Title: A Sampling Framework for Solving Physics-driven Inverse Source Problems,
Abstract: 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,
generalised measurements by using multidimensional frequency estimation
techniques. Next we show that---for physics-driven fields---this sequence of
generalised 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 formulae are then derived for the
sequence of weights, that map sensor samples to the exact sequence of
generalised measurements when the Green's function satisfies the generalised
Strang-Fix condition. Otherwise, the same mapping yields a close approximation
of the generalised 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.