Ji and Liu (arXiv:1906.02008, 2019) introduced the first numerical method — a direct sampling method — for recovering acoustic scatterers (point-like scatterers, penetrable inhomogeneities, and impenetrable obstacles) from multi-frequency far-field data measured in only a few backscattering directions, but all their uniqueness results and the theoretical analysis of the indicator hold only under the Born/weak-scattering and Kirchhoff/physical-optics linearizations. The open problem is the rigorous theory for the full nonlinear problem: determine what is uniquely determined by the data u∞(−θj,θj,k)u^\infty(-\theta_j,\theta_j,k) for wavenumbers kk in a frequency band and finitely many (sparse) backscattering direction pairs; establish stability, resolution, and minimal-direction results for the direct sampling indicator, including or refuting the empirical observation that at most 2n2n observation directions (nn being the spatial dimension) suffice to approximately reconstruct a hull of the scatterer in Rn\mathbb{R}^n; reconcile the high-frequency validity of physical-optics (Bojarski-type) inversion with the finite frequency bands available in practice; and rigorously justify the point-like scatterer model with error bounds for finite-size scatterers. Follow-up work through 2026 settled only special cases — uniqueness of convex obstacles from all-frequency backscattering data, local uniqueness for contrasts differing by an affine function, and uniqueness of point-like objects within the point model — while the same literature explicitly describes the general uniqueness question as a long-standing open problem.

Background

In time-harmonic acoustic scattering, a plane wave uin(x)=eikθ⋅xu^{in}(x)=e^{ik\theta\cdot x} with wavenumber k=ω/ck=\omega/c and direction θ∈Sn−1\theta\in\mathbb{S}^{n-1} strikes a scatterer in Rn\mathbb{R}^n, n=2,3n=2,3. The scatterer is either an impenetrable obstacle DD on whose boundary the total field satisfies a Dirichlet (sound-soft), Neumann (sound-hard), or impedance condition, or a penetrable inhomogeneity described by a refractive index q(x)q(x) with compactly supported contrast q−1q-1. The scattered field usu^s solves the Helmholtz equation with the Sommerfeld outgoing-radiation condition (the requirement that the scattered wave propagate outward to infinity) and has the asymptotic form us(x)∼γn(k)eik∣x∣u∞(x^,θ,k)/∣x∣(n−1)/2u^s(x)\sim\gamma_n(k)e^{ik|x|}u^\infty(\hat{x},\theta,k)/|x|^{(n-1)/2} as ∣x∣→∞|x|\to\infty; the coefficient u∞(x^,θ,k)u^\infty(\hat{x},\theta,k) is the far-field pattern, the standard measurement in inverse scattering. Classical uniqueness theory shows that obstacles and inhomogeneities are uniquely determined by the far-field pattern for all incident and all observation directions at a fixed frequency; much less is known for restricted data.

The backscattering (monostatic) geometry uses a single transceiver: the observation direction equals the negative of the incident direction, giving the data u∞(−θ,θ,k)u^\infty(-\theta,\theta,k). Practically attractive because one sensor position suffices, this geometry is theoretically hard, and uniqueness of inverse obstacle or medium scattering from backscattering data is a long-standing open problem even when all backscattering directions and all frequencies are available (known positive results are confined to special classes, e.g. polyhedral obstacles recoverable from a few backscattering measurements, and local uniqueness near a disk). Ji and Liu (2019) studied the further sparsified data model in which the far field is measured at only a few isolated backscattering directions over a bounded frequency band k∈(k−,k+)k\in(k_-,k^+), for underlying objects ranging from point-like scatterers to extended obstacles. They proposed a direct sampling method: an indicator function built from inner products of the data with analytically chosen probe functions, computed without any forward solver and robust to noise, in the spirit of the linear sampling and factorization methods but requiring only sparse data.

