Determining evolutionary equations by a single passive boundary observation

This paper addresses the longstanding inverse problem of simultaneously recovering both causal sources and medium parameters in evolutionary PDEs from a single passive boundary observation. We develop a mathematical framework focusing on second-order hyperbolic systems, where the measurement consists of the Cauchy pair of the wave field restricted to an open subset of the boundary over all positive time. Under a structural invariance condition on the source-to-speed ratio, we prove that the initial source (f) and wave speed (c) of the wave equation are uniquely determined by such boundary data. The proof combines intricate integral identities, Fourier and harmonic analysis, and subtle high-frequency asymptotics, avoiding artificial decoupling assumptions and accommodating physically realistic scenarios. As key extensions, we demonstrate the recovery of multiple unknowns in more general hyperbolic systems and establish analogous unique determination results for parabolic and Schrödinger-type equations, showcasing the versatility of the methodology. These results resolve a major open problem in coupled-physics imaging and provide a rigorous mathematical foundation for similar inverse problems arising in more sophisticated evolutionary settings.

💡 Research Summary

The paper tackles a classic and notoriously difficult inverse problem: recovering both the initial source and the medium parameters of an evolutionary partial differential equation from a single passive boundary measurement. Focusing on second‑order hyperbolic systems (the wave equation with spatially varying speed (c(x)) and possibly anisotropic conductivity tensor (\sigma(x))), the authors consider the Cauchy data ((u,\partial_\nu u)) recorded on an open subset (\Gamma) of the boundary for all positive times. The sources (f) (initial displacement) and (h) (initial velocity) are compactly supported in a bounded domain (\Omega), while (c) and (\sigma) coincide with the constant background outside (\Omega).

Two structural “admissibility” conditions are introduced. The first requires the existence of a unit vector (\omega) such that (\omega\cdot\nabla\bigl(f/c^{2}\bigr)=0) almost everywhere; this means the ratio (f/c^{2}) is constant along one direction. Under this condition and a non‑trapping hypothesis (ensuring that high‑frequency rays leave (\Omega) in finite time), Theorem 1.2 proves that the boundary data uniquely determines the function (f/c^{2}). No separate knowledge of (c) is needed.

A stronger admissibility condition assumes a unit vector (\hat\omega) with (\hat\omega\cdot\nabla f=0). Together with the first condition, Theorem 1.3 shows that the same boundary measurements uniquely determine both the source (f) and the speed (c) (the latter on the support of (f)). This is, to the best of the authors’ knowledge, the first rigorous uniqueness result for simultaneous recovery of source and wave speed from a single passive measurement without artificial decoupling.

The proof strategy departs from traditional low‑frequency Carleman‑estimate approaches. First, a temporal Fourier transform converts the hyperbolic problem into a family of Helmholtz equations parameterized by frequency. Then, high‑frequency asymptotics are analyzed via stationary phase and microlocal techniques, yielding precise decay estimates for the transformed solutions. Two integral identities are derived: one linking the boundary data to the unknown ratio (f/c^{2}), the other separating (f) and (c). The non‑trapping condition guarantees that the high‑frequency contributions dominate, forcing the interior integrals to vanish when the boundary data coincide, which in turn forces the unknowns to be equal.

The methodology is shown to be robust: the authors extend the same framework to parabolic (heat) and Schrödinger equations, obtaining analogous uniqueness statements under analogous structural conditions. They also discuss the role of anisotropic (\sigma) and the associated gauge invariance, noting that the admissibility conditions effectively suppress the gauge freedom.

In the literature review, the paper contrasts its results with earlier works that either required multiple independent measurements, strong analyticity or piecewise‑constant assumptions, or relied on low‑frequency expansions that discard much of the spectral information. By exploiting the full temporal spectrum, the present approach achieves uniqueness with minimal regularity assumptions (e.g., (c\in W^{2,\infty}), (f\in H^{4})) and realistic structural constraints, making it directly applicable to thermoacoustic and photoacoustic tomography, as well as geophysical exploration.

The paper concludes by emphasizing the theoretical breakthrough—solving a long‑standing open problem listed by G. Uhlmann—and outlines future directions, including relaxing the admissibility hypotheses, stability analysis in the presence of noise, and numerical implementation for practical imaging scenarios.