无限维线性逆问题中期望信息增益的子模性
Submodularity of the expected information gain in infinite-dimensional linear inverse problems
AI总结：
针对无限维线性高斯贝叶斯逆问题，证明期望信息增益（EIG）的子模性，并推广贪心传感器放置的近似保证到无限维极限。
摘要：
我们考虑具有不相关测量误差的无限维线性高斯贝叶斯逆问题，并专注于选择传感器放置以最大化期望信息增益（EIG）的问题。这项研究受由偏微分方程（PDE）约束的线性逆问题的最优传感器放置所启发。我们考虑每个传感器收集单快照测量的测量模型。这涵盖了由线性稳态PDE或具有最终时间观测的演化方程所支配的逆问题的传感器放置。众所周知，在有限维（离散化）形式的此类逆问题中，EIG是单调子模函数。这也为离散化设置中的贪心传感器放置提供了理论保证。我们将EIG的子模性结果推广到无限维设置，证明贪心传感器放置的近似保证在无限维极限下仍然有效。我们还讨论了计算方面的考虑，并提出了利用问题结构和子模性来实现贪心过程高效计算的策略。
英文摘要：
We consider infinite-dimensional linear Gaussian Bayesian inverse problems with uncorrelated measurement errors and focus on the problem of selecting sensor placements that maximize the expected information gain (EIG). This study is motivated by optimal sensor placement for linear inverse problems constrained by partial differential equations (PDEs). We consider measurement models where each sensor collects a single-snapshot measurement. This covers sensor placement for inverse problems governed by linear steady PDEs or evolution equations with final-in-time observations. It is well-known that in the finite-dimensional (discretized) formulations of such inverse problems, the EIG is a monotone submodular function. This also entails a theoretical guarantee for greedy sensor placement in the discretized setting. We extend the result on submodularity of the EIG to the infinite-dimensional setting, proving that the approximation guarantee of greedy sensor placement remains valid in the infinite-dimensional limit. We also discuss computational considerations and present strategies that exploit problem structure and submodularity to yield efficient implementations of the greedy procedure.
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