Heuristic parameter selection based on functional minimization: optimality and model function approach (Q2840624)

Two noise-level-free parameter choices for Tikhonov regularization to solve linear ill-posed problems in Hilbert spaces are analyzed. These are the modified L-curve method and a variant of the Hanke-Raus type rule. We present some details, and for this purpose we consider a linear compact operator \( A: X \to Y \) between Hilbert spaces \( X \) and \( Y \). In addition, let \( y_\delta = Ax + \delta \xi \) be noisy data, where \( x \in X \) denotes the unknown solution, the noise \( \xi \in Y \) satisfies \( \| \xi \| \leq 1 \), and \( \delta > 0 \) represents some noise level which may be unknown, in general. For Tikhonov regularization \( x_{\alpha,\delta} = (A^*A + \alpha I)^{-1} A^* y_\delta\), \(\alpha > 0 \), the authors consider the modified L-curve criterion \( \Psi_\mu(\alpha) = \| A x_{\alpha,\delta} - y_\delta \|^2 \| x_{\alpha,\delta} \|^{2\mu} \to \min \), as well as the Hanke-Raus type rule \( \Psi(\alpha) = \| A x_{\alpha,\delta} - y_\delta \|^2/\alpha \to \min \). Here, \( \mu > 0 \) denotes some parameter. For both parameter choice strategies, necessary and sufficient conditions for minimizers are provided. In addition, estimates for \( \| x-x_{\alpha_*,\delta} \| \) in terms of the noise level \( \delta \) and the realized discrepancy \( \delta_* = \| A x_{\alpha_*,\delta} - y_\delta \| \) are presented, where \( \alpha_* \) denotes a global minimizer of the parameter choice strategy under consideration. In those estimates it is assumed that a general source condition of the form \( x = \varphi(A^*A) v \) with \( v \in X\), \(\| v \| \leq 1 \), holds. For the L-curve method this in fact is the first available error bound. The next section of the paper is devoted to the analysis of model function approaches. Here, \( \Psi_\mu(\alpha) \) and \( \Psi(\alpha) \) are replaced by some model counter parts, respectively. Those models are based on approximations of the functional \( h(\alpha) := \| A x_{\alpha,\delta} \|^2 + \alpha \| x_{\alpha,\delta} \|^2 \) of the form \( h(\alpha) \approx C/(T+\alpha) \) with some parameters \( C \) and \( T \). Finally, results of some numerical experiments are presented.