SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page A1714-A1740, January 2020.
We consider optimal design of PDE-based Bayesian linear inverse problems with infinite-dimensional parameters. We focus on the A-optimal design criterion, defined as the average posterior variance and quantified by the trace of the posterior covariance operator. We propose using structure exploiting randomized methods to compute the A-optimal objective function and its gradient, and we provide a detailed analysis of the error for the proposed estimators. To ensure sparse and binary design vectors, we develop a novel reweighted $\ell_1$-minimization algorithm. We also introduce a modified A-optimal criterion and present randomized estimators for its efficient computation. We present numerical results illustrating the proposed methods on a model contaminant source identification problem, where the inverse problem seeks to recover the initial state of a contaminant plume using discrete measurements of the contaminant in space and time.

中文翻译：

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线性反问题的A-最优设计的随机化和加权加权\\ ell_1 $-最小化

SIAM科学计算杂志，第42卷，第3期，第A1714-A1740页，2020年1月。
我们考虑具有无限维参数的基于PDE的贝叶斯线性逆问题的优化设计。我们关注于A最优设计标准，该标准定义为平均后验方差，并通过后协方差算子的痕迹进行量化。我们提出使用结构利用随机方法来计算A最优目标函数及其梯度，并为所提出的估计量提供误差的详细分析。为了确保稀疏和二进制设计向量，我们开发了一种新颖的加权$ \ ell_1 $最小化算法。我们还介绍了一种改进的A最优准则，并提出了用于其高效计算的随机估计器。我们提供的数值结果说明了针对模型污染物源识别问题的拟议方法，