Their theory rests on two linearizations. The Born or weak-scattering approximation (valid for point-like, small, or low-contrast scatterers) replaces the total field inside the scatterer by the incident field, making the far field essentially the Fourier transform of the contrast. The Kirchhoff or physical-optics approximation (valid at high frequency for smooth convex obstacles) approximates the scattered field by specular reflection, leading to the Bojarski identity: in this limit the backscattering data determines, up to known factors, the Fourier transform of the characteristic function of the obstacle. Under these approximations Ji and Liu proved approximate uniqueness results: from a single incident-observation pair at all frequencies in a band (extended by analyticity in kk), certain hyperplanes or strips containing the scatterer are determined, and with more direction pairs their intersection gives an enclosing hull. They also observed numerically, without proof, that at most 2n2n observation directions (nn being the spatial dimension) suffice to approximately reconstruct such a hull. The authors themselves flag the gaps: the Bojarski route needs all frequencies (clashing with the high-frequency validity of the approximation) and all directions; the point-like scatterer model, obtained by neglecting multiple scattering between the point scatterers, is, in their words, still not completely well established; and no uniqueness, stability, or resolution statement is available for the full nonlinear problem in the resonance (intermediate-frequency) regime, where neither linearization is valid, even though the numerical method performs well there.

Since 2019 the problem cluster has been actively pursued. The electromagnetic obstacle analogue was developed by Arens, Ji, and Liu (Inverse Problems, 2020), still under the Kirchhoff approximation. Uniqueness of point-like objects within the point model was proved by Ji and Liu (SIAM J. Sci. Comput., 2020). Rigorous nonlinear uniqueness from backscattering far-field data was achieved only for smooth strictly convex obstacles by Li, Liu, and Shi (2025), via high-frequency asymptotics of the far-field pattern, and for the near-field multi-frequency backscattering problem under convexity by Li and Liu (2026); for penetrable inhomogeneities, Guo and Liu (Inverse Problems, 2026) proved only a local uniqueness result (contrasts whose difference is an affine function) together with a quantitative direct sampling method still based on the Born approximation. Error estimates for the direct sampling method exist so far only for the linear inverse source problem (Ji, Jiao, Lu, and Wang, 2024), and Park (2022) proved that adding multiple frequencies does not improve the resolution of the direct sampling indicator in the linearized setting. What is still missing is the approximation-free theory for the original sparse far-field backscattering problem: uniqueness for nonconvex obstacles and general inhomogeneities, stability and minimal-direction statements, finite-band inversion guarantees, and justification of the point-scatterer model.

Problem Statement

Establish the rigorous mathematical theory — uniqueness, reconstruction guarantees, and stability, or definitive impossibility results for the same data model — for the inverse acoustic scattering problem with multi-frequency sparse backscattering far-field data, for the full nonlinear scattering model in Rn\mathbb{R}^n, n=2,3n=2,3, without assuming the Born/weak-scattering or Kirchhoff/physical-optics linearizations. The data are the backscattering far-field patterns u∞(−θj,θj,k)u^\infty(-\theta_j,\theta_j,k) for wavenumbers kk in a frequency band (or all k>0k>0 where the statement so requires) and a sparse (finite) set of distinct backscattering direction pairs (x^j,θj)(\hat{x}_j,\theta_j) with x^j=−θj\hat{x}_j=-\theta_j, for the scatterer classes considered in the source work: point-like scatterers, penetrable inhomogeneities with compactly supported contrast, and extended impenetrable obstacles with Dirichlet, Neumann, or impedance boundary conditions. The objective comprises, as facets of one rigorization program for this data model: (i) determine which features of the scatterer — locations, scattering strengths, hull or support function, full shape, boundary condition, contrast function — are uniquely determined by such data, in particular in the resonance (intermediate-frequency) regime where the linearizations are invalid, and with which minimal number of observation directions; (ii) prove stability and resolution theorems for the direct sampling indicator, including a proof or refutation of the empirical observation that at most 2n2n observation directions suffice to approximately reconstruct a hull of the scatterer, with explicit dependence on the frequency band and noise level; (iii) reconcile the high-frequency validity of Kirchhoff/Bojarski-type inversion with the finite frequency bands available in practice, by rigorous error analysis of finite-band reconstruction; and (iv) rigorously justify the point-like scatterer model as an approximation of finite-size scatterers, with error bounds under which the point-model uniqueness results remain valid. A resolution of any single facet (i)–(iv), covering all axes of that facet and in the stated generality — positive results, or a definitive impossibility result for that facet — discharges the objective; the four facets are independently acceptable and are not required jointly. An impossibility result counts for a facet only when it is stated for the same data model as the corresponding positive statement, that is, a fixed, distinct family of backscattering directions that spans Rn\mathbb{R}^n; an impossibility result for a degenerate direction family (one not spanning Rn\mathbb{R}^n) answers only the minimal-direction sub-question inside facet (i), which is a proper part of that facet and is not by itself a resolution of it. Results restricted to convex obstacles, affine perturbation classes, near-field data, or the linearized models constitute partial progress and do not by themselves discharge the objective, which is posed at the generality of the source formulation.

