This paper tackles optimal sensor placement for Bayesian linear inverse problems, a popular version of the more general Optimal Experiment Design (OED) problem, using the D-optimality criterion. This is done by establishing connections between sensor placement and Column Subset Selection Problem (CSSP), which is a well-studied problem in Numerical Linear Algebra (NLA). In particular, we use the Golub-Klema-Stewart (GKS) approach which involves computing the truncated Singular Value Decomposition (SVD) followed by a pivoted QR factorization on the right singular vectors. The algorithms are further accelerated by using randomization to compute the low-rank approximation as well as for sampling the indices. The resulting algorithms are robust, computationally efficient, require virtually no parameter tuning, and come with strong theoretical guarantees. We also propose a new approach for OED, called reweighted sensors, that selects $k$ sensors but judiciously recombines sensor information to dramatically improve the D-optimality criterion. Additionally, we develop a method for data completion without solving the inverse problem. Numerical experiments on model inverse problems involving the heat equation and seismic tomography in two spatial dimensions demonstrate the performance of our approaches.

翻译：本文针对贝叶斯线性反问题中的最优传感器布局问题（这是更一般的实验最优设计（OED）问题的一个常见形式），采用D最优性准则进行处理。通过建立传感器布局与数值线性代数（NLA）中已深入研究的问题——列子集选择问题（CSSP）——之间的联系，我们采用Golub-Klema-Stewart（GKS）方法，该方法涉及计算截断奇异值分解（SVD），随后对右奇异向量进行主元QR分解。利用随机化技术加速低秩近似计算和指标采样，进一步优化了算法。所提出的算法鲁棒性强、计算效率高、几乎无需参数调优，并具有坚实的理论保证。同时，我们提出了一种名为"重加权传感器"的新型OED方法，该方法选取k个传感器，但通过智能地重组传感器信息以显著改善D最优性准则。此外，我们开发了一种无需解决反问题即可完成数据补全的方法。针对涉及热方程和二维空间地震层析成像的模型反问题进行的数值实验，验证了我们方法的有效性。