Dual gradient method for ill-posed problems using multiple repeated measurement data

Abstract

We consider determining $\mathbb{R}$-minimizing solutions of linear ill-posed problems $Ax=y$, where $A:\mathcal{X}\to \mathcal{Y}$ is a bounded linear operator from a Banach space $\mathcal{X}$ to a Hilbert space $\mathcal{Y}$ and $\mathcal{R}:\mathcal{X}\to [0,\infty ]$ is a proper strongly convex penalty function. Assuming that multiple repeated independent identically distributed unbiased data of $y$ are available, we consider a dual gradient method to reconstruct the $\mathcal{R}$-minimizing solution using the average of these data. By terminating the method by either an {\it a priori} stopping rule or a statistical variant of the discrepancy principle, we provide the convergence analysis and derive convergence rates when the sought solution satisfies certain variational source conditions. Various numerical results are reported to test the performance of the method.

References in corpus (9)

• Characterizations of variational source conditions, converse results, and maxisets of spectral regularization methods
• Verification of a variational source condition for acoustic inverse medium scattering problems
• Hanke-Raus heuristic rule for variational regularization in Banach spaces
• Stochastic mirror descent method for linear ill-posed problems in Banach spaces
• Beyond the Bakushinskii veto: Regularising linear inverse problems without knowing the noise distribution
• Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems
• Convergence of Heuristic Parameter Choice Rules for Convex Tikhonov Regularisation
• A modified discrepancy principle to attain optimal convergence rates under unknown noise
• Regularising linear inverse problems under unknown non-Gaussian white noise allowing repeated measurements