Sampled Tikhonov regularization for large linear inverse problems
Sampled Tikhonov regularization for large linear inverse problems
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DOI：
10.1088/1361-6420/ab2787
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发表时间：
2018-12
期刊：
影响因子：
2.1
通讯作者：
J. T. Slagel;Julianne Chung;Matthias Chung;David Kozak;L. Tenorio
中科院分区：
文献类型：
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作者：
J. T. Slagel;Julianne Chung;Matthias Chung;David Kozak;L. Tenorio
在本文中，我们研究基于数据采样的迭代方法来计算吉洪诺夫正则化解。我们专注于非常大的逆问题，其中不可能一次性访问整个数据集（例如，对于流或海量数据集的问题）。行访问方法为解决此类问题提供了一个理想的框架，因为它们只需要在任何给定时间访问数据“块”。然而，当使用这些迭代采样方法来解决逆问题时，主要挑战包括正确选择正则化参数、适当的采样策略和收敛分析。为了解决这些挑战，我们描述了一系列采样迭代方法，这些方法可以在数据可用时合并数据（例如随机采样）。我们考虑两种采样迭代方法，其中迭代可以被描述为一系列近似吉洪诺夫问题的解。第一种方法要求先验固定正则化参数，并渐近收敛到随机采样数据的非正则化解。这对于逆问题来说是不希望的。因此，我们关注第二种方法，其主要优点是可以在迭代过程中更新正则化参数，并且迭代渐近收敛到吉洪诺夫正则化解。我们描述了基于采样残差更新正则化参数的自适应方法，并且我们为更大的问题提供了有限内存变体。包括大规模超分辨率成像示例在内的数值示例证明了这些方法的潜力。
In this paper we investigate iterative methods that are based on sampling of the data for computing Tikhonov-regularized solutions. We focus on very large inverse problems where access to the entire data set is not possible all-at-once (e.g. for problems with streaming or massive datasets). Row-access methods provide an ideal framework for solving such problems since they only require access to ‘blocks’ of the data at any given time. However, when using these iterative sampling methods to solve inverse problems, the main challenges include a proper choice of the regularization parameter, appropriate sampling strategies, and a convergence analysis. To address these challenges, we describe a family of sampled iterative methods that can incorporate data as they become available (e.g. randomly sampled). We consider two sampled iterative methods where the iterates can be characterized as solutions to a sequence of approximate Tikhonov problems. The first method requires the regularization parameter to be fixed a priori and converges asymptotically to an unregularized solution for randomly sampled data. This is undesirable for inverse problems. Thus, we focus on the second method where the main benefits are that the regularization parameter can be updated during the iterative process and the iterates converge asymptotically to a Tikhonov-regularized solution. We describe adaptive approaches to update the regularization parameter that are based on sampled residuals, and we provide a limited-memory variant for larger problems. Numerical examples, including a large-scale super-resolution imaging example, demonstrate the potential for these methods.