Sensing physical fields: Inverse problems for the diffusion equation and beyond

Abstract

Due to significant advances made over the last few decades in the areas of (wireless) networking, communications and microprocessor fabrication, the use of sensor networks to observe physical phenomena is rapidly becoming commonplace. Over this period, many aspects of sensor networks have been explored, yet a thorough understanding of how to analyse and process the vast amounts of sensor data collected, remains an open area of research. This work therefore, aims to provide theoretical, as well as practical, advances this area. In particular, we consider the problem of inferring certain underlying properties of the monitored phenomena, from our sensor measurements. Within mathematics, this is commonly formulated as an inverse problem; whereas in signal processing it appears as a (multidimensional) sampling and reconstruction problem. Indeed it is well known that inverse problems are notoriously ill-posed and very demanding to solve; meanwhile viewing it as the latter also presents several technical challenges. In particular, the monitored field is usually nonbandlimited, the sensor placement is typically non-regular and the space-time dimensions of the field are generally non-homogeneous. Furthermore, although sensor production is a very advanced domain, it is near impossible and/or extremely costly to design sensors with no measurement noise. These challenges therefore motivate the need for a stable, noise robust, yet simple sampling theory for the problem at hand. In our work, we narrow the gap between the domains of inverse problems and modern sampling theory, and in so doing, extend existing results by introducing a framework for solving the inverse source problems for a class of some well-known physical phenomena. Some examples include: the reconstruction of plume sources, thermal monitoring of multi-core processors and acoustic source estimation, to name a few. We assume these phenomena and their sources can be described using partial differential equation (PDE) and parametric source models, respectively. Under this assumption, we obtain a well-posed inverse problem. Initially, we consider a phenomena governed by the two-dimensional diffusion equation -- i.e. 2-D diffusion fields, and assume that we have access to its continuous field measurements. In this setup, we derive novel exact closed-form inverse formulae that solve the inverse diffusion source problem, for a class of localized and non-localized source models. In our derivation, we prove that a particular 1-D sequence of, so called, generalized measurements of the field is governed by a power-sum series, hence it can be efficiently solved using existing algebraic methods such as Prony's method. Next, we show how to obtain these generalized measurements, by using Green's second identity to combine the continuous diffusion field with a family of well-chosen sensing functions. From these new inverse formulae, we therefore develop novel noise robust centralized and distributed reconstruction methods for diffusion fields. Specifically, we extend these inverse formulae to centralized sensor networks using numerical quadrature; conversely for distributed networks, we propose a new physics-driven consensus scheme to approximate the generalized measurements through localized interactions between the sensor nodes. Finally we provide numerical results using both synthetic and real data to validate the proposed algorithms. Given the insights gained, we eventually turn to the more general problem. That is, the two- and three-dimensional inverse source problems for any linear PDE with constant coefficients. Extending the previous framework, we solve the new class of inverse problems by establishing an otherwise subtle link with modern sampling theory. We achieved this by showing that, the desired generalized measurements can be computed by taking linear weighted-sums of the sensor measurements. The advantage of this is two-fold. First, we obtain a more flexible framework that permits the use of more general sensing functions, this freedom is important for solving the 3-D problem. Second, and remarkably, we are able to analyse many more physical phenomena beyond diffusion fields. We prove that computing the proper sequence of generalized measurements for any such field, via linear sums, reduces to approximating (a family of) exponentials with translates of a particular prototype function. We show that this prototype function depends on the Green's function of the field, and then derive an explicit formula to evaluate the proper weights. Furthermore, since we now have more freedom in selecting the sensing functions, we discuss how to make the correct choice whilst emphasizing how to retrieve the unknown source parameters from the resulting (multidimensional) Prony-like systems. Based on this new theory we develop practical, noise robust, sensor network strategies for solving the inverse source problem, and then present numerical simulation results to verify the performance of our proposed schemes.Open Acces