On convergence rates of adaptive ensemble Kalman inversion for linear ill-posed problems

Parzer Fabian; Scherzer Otmar

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On convergence rates of adaptive ensemble Kalman inversion for linear ill-posed problems

机译：线性病态问题的自适应集成卡尔曼反演的收敛率研究

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In this paper we discuss a deterministic form of ensemble Kalman inversion as a regularization method for linear inverse problems. By interpreting ensemble Kalman inversion as a low-rank approximation of Tikhonov regularization, we are able to introduce a new sampling scheme based on the Nystrom method that improves practical performance. Furthermore, we formulate an adaptive version of ensemble Kalman inversion where the sample size is coupled with the regularization parameter. We prove that the proposed scheme yields an order optimal regularization method under standard assumptions if the discrepancy principle is used as a stopping criterion. The paper concludes with a numerical comparison of the discussed methods for an inverse problem of the Radon transform.

机译：在本文中，我们讨论了集成卡尔曼反演的确定性形式作为线性逆问题的正则化方法。通过将集成卡尔曼反演解释为Tikhonov正则化的低秩近似，我们能够引入一种基于Nystrom方法的新采样方案，以提高实际性能。此外，我们制定了集成卡尔曼反演的自适应版本，其中样本量与正则化参数耦合。证明了在标准假设下，如果以差异原理为停止准则，则所提方案产生了一种阶最优正则化方法。最后，本文对所讨论的氡变换逆问题的方法进行了数值比较。

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来源
《Numerische Mathematik》 |2022年第2期|371-409|共39页
作者
Parzer Fabian; Scherzer Otmar;
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作者单位

Univ Vienna;

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原文格式 PDF
正文语种英语
中图分类数值分析;
关键词
47A52; 65J20; 65C05; TIKHONOV REGULARIZATION; LANDWEBER ITERATION; FILTER; MATRIX; APPROXIMATION; ALGORITHMS; EQUATIONS; BOUNDS;

机译：47A52;65J20;65C05;TIKHONOV 正则化;LANDWEBER 迭代;过滤器;矩阵;近似值;算法;方程式;边界;

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2. Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed Problems [J] . Boţ Radu, Dong Guozhi, Elbau PeterScherzer Otmar Foundations of computational mathematics . 2022,第5期

机译：Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed Problems
3. Ill-posed problems and the conjugate gradient method: Optimal convergence rates in the presence of discretization and modelling errors [J] . Neubauer Andreas Journal of inverse and ill-posed problems . 2022,第6期

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On convergence rates of adaptive ensemble Kalman inversion for linear ill-posed problems

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