SIAM Journal on Scientific Computing, Ahead of Print.
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.

中文翻译：

---

反问题的带循环的混合投影方法

《 SIAM科学计算杂志》，预印本。
由于其固有的正则化特性以及自适应地选择正则化参数的额外灵活性，迭代混合投影方法已被证明对于解决大型线性逆问题非常有效。在这项工作中，我们开发了基于Golub-Kahan的混合投影方法，该方法可以利用压缩和循环技术来解决一类广泛的反问题，而在这些问题中，内存需求或高计算成本可能会被禁止。对于具有许多未知参数且需要多次迭代的问题，可以使用具有循环功能的混合投影方法压缩和循环使用解决方案基础向量，以减少必须存储的解决方案基础向量的数量，同时获得可与之媲美的解决方案精度。标准方法。如果需要重新正交化，这也可以大大降低计算成本。在其他情况下，例如流数据问题或具有多个数据集的逆问题，具有回收功能的混合投影方法可用于有效地集成先前计算的信息，以便更快更好地进行重建。所提出的方法的额外好处是可以合并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这些方法。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解决方案仍然接近标准混合方法的正则化解决方案，并且揭示了所得投影矩阵之间的重要联系。图像处理中的数值示例显示了将回收与混合投影方法相结合的潜在好处。