偏微分方程控制的贝叶斯逆问题中超参数边缘化的摊销低秩近似
Amortized low-rank approximation for hyperparameter marginalization in PDE-governed Bayesian inverse problems
AI总结：
研究含高维或无限维参数场及少量超参数的分层贝叶斯逆问题，聚焦参数到观测映射为线性PDE的问题。核心方法是基于广义低秩近似，提出摊销变体并比较性能。主要贡献是在数值实验中实现加速。
摘要：
本文解决具有高维或无限维参数场和适量超参数的分层贝叶斯逆问题。聚焦参数到观测映射为线性PDE的一类问题，固定超参数时问题为条件高斯。超参数边缘化需重复评估其边际密度，涉及重复大规模对数行列式比计算和最大后验估计。为应对计算挑战，引入基于先验到后验精度矩阵更新的广义低秩近似的可扩展框架。针对非线性先验超参数，提出该框架的两种摊销变体并比较其理论性质和计算复杂度与直接方法。在数值实验中，评估它们在二维和三维平流扩散初始条件逆问题上的性能，超参数在先验和噪声协方差中均有。发现摊销方法在三维问题中对100次边际密度评估相对于直接方法实现30 - 45倍加速。
英文摘要：
This paper addresses the efficient solution of hierarchical Bayesian inverse problems with a high- or infinite-dimensional parameter field and a moderate number of hyperparameters. We focus on a class of problems in which the parameter-to-observable mapping is a linear PDE, so that, for fixed hyperparameters, the problem becomes conditionally Gaussian. Marginalizing the hyperparameters entails repeated evaluation of their marginal density, which in turn entails repeated large-scale log-determinant ratio computations and maximum a posteriori (MAP) estimations. To address these computational challenges, we introduce a scalable framework that relies on generalized low-rank approximations of the update from the prior to the posterior precision matrix. Though such a framework is state-of-the-art in non-hierarchical settings, in the case of nonlinear prior hyperparameters such as covariance length scales, a direct application to hierarchical problems is inefficient. We propose two amortized variants of this framework, comparing their theoretical properties and computational complexity to the direct method. In numerical experiments, we evaluate their performance on an advection-diffusion initial condition inverse problem in two and three spatial dimensions, with hyperparameters in both the prior and the noise covariance. We find that our amortized methods achieve a factor of 30-45 speedup for 100 marginal density evaluations relative to the direct method in the 3D problem.
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