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Discretisation-adaptive regularisation of statistical inverse problems
Tim Jahn
We consider linear inverse problems under white noise. These types of problems can be tackled with, e.g., iterative regularisation methods and the main challenge is to determine a…
A Probabilistic Oracle Inequality and Quantification of Uncertainty of a modified Discrepancy Principle for Statistical Inverse Problems
Tim Jahn
In this note we consider spectral cut-off estimators to solve a statistical linear inverse problem under arbitrary white noise. The truncation level is determined with a recently i…
Optimal Convergence of the Discrepancy Principle for polynomially and exponentially ill-posed Operators under White Noise
Tim Jahn
We consider a linear ill-posed equation in the Hilbert space setting under white noise. Known convergence results for the discrepancy principle are either restricted to Hilbert-Sch…
A modified discrepancy principle to attain optimal convergence rates under unknown noise
Tim Jahn
We consider a linear ill-posed equation in the Hilbert space setting. Multiple independent unbiased measurements of the right hand side are available. A natural approach is to take…
On the Discrepancy Principle for Stochastic Gradient Descent
Tim Jahn, Bangti Jin
Stochastic gradient descent (SGD) is a promising numerical method for solving large-scale inverse problems. However, its theoretical properties remain largely underexplored in the…
Beyond the Bakushinskii veto: Regularising linear inverse problems without knowing the noise distribution
Bastian Harrach, Tim Jahn, Roland Potthast
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 Bakushinski…