Volume 58, pp. 450-469, 2023.

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

Pallavi Mahale and Ankit Singh


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 $F$.

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Key words

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

AMS subject classifications

65J20, 47J06

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