Beyond the Bakushinskii veto: Regularising linear inverse problems without knowing the noise distribution

Abstract

This article deals with the solution of linear ill-posed equations in Hilbert spaces. Often, one only has a corrupted measurement of the right hand side at hand and the Bakushinskii veto tells us, that we are not able to solve the equation if we do not know the noise level. But in applications it is ad hoc unrealistic to know the error of a measurement. In practice, the error of a measurement may often be estimated through averaging of multiple measurements. We integrated that in our anlaysis and obtained convergence to the true solution, with the only assumption that the measurements are unbiased, independent and identically distributed according to an unknown distribution.

22 pages, 3 figures

References in corpus (6)

• First M87 Event Horizon Telescope Results. III. Data Processing and Calibration
• Methods of Solving Ill-Posed Problems
• Risk hull method and regularization by projections of ill-posed inverse problems
• Regularization independent of the noise level: an analysis of quasi-optimality
• Regularization of statistical inverse problems and the Bakushinskii veto
• Monte Carlo pathwise sensitivities for barrier options

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• On the asymptotical regularization for linear inverse problems in presence of white noise
• On the Saturation Phenomenon of Stochastic Gradient Descent for Linear Inverse Problems
• Dual-grid parameter choice method with application to image deblurring