Efficient Krylov subspace methods for uncertainty quantification in large Bayesian linear inverse problems
Efficient Krylov subspace methods for uncertainty quantification in large Bayesian linear inverse problems
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DOI：
10.1002/nla.2325
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发表时间：
2020-08
影响因子：
4.3
通讯作者：
A. Saibaba;Julianne Chung;Katrina Petroske
中科院分区：
文献类型：
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作者：
A. Saibaba;Julianne Chung;Katrina Petroske
线性逆问题的不确定性量化仍然是一项具有挑战性的任务，特别是对于具有非常大量未知参数的问题（例如，动态逆问题）以及用于计算平方根和先验协方差矩阵的逆不可行的问题。这项工作利用Krylov子空间方法来开发和分析逆问题中大规模不确定性量化的新技术。在这项工作中，我们假设广义Golub-Kahan-based方法已被用来计算估计的解决方案，我们描述了有效的方法来探索后验分布。特别地，我们使用广义Golub-Kahan双对角化来推导后验协方差矩阵的近似，并且我们提供了量化近似后验协方差矩阵和所得后验分布的准确性的理论结果。然后，我们描述了使用近似计算不确定性度量的有效方法，包括Kullback-Liebler散度。我们提出了两种方法，使用预处理Lanczos算法有效地生成样本的后验分布。动态光声层析成像的数值例子证明了所述方法的有效性。
Uncertainty quantification for linear inverse problems remains a challenging task, especially for problems with a very large number of unknown parameters (e.g., dynamic inverse problems) and for problems where computation of the square root and inverse of the prior covariance matrix are not feasible. This work exploits Krylov subspace methods to develop and analyze new techniques for large‐scale uncertainty quantification in inverse problems. In this work, we assume that generalized Golub‐Kahan‐based methods have been used to compute an estimate of the solution, and we describe efficient methods to explore the posterior distribution. In particular, we use the generalized Golub‐Kahan bidiagonalization to derive an approximation of the posterior covariance matrix, and we provide theoretical results that quantify the accuracy of the approximate posterior covariance matrix and of the resulting posterior distribution. Then, we describe efficient methods that use the approximation to compute measures of uncertainty, including the Kullback‐Liebler divergence. We present two methods that use the preconditioned Lanczos algorithm to efficiently generate samples from the posterior distribution. Numerical examples from dynamic photoacoustic tomography demonstrate the effectiveness of the described approaches.