Sensor Placement Sensitivity and Robust Reconstruction of Wave Dynamics from Multiple Sensors

This paper presents a rigorous analysis of circumstances under which wave dynamics can be fully reconstructed from sensor networks. We prove that, in many cases, wave dynamics can be fully reconstructed from a single sensor, but that such reconstructions can be sensitive to small perturbations in sensor placement. In general, multiple sensors are necessary, and we outline a procedure for reconstructing the dynamics from them. We also show that if a network of n sensors suffers from Gaussian noise, then the reconstruction error decreases as n ^-1/². We illustrate and verify these results numerically by simulating wave dynamics and then reconstructing from sensor data the initial state of the system and time series of the system at randomly chosen locations.