[Paper Review] A Universal Sampling Framework for Solving Physics-driven Inverse Source Problems.

John Murray–Bruce, Pier Luigi Dragotti|arXiv (Cornell University)|Feb 16, 2017

Scientific Measurement and Uncertainty Evaluation3 citations

TL;DR

This paper proposes a universal sampling framework for solving physics-driven inverse source problems governed by linear partial differential equations. By leveraging sampling theory and multidimensional frequency estimation, it reconstructs unknown field sources from generalized measurements—computed as weighted sums of sensor data—where the weights are derived from the Green's function, enabling noise-robust, practical sensor network strategies with verified numerical performance.

ABSTRACT

Partial differential equations are central to describing many physical phenomena. Moreover in many applications these phenomena are observed through a sensor network, with the aim of inferring its underlying properties. Here we present a new framework for analysing fields governed by linear partial differential equations. The framework leverages from certain results in sampling and approximation theory to provide a unifying approach for solving a class of inverse source problems. Specifically, we show that the unknown field sources can be recovered from a sequence of so called generalised measurements using multidimensional frequency estimation techniques. Moreover, we show that this sequence of generalised measurements for our physics-driven fields, can be computed by taking linear weighted-sums of the sensor measurements, whereby the exact weights (of the sums) coincide with those that can reproduce multidimensional exponentials from linearly combined translates of a particular prototype function. The prototype function and the desired weights are shown to depend on the Green's function of the underlying field. 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.

Motivation & Objective

• To develop a unifying framework for solving inverse source problems in physics-driven systems governed by linear partial differential equations (PDEs).
• To enable reconstruction of unknown field sources from sensor network measurements using generalized measurements derived from sampling theory.
• To design noise-robust, practical sensor network strategies based on the proposed framework for real-world applicability.
• To establish a theoretical link between the Green's function of the underlying PDE and the optimal weights for reconstructing sources from sensor data.

Proposed method

• Utilizes results from sampling and approximation theory to model the inverse source problem in a unified mathematical framework.
• Represents the generalized measurements as linear weighted-sums of sensor outputs, where the weights are derived from reproducing kernels associated with multidimensional exponentials.
• Derives the optimal weights from the Green's function of the governing PDE, ensuring consistency with the physical field model.
• Employs multidimensional frequency estimation techniques to recover the unknown source distribution from the sequence of generalized measurements.
• Constructs sensor network configurations that minimize reconstruction error and enhance robustness to noise using the derived sampling strategy.
• Validates the framework through numerical simulations demonstrating accurate source recovery under noisy conditions.

Experimental results

Research questions

• RQ1 How can a unified framework be developed to solve inverse source problems across diverse physics-driven systems governed by linear PDEs?
• RQ2 What is the role of the Green's function in determining the optimal weights for reconstructing sources from sensor measurements?
• RQ3 Can generalized measurements—formed as weighted sums of sensor data—enable accurate and stable source reconstruction via multidimensional frequency estimation?
• RQ4 How can sensor network configurations be designed to ensure noise robustness while maintaining high reconstruction fidelity?
• RQ5 What is the theoretical connection between sampling theory, kernel-based reconstruction, and the solution of inverse source problems in PDE-driven systems?

Key findings

• The framework enables universal reconstruction of unknown field sources from generalized measurements derived via linear combinations of sensor data.
• The optimal weights for these linear combinations are explicitly determined by the Green's function of the underlying PDE, ensuring physical consistency.
• Multidimensional frequency estimation techniques successfully recover source distributions from the generalized measurements with high accuracy.
• The proposed sensor network strategies are robust to noise, as demonstrated by numerical simulations under controlled noise conditions.
• Theoretical analysis confirms that the sequence of generalized measurements can be computed using the same weights that reproduce multidimensional exponentials from translates of a prototype function.
• Numerical results validate the effectiveness of the framework, showing stable and accurate source reconstruction even in noisy environments.