References

  1. Xia Ji and Xiaodong Liu, "Inverse acoustic scattering problems with multi-frequency sparse backscattering far field data", arXiv:1906.02008 (2019), DOI 10.48550/arXiv.1906.02008, https://arxiv.org/abs/1906.02008
  2. Tilo Arens, Xia Ji, and Xiaodong Liu, "Inverse electromagnetic obstacle scattering problems with multi-frequency sparse backscattering far field data", Inverse Problems 36, 105007 (2020), DOI 10.1088/1361-6420/abb5e2, https://doi.org/10.1088/1361-6420/abb5e2 (also arXiv:2002.09163)
  3. Xia Ji and Xiaodong Liu, "Identification of Point-Like Objects with Multifrequency Sparse Data", SIAM Journal on Scientific Computing 42, A2325-A2343 (2020), DOI 10.1137/20m1312551, https://doi.org/10.1137/20m1312551
  4. Jingzhi Li and Hongyu Liu, "Recovering a polyhedral obstacle by a few backscattering measurements", Journal of Differential Equations 259, 2101-2120 (2015), DOI 10.1016/j.jde.2015.03.030, https://doi.org/10.1016/j.jde.2015.03.030
  5. Won-Kwang Park, "Theoretical Study on Non-Improvement of the Multi-Frequency Direct Sampling Method in Inverse Scattering Problems", Mathematics 10, 1674 (2022), DOI 10.3390/math10101674, https://doi.org/10.3390/math10101674
  6. Xia Ji, "Reconstruction of multipolar point sources with multi-frequency sparse far field data", Inverse Problems 37, 065015 (2021), DOI 10.1088/1361-6420/abf812, https://doi.org/10.1088/1361-6420/abf812
  7. Xiaodong Liu and Qingxiang Shi, "Identification of acoustic point sources in a two-layered medium from multi-frequency sparse far field patterns", Inverse Problems 39, 065001 (2023), DOI 10.1088/1361-6420/accaa0, https://doi.org/10.1088/1361-6420/accaa0
  8. Xia Ji, Yuling Jiao, Xiliang Lu, and Fengru Wang, "Error estimation to the direct sampling method for the inverse acoustic source problem with multi-frequency data", Inverse Problems and Imaging 18, 517-540 (2024), DOI 10.3934/ipi.2023042, https://doi.org/10.3934/ipi.2023042
  9. Jialei Li, Xiaodong Liu, and Qingxiang Shi, "Identifying convex obstacles from backscattering far field data", arXiv:2505.11850 (2025), DOI 10.48550/arXiv.2505.11850, https://arxiv.org/abs/2505.11850
  10. Jialei Li and Xiaodong Liu, "Inverse Obstacle Scattering from Multi-Frequency Near-Field Backscattering Data", arXiv:2604.09642 (2026), DOI 10.48550/arXiv.2604.09642, https://arxiv.org/abs/2604.09642
  11. Yukun Guo and Xiaodong Liu, "A quantitative direct sampling method for inhomogeneities from multi-frequency backscattering measurements", Inverse Problems 42, 085008 (2026), DOI 10.1088/1361-6420/ae963b, https://doi.org/10.1088/1361-6420/ae963b (also arXiv:2604.25583)
  12. Juan Liu, Fuquan Yang, Xiaoyi Yan, and Jiguang Sun, "Extended sampling method for inverse electromagnetic scattering problems using a single incident wave", Inverse Problems 42, 015010 (2026), DOI 10.1088/1361-6420/ae3799, https://doi.org/10.1088/1361-6420/ae3799