The article focuses on the convergence analysis of multi-step one-shot inversion methods for solving linear inverse problems. The key highlights and insights are:
One-shot methods iterate simultaneously on the inverse problem unknown and the forward/adjoint problem solutions, which can be advantageous for large-scale problems where the forward and adjoint problems are solved iteratively rather than exactly.
The authors analyze two variants of multi-step one-shot methods: the k-step one-shot method and the semi-implicit k-step one-shot method, where k inner iterations are performed on the state and adjoint state before updating the parameter.
The convergence analysis is performed by studying the eigenvalues of the block iteration matrix of the coupled iterations. Sufficient conditions on the descent step size are derived to ensure that all eigenvalues lie inside the unit circle, guaranteeing the convergence of the one-shot methods.
The analysis considers the case where the inner iterations on the forward and adjoint problems are incomplete, i.e., stopped before achieving high accuracy. This is motivated by the fact that solving these problems exactly can be very time-consuming with little improvement in the accuracy of the inverse problem solution.
Numerical experiments on a 2D Helmholtz inverse problem demonstrate that very few inner iterations are enough to guarantee good convergence of the one-shot inversion algorithms, even in the presence of noisy data.
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