Hybrid Projection Methods with Recycling for Inverse Problems
Hybrid Projection Methods with Recycling for Inverse Problems
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DOI：
10.1137/20m1349515
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发表时间：
2020-07
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影响因子：
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通讯作者：
Julianne Chung;E. D. Sturler;Jiahua Jiang
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作者：
Julianne Chung;E. D. Sturler;Jiahua Jiang
迭代混合投影方法由于其固有的正则化特性以及自适应选择正则化参数的灵活性，已被证明是非常有效的解决大型线性反问题。在这项工作中，我们开发了基于Golub-Kahan的混合投影方法，可以利用压缩和回收技术，以解决一类广泛的逆问题，其中内存需求或高计算成本可能会令人望而却步。对于具有许多未知参数且需要多次迭代的问题，可以使用具有回收的混合投影方法来压缩和回收解基向量，以减少必须存储的解基向量的数量，同时获得与标准方法相当的解精度。如果需要重新正交化，这也可以大大降低计算成本。在其他场景中，例如流数据问题或多个数据集的逆问题，具有再循环的混合投影方法可以用于有效地整合先前计算的信息，以实现更快和更好的重建。所提出的方法的额外益处在于，可以并入各种子空间选择和压缩技术，可以使用用于自动正则化参数选择的标准技术，并且可以以迭代方式多次应用所述方法。理论结果表明，在合理的条件下，我们提出的回收混合方法的正则化的解决方案仍然接近标准的混合方法的正则化的解决方案，并揭示了重要的连接所产生的投影矩阵。从图像处理的数值例子显示了结合回收与混合投影方法的潜在好处。
Iterative hybrid projection methods have proven to be very effective for solving large linear inverse problems due to their inherent regularizing properties as well as the added flexibility to select regularization parameters adaptively. In this work, we develop Golub-Kahan-based hybrid projection methods that can exploit compression and recycling techniques in order to solve a broad class of inverse problems where memory requirements or high computational cost may otherwise be prohibitive. For problems that have many unknown parameters and require many iterations, hybrid projection methods with recycling can be used to compress and recycle the solution basis vectors to reduce the number of solution basis vectors that must be stored, while obtaining a solution accuracy that is comparable to that of standard methods. If reorthogonalization is required, this may also reduce computational cost substantially. In other scenarios, such as streaming data problems or inverse problems with multiple datasets, hybrid projection methods with recycling can be used to efficiently integrate previously computed information for faster and better reconstruction. Additional benefits of the proposed methods are that various subspace selection and compression techniques can be incorporated, standard techniques for automatic regularization parameter selection can be used, and the methods can be applied multiple times in an iterative fashion. Theoretical results show that, under reasonable conditions, regularized solutions for our proposed recycling hybrid method remain close to regularized solutions for standard hybrid methods and reveal important connections among the resulting projection matrices. Numerical examples from image processing show the potential benefits of combining recycling with hybrid projection methods.