无限维贝叶斯反问题计算框架。第一部分：线性化情况，及其在全球地震反演中的应用
A computational framework for infinite-dimensional Bayesian inverse problems. Part I: The linearized case, with application to global seismic inversion
AI总结：
本文提出一种计算框架，用于估计线性化无限维统计反问题数值解的不确定性，通过贝叶斯推断方法解决参数场的后验分布问题，并通过低秩近似和高效算法实现高维参数的可扩展计算。
摘要：
我们提出了一种计算框架，用于估计线性化无限维统计反问题数值解的不确定性。我们采用贝叶斯推断方法：给定观测数据及其不确定性，正向问题及其不确定性，以及描述参数场不确定性的先验概率分布，找到参数场上的后验概率分布。先验必须适当选择以保证无限维反问题的适定性，并促进后验计算。此外，直接离散化可能无法导致无限维问题的收敛近似。最后，通过显式构造协方差矩阵来求解离散反问题是不可行的，因为需要求解正向问题的次数与参数数量相同。我们的计算框架基于Stuart提出的无限维公式，并结合了多个组件以确保无限维反问题的收敛离散化。该框架还结合了处理先验、构造数据驱动的后验协方差算子的低秩近似以及探索后验的方法，确保整个框架能够扩展到非常高的参数维度。我们在此计算框架上展示了在具有数万参数的3D全球地震波传播反问题中的贝叶斯解法。
英文摘要：
We present a computational framework for estimating the uncertainty in the numerical solution of linearized infinite-dimensional statistical inverse problems. We adopt the Bayesian inference formulation: given observational data and their uncertainty, the governing forward problem and its uncertainty, and a prior probability distribution describing uncertainty in the parameter field, find the posterior probability distribution over the parameter field. The prior must be chosen appropriately in order to guarantee well-posedness of the infinite-dimensional inverse problem and facilitate computation of the posterior. Furthermore, straightforward discretizations may not lead to convergent approximations of the infinite-dimensional problem. And finally, solution of the discretized inverse problem via explicit construction of the covariance matrix is prohibitive due to the need to solve the forward problem as many times as there are parameters. Our computational framework builds on the infinite-dimensional formulation proposed by Stuart (A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), pp. 451-559), and incorporates a number of components aimed at ensuring a convergent discretization of the underlying infinite-dimensional inverse problem. The framework additionally incorporates algorithms for manipulating the prior, constructing a low rank approximation of the data-informed component of the posterior covariance operator, and exploring the posterior that together ensure scalability of the entire framework to very high parameter dimensions. We demonstrate this computational framework on the Bayesian solution of an inverse problem in 3D global seismic wave propagation with hundreds of thousands of parameters.
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