Bayes Meets Krylov: Statistically Inspired Preconditioners for CGLS
Bayes Meets Krylov: Statistically Inspired Preconditioners for CGLS
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DOI：
10.1137/15m1055061
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发表时间：
2018-06-01
期刊：
影响因子：
10.2
通讯作者：
Vantaggi, B.
中科院分区：
文献类型：
--
作者：
Calvetti, D.;Pitolli, F.;Vantaggi, B.
当未知参数超过数据时，线性反问题的解需要解决非平凡零空间的问题。在贝叶斯框架内重述问题后，可以利用关于未知的先验信息来确定零空间对解的贡献。更具体地说，如果相关的线性系统的解决方案是由共轭梯度最小二乘（CGLS）方法计算，额外的信息可以编码的形式，一个正确的预处理。本文研究了右预条件子如何改变CGLS迭代所处的Krylov子空间，从而将贝叶斯推理与Krylov子空间方法联系起来。用算例说明了Bayes Meets Krylov方法求解欠定线性反问题的优越性。
The solution of linear inverse problems when the unknown parameters outnumber data requires addressing the problem of a nontrivial null space. After restating the problem within the Bayesian framework, a priori information about the unknown can be utilized for determining the null space contribution to the solution. More specifically, if the solution of the associated linear system is computed by the conjugate gradient for least squares (CGLS) method, the additional information can be encoded in the form of a right preconditioner. In this paper we study how the right preconditioner changes the Krylov subspaces where the CGLS iterates live, and we draw a tighter connection between Bayesian inference and Krylov subspace methods. The advantages of a Bayes-meets-Krylov approach to the solution of underdetermined linear inverse problems is illustrated with computed examples.