Efficient generalized Golub–Kahan based methods for dynamic inverse problems
Efficient generalized Golub–Kahan based methods for dynamic inverse problems
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DOI：
10.1088/1361-6420/aaa0e1
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发表时间：
2017-05
期刊：
影响因子：
2.1
通讯作者：
Julianne Chung;A. Saibaba;Matthew Brown;E. Westman
中科院分区：
文献类型：
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作者：
Julianne Chung;A. Saibaba;Matthew Brown;E. Westman
我们考虑计算动态反问题的解和估计不确定性的有效方法，其中感兴趣的参数可能在测量过程中发生变化。与静态逆问题相比，在贝叶斯框架中合并空间和时间的先验信息可能会变得计算密集，部分原因是由于存在大量未知参数。在这些问题中，不可能显式计算先验协方差矩阵的平方根和/或逆矩阵，因此我们考虑基于广义 Golub-Kahan 双对角化的高效、迭代、无矩阵方法，该方法允许自动正则化参数和方差估计。我们证明这些动态反演方法可以比标准方法更灵活，并开发可以利用先验结构以及前向模型中可能的结构的有效实现。光声断层扫描、时空去模糊和被动地震断层扫描的数值示例证明了所描述方法的适用范围和有效性。具体来说，在被动地震层析成像中，我们展示了我们对合成数据和真实数据的方法。为了证明我们算法的可扩展性，我们在标准桌面上在 40 秒内解决了大约 43000 个测量值和 780 万个未知数的动态反问题。
We consider efficient methods for computing solutions to and estimating uncertainties in dynamic inverse problems, where the parameters of interest may change during the measurement procedure. Compared to static inverse problems, incorporating prior information in both space and time in a Bayesian framework can become computationally intensive, in part, due to the large number of unknown parameters. In these problems, explicit computation of the square root and/or inverse of the prior covariance matrix is not possible, so we consider efficient, iterative, matrix-free methods based on the generalized Golub–Kahan bidiagonalization that allow automatic regularization parameter and variance estimation. We demonstrate that these methods for dynamic inversion can be more flexible than standard methods and develop efficient implementations that can exploit structure in the prior, as well as possible structure in the forward model. Numerical examples from photoacoustic tomography, space-time deblurring, and passive seismic tomography demonstrate the range of applicability and effectiveness of the described approaches. Specifically, in passive seismic tomography, we demonstrate our approach on both synthetic and real data. To demonstrate the scalability of our algorithm, we solve a dynamic inverse problem with approximately 43000 measurements and 7.8 million unknowns in under 40 s on a standard desktop.