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Convergence Analysis of Iterative One-Shot Inversion Methods for Linear Inverse Problems


Core Concepts
The core message of this article is to establish sufficient conditions on the descent step for the convergence of multi-step one-shot inversion methods, where the forward and adjoint problems are solved iteratively with incomplete inner iterations, for general linear inverse problems.
Abstract
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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Key Insights Distilled From

by Marcella Bon... at arxiv.org 04-12-2024

https://arxiv.org/pdf/2404.07526.pdf
On the convergence analysis of one-shot inversion methods

Deeper Inquiries

How can the convergence analysis be extended to nonlinear inverse problems

To extend the convergence analysis to nonlinear inverse problems, we need to consider the nonlinearity in the forward and adjoint problems, as well as in the parameter identification process. In the context of one-shot inversion methods, this would involve incorporating nonlinear operators and functions into the iterative process. One approach could be to use iterative optimization algorithms that can handle nonlinearities, such as Newton's method or Levenberg-Marquardt algorithm. These methods involve computing the Jacobian or Hessian of the forward and adjoint problems, which can be challenging in the nonlinear case but essential for convergence analysis. Additionally, the convergence analysis for nonlinear inverse problems may require more sophisticated mathematical tools, such as variational analysis and convex optimization theory. The analysis would need to consider the stability and convergence properties of the iterative process in the presence of nonlinearity, potentially leading to more complex convergence criteria. Overall, extending the convergence analysis to nonlinear inverse problems would involve adapting the existing framework to handle the challenges posed by nonlinearity and ensuring the convergence of the iterative optimization process in this more complex setting.

What are the potential drawbacks or limitations of the one-shot inversion approach compared to other optimization-based methods for inverse problems

One potential drawback of the one-shot inversion approach compared to other optimization-based methods for inverse problems is the computational cost. The one-shot method involves simultaneously iterating on the forward and adjoint problems along with the parameter estimation, which can be computationally intensive, especially for large-scale problems. Another limitation is the sensitivity to the choice of regularization parameters in the cost function. The one-shot inversion approach often involves regularization terms to stabilize the optimization process and prevent overfitting. However, selecting the appropriate regularization parameters can be challenging and may require manual tuning, leading to suboptimal results. Furthermore, the one-shot inversion approach may struggle with highly nonlinear or ill-posed inverse problems where the optimization landscape is complex and non-convex. In such cases, the method may get stuck in local minima or struggle to converge to the global solution. Overall, while the one-shot inversion approach offers a simultaneous and efficient way to solve inverse problems, it is important to consider these drawbacks and limitations when choosing the appropriate method for a specific problem.

How can the one-shot inversion framework be adapted to handle constraints or additional regularization terms in the inverse problem formulation

To adapt the one-shot inversion framework to handle constraints or additional regularization terms in the inverse problem formulation, we can modify the cost function and optimization process accordingly. For handling constraints, we can incorporate them directly into the optimization problem by adding constraint functions to the cost function. This can be done using techniques like Lagrange multipliers or penalty methods to enforce the constraints during the optimization process. To include additional regularization terms, we can modify the cost function to include these terms, such as Tikhonov regularization or sparsity-inducing penalties. By adjusting the regularization parameters in the cost function, we can control the trade-off between data fidelity and regularization in the inversion process. Moreover, for constrained optimization, we can utilize optimization algorithms that are specifically designed to handle constraints, such as the projected gradient method or interior-point methods. These algorithms ensure that the solutions satisfy the constraints while optimizing the cost function. Overall, by adapting the cost function and optimization process to incorporate constraints and additional regularization terms, the one-shot inversion framework can be enhanced to handle a wider range of inverse problems with specific requirements and constraints.
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