Partial differential equations (PDEs) describe several physical phenomena, from light transport and transport of atmospheric pollutants to the spread of diseases and rumors in a network. Typically, the aim in these scenarios is to recover certain properties of a physical phenomenon from observations, collected by, for instance, an array of optical detectors or a sensor network.

In this talk, we will briefly explore two applications. The first demonstrates the utility of a PDE model in computing images from extremely weak optical signals: such as in non-line-of-sight imaging, where measurements of diffusely reflected light, containing seemingly little spatial information, are used to reconstruct images of hidden scenes. The second is on inverse source problems in environmental monitoring using sensor networks. Here, we will briefly outline a sampling-theoretic framework to solve a class of inverse source problems for fields governed by linear PDEs. Specifically, our framework maps the inverse source problem to a multidimensional frequency estimation problem and yields demonstrably efficient and noise-robust, sensor network strategies for solving linear PDE-constrained inverse source problems.