In Bayesian inverse problems, it is common to consider several hyperparameters that define the prior and the noise model that must be estimated from the data. In particular, we are interested in linear inverse problems with additive Gaussian noise and Gaussian priors defined using Mat\'{e}rn covariance models. In this case, we estimate the hyperparameters using the maximum a posteriori (MAP) estimate of the marginalized posterior distribution. However, this is a computationally intensive task since it involves computing log determinants. To address this challenge, we consider a stochastic average approximation (SAA) of the objective function and use the preconditioned Lanczos method to compute efficient approximations of the function and gradient evaluations. We propose a new preconditioner that can be updated cheaply for new values of the hyperparameters and an approach to compute approximations of the gradient evaluations, by reutilizing information from the function evaluations. We demonstrate the performance of our approach on static and dynamic seismic tomography problems.

翻译：在贝叶斯反问题中，通常需考虑若干定义先验分布与噪声模型的超参数，这些参数必须通过数据估计得到。本文特别关注具有加性高斯噪声的线性反问题，以及采用Matérn协方差模型定义的高斯先验分布。在此框架下，我们通过边缘化后验分布的最大后验概率估计来确定超参数。然而，由于涉及对数行列式计算，该任务计算量极大。为应对这一挑战，我们采用目标函数的随机平均逼近方法，并利用预处理Lanczos算法高效逼近目标函数值与梯度计算。我们提出了一种新型预处理子，该预处理子能够以较低计算成本随超参数值更新，同时通过复用函数计算中的信息来实现梯度计算的高效逼近。我们在静态与动态地震层析成像问题上验证了所提方法的性能。