Iterative Lavrentiev regularization method under a heuristic rule for nonlinear ill-posed operator equations

Pallavi  Mahale; Ankit  Singh

doi:doi:10.1553/etna_vol58s450

Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

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Iterative Lavrentiev regularization method under a heuristic rule for nonlinear ill-posed operator equations

Pallavi Mahale,

Ankit Singh

ETNA - Electronic Transactions on Numerical Analysis, pp. 450-469, 2023/07/03

doi: 10.1553/etna_vol58s450

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doi:10.1553/etna_vol58s450

Abstract

In this paper, we consider the iterative Lavrentiev regularization method for obtaining a stable approximate solution for a nonlinear ill-posed operator equation $F(x)=y$, where $ F:D(F) \subset X \rightarrow X$ is a nonlinear monotone operator on the Hilbert spaces $X$. In order to obtain a stable approximate solution using iterative regularization methods, it is important to use a suitable stopping rule to terminate the iterations at the appropriate step. Recently, Qinian Jin and Wei Wang (2018) have proposed a heuristic rule to stop the iterations for the iteratively regularized Gauss-Newton method. The advantage of a heuristic rule over the existing a priori and a posteriori rules is that it does not require accurate information on the noise level, which may not be available or reliable in practical applications. In this paper, we propose a heuristic stopping rule for an iterated Lavrentiev regularization method.We derive error estimates under suitable nonlinearity conditions on the operator $F$.

Keywords: Lavrentiev regularization, nonlinear ill-posed problems, heuristic parameter choice rules

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Published Online: 2023/07/03 09:34:27

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Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

Iterative Lavrentiev regularization method under a heuristic rule for nonlinear ill-posed operator equations

Abstract