On the a posteriori choice of a regularization parameter in the solution of severely ill-posed problems
For more than 30 years, results on convergence rates of stable approximate solutions to ill-posed linear operator equations \(Ax=y\) in Hilbert spaces, obtained by the basic version \(\|Ax-y_\delta\|^2+\alpha\|x\|^2 \to \min\) of Tikhonov's regularization method, have been published. Embedded in a series of similar papers published by the first author with coworkers, the authors present in this context some further results on logarithmic convergence rates \(O\left(\log^{-p}(1/\delta)\right)\) under a deterministic noise model \(\|y-y_\delta\| \leq \delta\) and under associated logarithmic source conditions for some \(p>0\). The crucial point seems to be that rates of optimal order can be proven for arbitrary \(p>0\) when the regularization parameter \(\alpha>0\) is chosen according to the balancing principle. Also, the influence of discretization errors is discussed. As an application background, the authors refer to a specific example from gravitational gradiometry with data supplied by satellites. The interpretation of those indirect measuring data consists in the solution of a severely ill-posed integral equation of the first kind, where the corresponding solution is an element of the Sobolev spherical space \(\mathcal{H}_s\). The decay rate of the singular values \(\sigma_n\) of the compact integral operator \(A\) is characterized by the asymptotics \(\log \sigma_n^{-2} \sim \left(n+\frac{1}{2}\right)\). Moreover, there are constants \(c_2>c_1>0\) such that \( c_1 \|x\|_s \leq \|\log^s(A^*A)^{-1}x\| \leq c_2 \|x\|_s\) for all \(x \in \mathcal{H}_s \).
scientific article; zbMATH DE number 6110542