On a class of damped Morozov principles
Regularization methods for solving nonlinear ill-posed inverse problems are considered. The equation \(F(x) = y\) with \(F: D_ F \subset X \to Y\) is formulated as a nonlinear regularized least-squares problem \(\min_{x \in D_ F}| F(x)-y|^ 2_ Y + \beta | x - x_ e|^ 2_ X\) with \(X\), \(Y\) Hilbert spaces, \(x_ e\) an a priori guess to a solution and \(\beta > 0\) a regularization parameter. The author studies a family of parameter choice strategies given by \(| F(x_ \beta) - y^ \delta|^ 2 + \beta^ \gamma| x_ \beta - x_ e|^ 2 = \delta^ 2\), where \(y^ \delta\) are data with noise level \(\delta\), and \(\gamma \in [1,\infty)\) is an arbitrary parameter. The strategy is the `criterion-\(\varphi\)' for \(\gamma = 1\) and `Morozov's principle' for \(\gamma = \infty\). Asymptotic results for \(x_{\beta(\delta)}\) with \(\delta \to 0\) are given. As a numerical example a parameter estimation problem is considered in detail. It shows that within the class considered the choice of the regularization parameter is optimal most often for values \(1<\gamma < \infty\). But the results show as well that the choice of \(\gamma\) is crucial.
nonlinear regularized least squares problem
scientific article; zbMATH DE number 425517
scientific article