Group greedy method for sensor placement

This paper discusses greedy methods for sensor placement in linear inverse problems. We comprehensively review the greedy methods in the sense of optimizing the mean squared error (MSE), the volume of the confidence ellipsoid, and the worst-case error variance. We show that the greedy method of optimizing an MSE related cost function can find a near-optimal solution. We then provide a new fast algorithm to optimize the MSE. In greedy methods, we select the sensing location one by one. In this way, the searching space is greatly reduced but many valid solutions are ignored. To further improve the current greedy methods, we propose a group-greedy strategy, which can be applied to optimize all the three criteria. In each step, we reserve a group of suboptimal sensor configurations which are used to generate the potential sensor configurations of the next step and the best one is used to check the terminal condition. Compared with the current greedy methods, the group-greedy strategy increases the searching space but greatly improve the solution performance. We find the necessary and sufficient conditions that the current greedy methods and the proposed group greedy method can obtain the optimal solution. The illustrative examples show that the group greedy method outperforms the corresponding greedy method. We also provide a practical way to find a proper group size with which the proposed group greedy method can find a solution that has almost the same performance as the optimal solution.