Noise level free regularisation of general linear inverse problems under unconstrained white noise

TL;DR

This paper introduces a noise level free regularization method for linear inverse problems with unknown noise characteristics, using an adaptive discretization approach and the heuristic discrepancy principle to ensure convergence.

Contribution

It presents a novel regularization technique that does not require noise level knowledge, applicable to arbitrary compact operators, with convergence guarantees and uncertainty quantification.

Findings

Convergence shown for arbitrary compact operators and solutions.

Adaptive discretization controls non-Gaussian, unbounded noise effectively.

Uncertainty in convergence rate is quantified in a Bayesian-like framework.

Abstract

In this note we solve a general statistical inverse problem under absence of knowledge of both the noise level and the noise distribution via application of the (modified) heuristic discrepancy principle. Hereby the unbounded (non-Gaussian) noise is controlled via introducing an auxiliary discretisation dimension and choosing it in an adaptive fashion. We first show convergence for completely arbitrary compact forward operator and ground solution. Then the uncertainty of reaching the optimal convergence rate is quantified in a specific Bayesian-like environment.