On the acceleration of optimal regularization algorithms for linear ill-posed inverse problems

Article Computer Science, Theory & Methods

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

Radu Bot et al.

Summary: This paper examines solutions to ill-posed linear problems using dynamical flows, leveraging results from convex analysis to improve these solutions and introduce optimal regularization methods and convergence rates. The proposed flows for minimizing the norm of the residual of a linear operator equation are shown to provide optimal convergence rates for regularized solutions, setting benchmarks for further studies in convex analysis.

FOUNDATIONS OF COMPUTATIONAL MATHEMATICS (2022)

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Article Mathematics, Applied

Optimal-order convergence of Nesterov acceleration for linear ill-posed problems*

Stefan Kindermann

Summary: This paper proves that Nesterov acceleration is an optimal-order iterative regularization method for linear ill-posed problems when a parameter is chosen based on the smoothness of the solution. This method works for both a priori stopping rule and discrepancy principle under Holder source conditions. The key tool to achieve these results is a representation of the residual polynomials using Gegenbauer polynomials.

INVERSE PROBLEMS (2021)

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Article Mathematics, Applied

TWO NEW NON-NEGATIVITY PRESERVING ITERATIVE REGULARIZATION METHODS FOR ILL-POSED INVERSE PROBLEMS

Ye Zhang et al.

Summary: This paper introduces two novel non-negativity preserving iterative regularization methods for solving ill-posed linear operator equations in a Hilbert space setting. These methods demonstrate stronger convergence compared to traditional methods, with adapted discrepancy principles used as stopping rules. Additionally, the accuracy and acceleration effects of the new methods are showcased through application to a biosensor problem and numerical examples.

INVERSE PROBLEMS AND IMAGING (2021)

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Article Mathematics, Applied