g_\alpha (A^*A)A^*y_\delta\), \(\alpha>0\). He assumes that the sought solution has a sourcewise representation \(x=\varphi(A^*A)v\) for some \(v\in X\) with \(\| v\| \leq R\) where \(\varphi: (0,\| A^*A\| ] \rightarrow \mathbb{R}^+\) is the index function. The regularizing parameter \(\alpha\) is determined by the discrepancy principle: \(\alpha_*:= \sup\{\alpha\leq\| A^*A\| \), \(\| Ax_{\alpha, \delta}-y_\delta\| \leq\tau\delta\}\). The main results of the paper are upper and lower estimations of a deviation between the exact and the regularized solutions \(\| x_{\alpha_*, \delta}-x\| \) expressed in terms of a qualification of the regularization method, and the index function \(\varphi\).