A Sampling Framework for Solving Physics-Driven Inverse Source Problems
Partial differential equations are central to describing many physical phenomena. In many applications, these phenomena are observed through a sensor network, with the aim of inferring their underlying properties. Leveraging from certain results in sampling and approximation theory, we present a new framework for solving a class of inverse source problems for physical fields governed by linear partial differential equations. Specifically, we demonstrate that the unknown field sources can be recovered from a sequence of, so called, generalized measurements by using multidimensional frequency estimation techniques. Next we show that—for physics-driven fields—this sequence of generalized measurements can be estimated by computing a linear weighted sum of the sensor measurements; whereby the exact weights (of the sums) correspond to those that reproduce multidimensional exponentials, when used to linearly combine translates of a particular prototype function related to the Green's function of our underlying field. Explicit formulas are then derived for the sequence of weights, which map sensor samples to the exact sequence of generalized measurements when the Green's function satisfies the generalized Strang-Fix condition. Otherwise, the same mapping yields a close approximation of the generalized measurements. Based on this new framework, we develop practical, noise robust, sensor network strategies for solving the inverse source problem, and then present numerical simulation results to verify their performance.