A Fast and Scalable Computational Framework for Large-Scale High-Dimensional Bayesian Optimal Experimental Design
A Fast and Scalable Computational Framework for Large-Scale High-Dimensional Bayesian Optimal Experimental Design
复制标题
用于大规模高维贝叶斯最优实验设计的快速且可扩展的计算框架
DOI：
10.1137/21m1466499
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发表时间：
2023
期刊：
影响因子：
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通讯作者：
Ghattas, Omar
中科院分区：
文献类型：
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作者：
Wu, Keyi;Chen, Peng;Ghattas, Omar
我们开发了一个快速和可扩展的计算框架来解决偏微分方程组(PDE)控制的贝叶斯最优试验设计问题，并将其应用于最大化期望信息增益(EIG)的传感器优化布置。由于高维参数的维度灾难和大规模偏微分方程组的昂贵解决方案，这些问题特别具有挑战性。为了应对这些挑战，我们利用了两个基本性质：(1)参数可观测映射的雅可比矩阵的低阶结构，以提取本质上低维的数据信息子空间；(2)EIG的一系列近似，在保持与真实EIG的高度相关性的同时，减少了PDE解的数量。基于这些性质，我们提出了一种有效的离线-在线分解方法来求解优化问题。离线阶段控制着成本，需要预先计算需要PDE解决方案的所有组件。在线阶段优化了传感器的布置，不需要任何偏微分方程组求解。对于在线阶段，我们提出了一种新的贪婪算法，该算法首先使用杠杆得分放置一组初始传感器，然后将选择的传感器与其他候选节点交换，直到满足一定的收敛标准，我们称之为交换贪婪算法。通过线性和非线性反问题验证了该方法的有效性和可伸缩性。特别地，我们证明了对于这两个问题，所需的偏微分方程解的数量很少，与参数维度无关，并且仅弱依赖于数据维度。
We develop a fast and scalable computational framework to solve Bayesian optimal experimental design problems governed by partial differential equations (PDEs) with application to optimal sensor placement by maximizing expected information gain (EIG). Such problems are particularly challenging due to the curse of dimensionality for high-dimensional parameters and the expensive solution of large-scale PDEs. To address these challenges, we exploit two fundamental properties: (1) the low-rank structure of the Jacobian of the parameter-to-observable map, to extract the intrinsically low-dimensional data-informed subspace, and (2) a series of approximations of the EIG that reduce the number of PDE solves while retaining high correlation with the true EIG. Based on these properties, we propose an efficient offline-online decomposition for the optimization problem. The offline stage dominates the cost and entails precomputing all components that require PDE solves. The online stage optimizes sensor placement and does not require any PDE solves. For the online stage, we propose a new greedy algorithm that first places an initial set of sensors using leverage scores and then swaps the selected sensors with other candidates until certain convergence criteria are met, which we call a swapping greedy algorithm. We demonstrate the efficiency and scalability of the proposed method by both linear and nonlinear inverse problems. In particular, we show that the number of required PDE solves is small, independent of the parameter dimension, and only weakly dependent on the data dimension for both problems.