Related papers: Regularization of statistical inverse problems and…
We consider the Bayesian approach to linear inverse problems when the underlying operator depends on an unknown parameter. Allowing for finite dimensional as well as infinite dimensional parameters, the theory covers several models with…
Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed (pseudo-)inverses feasible. In the last two decades interest has shifted from…
A common task in experimental sciences is to fit mathematical models to real-world measurements to improve understanding of natural phenomenon (reverse-engineering or inverse modeling). When complex dynamical systems are considered, such as…
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…
This paper deals with Lavrentiev regularization for solving linear ill-posed problems, mostly with respect to accretive operators on Hilbert spaces. We present converse and saturation results which are an important part in regularization…
The emergent field of probabilistic numerics has thus far lacked clear statistical principals. This paper establishes Bayesian probabilistic numerical methods as those which can be cast as solutions to certain inverse problems within the…
We establish a general principle which states that regularizing an inverse problem with a convex function yields solutions which are convex combinations of a small number of atoms. These atoms are identified with the extreme points and…
Inverse Problems in medical imaging and computer vision are traditionally solved using purely model-based methods. Among those variational regularization models are one of the most popular approaches. We propose a new framework for applying…
Fundamental inconsistencies of superstatistics are highlighted. There is no such thing as a superposition of Boltzmann factors; what is actually derived is a generating function and not a normalizable probability density. The beta density…
This paper addresses Tikhonov like regularization methods with convex penalty functionals for solving nonlinear ill-posed operator equations formulated in Banach or, more general, topological spaces. We present an approach for proving…
The axiomatic foundation of probability theory presented by Kolmogorov has been the basis of modern theory for probability and statistics. In certain applications it is, however, necessary or convenient to allow improper (unbounded)…
This paper addresses the issue of inversion in cases where (1) the observation system is modeled by a linear transformation and additive noise, (2) the problem is ill-posed and regularization is introduced in a Bayesian framework by an a…
We introduce and study a mathematical framework for a broad class of regularization functionals for ill-posed inverse problems: Regularization Graphs. Regularization graphs allow to construct functionals using as building blocks linear…
Convergence rates in spectral regularization methods quantify the approximation error in inverse problems as a function of the noise level or the number of sampling points. Classical strong convergence rate results typically rely on source…
We introduce quantum Markov categories as a structure that refines and extends a synthetic approach to probability theory and information theory so that it includes quantum probability and quantum information theory. In this broader…
We consider joint Tikhonov- and Lavrentiev-regularization of control problems with pointwise control- and state-constraints. We derive error estimates for the error which is introduced by the Tikhonov regularization. With the help of this…
We adopt Bayesian approach to consider the inverse problem of estimate a function from noisy observations. One important component of this approach is the prior measure. Total variation prior has been proved with no discretization invariant…
In this paper we demonstrate the possibility of a trend reversal in binary classification tasks between the dataset and a classification score obtained from a trained model. This trend reversal occurs for certain choices of the…
Reversal of the time direction in stochastic systems driven by white noise has been central throughout the development of stochastic realization theory, filtering and smoothing. Similar ideas were developed in connection with certain…
It is well-known in practice, that L^1 data fitting leads to improved robustness compared to standard L^2 data fitting. However, it is unclear whether resulting algorithms will perform as well in case of regular data without outliers. In…