Tim Jahn
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View article: Convergence of generalized cross-validation with applications to ill-posed integral equations
Convergence of generalized cross-validation with applications to ill-posed integral equations Open
In this article, we rigorously establish the consistency of generalized cross-validation as a parameter-choice rule for solving inverse problems. We prove that the index chosen by leave-one-out GCV achieves a non-asymptotic, order-optimal …
View article: Efficient solution of ill-posed integral equations through averaging
Efficient solution of ill-posed integral equations through averaging Open
This paper discusses the error and cost aspects of ill-posed integral equations when given discrete noisy point evaluations on a fine grid. Standard solution methods usually employ discretization schemes that are directly induced by the me…
View article: Noise level free regularisation of general linear inverse problems under unconstrained white noise
Noise level free regularisation of general linear inverse problems under unconstrained white noise Open
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…
View article: Discretisation-adaptive regularisation of statistical inverse problems
Discretisation-adaptive regularisation of statistical inverse problems Open
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 suitable stopping index for the iteration. Convergence resu…
View article: A Probabilistic Oracle Inequality and Quantification of Uncertainty of a modified Discrepancy Principle for Statistical Inverse Problems
A Probabilistic Oracle Inequality and Quantification of Uncertainty of a modified Discrepancy Principle for Statistical Inverse Problems Open
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 introduced adaptive method based on the classical discrepanc…
View article: A probabilistic oracle inequality and quantification of uncertainty of a modified discrepancy principle for statistical inverse problems
A probabilistic oracle inequality and quantification of uncertainty of a modified discrepancy principle for statistical inverse problems Open
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 introduced adaptive method based on the classical discrepanc…
View article: Optimal Convergence of the Discrepancy Principle for polynomially and exponentially ill-posed Operators under White Noise
Optimal Convergence of the Discrepancy Principle for polynomially and exponentially ill-posed Operators under White Noise Open
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-Schmidt operators (and they require a self-similarity conditio…
View article: Increasing the relative smoothness of stochastically sampled data.
Increasing the relative smoothness of stochastically sampled data. Open
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 the average of the measurements as an approximation of the…
View article: Regularising linear inverse problems under unknown non-Gaussian white noise.
Regularising linear inverse problems under unknown non-Gaussian white noise. Open
We deal with the solution of a generic linear inverse problem in the Hilbert space setting. The exact right hand side is unknown and only accessible through discretised measurements corrupted by white noise with unknown arbitrary distribut…
View article: Regularising linear inverse problems under unknown non-Gaussian white noise allowing repeated measurements
Regularising linear inverse problems under unknown non-Gaussian white noise allowing repeated measurements Open
We deal with the solution of a generic linear inverse problem in the Hilbert space setting. The exact right hand side is unknown and only accessible through discretised measurements corrupted by white noise with unknown arbitrary distribut…
View article: On the discrepancy principle for stochastic gradient descent
On the discrepancy principle for stochastic gradient descent Open
Stochastic gradient descent (SGD) is a promising numerical method for solving large-scale inverse problems. However, its theoretical properties remain largely underexplored in the lens of classical regularization theory. In this note, we s…
View article: Beyond the Bakushinkii veto: regularising linear inverse problems without knowing the noise distribution
Beyond the Bakushinkii veto: regularising linear inverse problems without knowing the noise distribution Open
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…
View article: On the Discrepancy Principle for Stochastic Gradient Descent
On the Discrepancy Principle for Stochastic Gradient Descent Open
Stochastic gradient descent (SGD) is a promising numerical method for solving large-scale inverse problems. However, its theoretical properties remain largely underexplored in the lens of classical regularization theory. In this note, we s…
View article: Neural Observer for Nonlinear State and Input Estimation in a Truck-Semitrailer Combination
Neural Observer for Nonlinear State and Input Estimation in a Truck-Semitrailer Combination Open
Driver assistance systems have become an indispensable part of today's vehicles technology. Especially in the commercial vehicle sector, the challenges in obtaining information increase with rising system complexity. Compared to trucks, tr…