This paper considers the optimal sensor allocation for estimating the emission rates of multiple sources in a two-dimensional spatial domain. Locations of potential emission sources are known (e.g., factory stacks), and the number of sources is much greater than the number of sensors that can be deployed, giving rise to the optimal sensor allocation problem. In particular, we consider linear dispersion forward models, and the optimal sensor allocation is formulated as a bilevel optimization problem. The outer problem determines the optimal sensor locations by minimizing the overall Mean Squared Error of the estimated emission rates over various wind conditions, while the inner problem solves an inverse problem that estimates the emission rates. Two algorithms, including the repeated Sample Average Approximation and the Stochastic Gradient Descent based bilevel approximation, are investigated in solving the sensor allocation problem. Convergence analysis is performed to obtain the performance guarantee, and numerical examples are presented to illustrate the proposed approach.

翻译：本文研究二维空间域中用于估计多个源排放速率的最优传感器分配问题。潜在排放源的位置已知（如工厂烟囱），但源的数量远大于可部署传感器的数量，由此产生了最优传感器分配问题。具体而言，我们考虑线性扩散前向模型，并将最优传感器分配问题表述为双层优化问题。外层问题通过最小化不同风况下估计排放速率的总体均方误差来确定最优传感器位置，内层问题则求解估计排放速率的反问题。本文研究了两种求解传感器分配问题的算法：重复样本平均近似法和基于随机梯度下降的双层近似法。通过收敛性分析获得了性能保证，并给出数值算例对所提方法进行验证。