Convergence rates in the Prokhorov metric for assessing uncertainty in ill-posed problems

2005 ◽  
Vol 21 (1) ◽  
pp. 399-412 ◽  
Author(s):  
Heinz W Engl ◽  
Andreas Hofinger ◽  
Stefan Kindermann
Keyword(s):  
Convergence Rates ◽  
Prokhorov Metric ◽  
Ill Posed

scholarly journals Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed Problems

2021 ◽  
Author(s):  
Radu Boţ ◽  
Guozhi Dong ◽  
Peter Elbau ◽  
Otmar Scherzer
Keyword(s):  
Linear Operator ◽  
Operator Equation ◽  
Convex Analysis ◽  
Convergence Rates ◽  
Linear Operator Equation ◽  
Optimal Convergence Rates ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Given ◽  
Theoretical Results

AbstractRecently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.

scholarly journals The Impact of the Discrepancy Principle on the Tikhonov-Regularized Solutions with Oversmoothing Penalties

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 331
Author(s):  
Bernd Hofmann ◽  
Christopher Hofmann
Keyword(s):  
Case Studies ◽  
Convergence Rates ◽  
Regularization Parameter ◽  
A Priori ◽  
Discrepancy Principle ◽  
Parameter Choice ◽  
Operator Equations ◽  
Analytical Results ◽  
Ill Posed ◽  
The Impact

This paper deals with the Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. One focus is on the application of the discrepancy principle for choosing the regularization parameter and its consequences. Numerical case studies are performed in order to complement analytical results concerning the oversmoothing situation. For example, case studies are presented for exact solutions of Hölder type smoothness with a low Hölder exponent. Moreover, the regularization parameter choice using the discrepancy principle, for which rate results are proven in the oversmoothing case in in reference (Hofmann, B.; Mathé, P. Inverse Probl. 2018, 34, 015007) is compared to Hölder type a priori choices. On the other hand, well-known analytical results on the existence and convergence of regularized solutions are summarized and partially augmented. In particular, a sketch for a novel proof to derive Hölder convergence rates in the case of oversmoothing penalties is given, extending ideas from in reference (Hofmann, B.; Plato, R. ETNA. 2020, 93).

scholarly journals Convergence Rates for Exponentially Ill-Posed Inverse Problems with Impulsive Noise

2016 ◽  
Vol 54 (1) ◽  
pp. 341-360 ◽  
Author(s):  
Claudia König ◽  
Frank Werner ◽  
Thorsten Hohage
Keyword(s):  
Inverse Problems ◽  
Convergence Rates ◽  
Impulsive Noise ◽  
Ill Posed

scholarly journals Quasi-boundary value method for non-well posed problem for a parabolic equation with integral boundary condition

2001 ◽  
Vol 7 (2) ◽  
pp. 129-145 ◽  
Author(s):  
M. Denche ◽  
K. Bessila
Keyword(s):  
Boundary Condition ◽  
Convergence Rates ◽  
Initial Conditions ◽  
Nonlocal Problem ◽  
Convergence Result ◽  
Integral Boundary Condition ◽  
Integral Boundary ◽  
Boundary Value Method ◽  
Ill Posed ◽  
Well Posed

In this paper we study the problem of control by the initial conditions of the heat equation with an integral boundary condition. This problem is ill-posed. Perturbing the final condition we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed. We also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions. Finally, we give explicit convergence rates.

Convergence rates for Tikhonov regularisation of non-linear ill-posed problems

1989 ◽  
Vol 5 (4) ◽  
pp. 523-540 ◽  
Author(s):  
H W Engl ◽  
K Kunisch ◽  
A Neubauer
Keyword(s):  
Convergence Rates ◽  
Non Linear ◽  
Ill Posed ◽  
Tikhonov Regularisation

A parameter choice strategy for a multilevel augmentation method in iterated Lavrentiev regularization

2018 ◽  
Vol 26 (2) ◽  
pp. 153-170 ◽  
Author(s):  
Chunmei Zeng ◽  
Xingjun Luo ◽  
Suhua Yang ◽  
Fanchun Li
Keyword(s):  
Integral Equation ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
Parameter Choice ◽  
Lavrentiev Regularization ◽  
Choice Strategy ◽  
Ill Posed ◽  
Parameter Choice Strategy

AbstractIn this paper we apply the multilevel augmentation method to solve an ill-posed integral equation via the iterated Lavrentiev regularization. This method leads to fast solutions of discrete iterated Lavrentiev regularization. The convergence rates of the iterated Lavrentiev regularization are achieved by using a certain parameter choice strategy. Finally, numerical experiments are given to illustrate the efficiency of the method.

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