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.
Highlights
S ENSOR NETWORKS, and the use thereof, for sensing and monitoring physical fields is receiving significant research attention due, in part, to the significant advances made over the last few decades in the fields of networking, communications and in the fabrication of microprocessors [3], [4]
The mode of transport governing the dispersion of plumes in environmental monitoring [7], spreading of fungal diseases in precision agriculture [8], biochemical and nuclear wastes [9] is well-known to be diffusion and the corresponding diffusion field is the variation in concentration of the released substance over space and time
The term physics-driven is used in this paper to describe physical phenomena, physical fields, that propagate through space and time according to some linear partial differential equation (PDE)
Summary
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. 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. 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 present numerical simulation results to verify their performance
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