Solving physics-driven inverse problems via structured least squares
Numerous physical phenomena are well modeled by partial differential equations (PDEs); they describe a wide range of phenomena across many application domains, from modeling EEG signals in electroencephalography to, modeling the release and propagation of toxic substances in environmental monitoring. In these applications it is often of interest to find the sources of the resulting phenomena, given some sparse sensor measurements of it. This will be the main task of this work. Specifically, we will show that finding the sources of such PDE-driven fields can be turned into solving a class of well-known multi-dimensional structured least squares problems. This link is achieved by leveraging from recent results in modern sampling theory - in particular, the approximate Strang-Fix theory. Subsequently, numerical simulation results are provided in order to demonstrate the validity and robustness of the proposed framework.