Bao, Lin and Xu (SIAM J. Numer. Anal. 58, 2020) proposed the MCCUQ method (Monte Carlo sampling of the probability space, continuation in the wavenumber, and a Karhunen-Loève expansion of the random structure) for reconstructing the expectation, root mean square, and correlation length of an unknown Λ-periodic random grating from multi-frequency measurements of the scattered field above the structure. The numerical experiments indicate convergence, but, as the authors state, no rigorous convergence and stability analysis is available. The open problem left by that work is to establish the identifiability, stability, and convergence theory for this statistical inverse scattering problem (in what sense and under which conditions the data determine the statistics of the random surface, and with what error the Monte-Carlo-continuation reconstruction converges), and to extend the reconstruction framework, with corresponding algorithmic development and analysis, to three-dimensional Maxwell scattering by random biperiodic structures and to non-periodic randomly rough surfaces. No rigorous analysis of this kind has appeared since: the later literature provides numerical algorithms for penetrable, elastic, and acoustic-elastic random periodic settings and forward-problem stability estimates, and Sun and Lin (arXiv:2507.19744, 2025) note that a rigorous theoretical analysis of the random-grating reconstruction algorithms is still lacking.

Background

A diffraction grating is a periodic surface structure that scatters time-harmonic electromagnetic waves; such structures are central to diffractive optics (antireflective interfaces, beam splitters, sensors). For media invariant in one horizontal direction, the three-dimensional time-harmonic Maxwell equations reduce to a two-dimensional Helmholtz equation on a domain above the grating profile. In transverse electric (TE) polarization the grating is perfectly reflecting, which corresponds to a homogeneous Dirichlet (sound-soft) boundary condition on the profile. The grating profile is Λ-periodic, so the scattered field is quasi-periodic (periodic up to the phase factor imposed by the incident wave) and, at each fixed frequency, admits a Rayleigh expansion — a Fourier series in which only finitely many propagating modes carry energy away from the structure while the remaining evanescent modes decay exponentially.

The classical inverse diffraction problem — recover a deterministic grating profile from measurements of the scattered field at a fixed distance above the structure — is well studied: uniqueness and stability results and a range of reconstruction algorithms (factorization-type, iterative regularization, and multi-frequency continuation methods) are known. In practice, however, manufactured gratings carry uncertainty: manufacturing defects and wear make the profile a random surface rather than a fixed function. Bao, Lin, and Xu modeled the unknown grating as a random periodic structure — a random field modulating the periodic profile — and posed the statistical inverse problem of recovering key properties of that random field, namely its expectation, root mean square, and correlation length, from multi-frequency boundary measurements of the scattered fields. The random field is represented through its Karhunen-Loève (KL) expansion, i.e. an eigenfunction expansion of its covariance operator that decorrelates the randomness into independent scalar coefficients.

Their method, called MCCUQ (Monte Carlo - continuation - uncertainty quantification), combines three ingredients: Monte Carlo sampling of the probability space to evaluate the statistics of the scattered field, a continuation method that solves a sequence of inverse problems at increasing wavenumbers (starting from low frequencies, where the inversion is more stable, and refining finer surface scales as the frequency grows), and the KL expansion to parametrize the random structure. Numerical examples demonstrate convergence of the reconstructed statistics, but the paper proves no theorem about it. Two features make the problem hard: the scattered field depends nonlinearly on the surface, so surface randomness propagates through the data in a non-explicit way, and it is unclear a priori to what extent the reconstruction can be made at all — which statistical quantities of the profile are actually determined by the measured data. In their concluding remarks the authors list four outstanding directions: (a) a rigorous convergence and stability analysis of the reconstruction (they write that a new theoretical framework must be developed); (b) algorithms for penetrable (non-perfectly-reflecting) randomly rough surfaces; (c) the more realistic inverse scattering for the full three-dimensional Maxwell equations, where the vectorial fields and the exponentially higher forward-solver cost demand algorithmic and analytic changes; and (d) numerical methods for non-periodic randomly rough surfaces, removing the Λ-periodicity assumption.

Problem Statement

Develop the rigorous mathematical theory of statistical inverse scattering by randomly rough surfaces, as left open by Bao-Lin-Xu (2020), and complete the framework's coverage of realistic settings. The common objective is that the recovery of the statistics of an unknown random scattering surface from scattered-field data — currently supported only by numerically convergent algorithms — be placed on a rigorous footing and extended to the settings the source identified as untreated. Concretely:

  1. Theory core (source direction a). For a time-harmonic plane wave incident on a perfectly reflecting Λ-periodic random grating (two-dimensional Helmholtz equation, TE polarization), establish: (i) identifiability — under which conditions on the stochastic surface model, the wavenumbers, and the measurement configuration the multi-frequency boundary data determine the statistics of the random profile — the mean profile and covariance operator, equivalently the Karhunen-Loève spectrum, from which the source's expectation, root mean square, and correlation length are obtained — and in what sense; (ii) stability — quantitative estimates bounding the error of the recovered statistics in terms of data noise and Monte Carlo sampling error; and (iii) convergence — convergence, with rates where attainable, of the Monte Carlo-continuation-Karhunen-Loève reconstruction (the MCCUQ algorithm and its successors) as the sample size, KL truncation order, and frequency grid are refined. The same theory is required for the penetrable variant, for which numerical algorithms now exist but no analysis does.

  2. Extensions (source directions c and d). Extend the statistical reconstruction — algorithmic development together with the corresponding analysis — to (a) inverse scattering governed by the three-dimensional Maxwell equations for random biperiodic structures (periodic in the two independent horizontal directions), handling vectorial fields and the substantially increased forward-solver cost, and (b) inverse scattering by non-periodic randomly rough surfaces, removing the Λ-periodicity assumption.

The objective is a rigorous theory of identifiability, stability, and convergence for the statistical reconstruction; results restricted to a single fixed surface realization fall outside the statistical problem. The theory core of item 1, including its penetrable variant, is part of the objective; the two extensions in item 2 are independent directions and each remains open on its own.

References

  1. Gang Bao, Yiwen Lin, Xiang Xu, Inverse Scattering by a Random Periodic Structure, SIAM Journal on Numerical Analysis 58(5) (2020) 2934-2952, DOI 10.1137/20M132167X; arXiv:2003.00642, https://arxiv.org/abs/2003.00642
  2. Gang Bao, Yiwen Lin, Determination of Random Periodic Structures in Transverse Magnetic Polarization, Communications in Mathematical Research 37(3) (2021) 271-296, DOI 10.4208/CMR.2021-0003, https://doi.org/10.4208/CMR.2021-0003
  3. Gang Bao, Yiwen Lin, Xiang Xu, Stability for the Helmholtz equation in deterministic and random periodic structures, Journal de Mathématiques Pures et Appliquées 187 (2024) 207-232, DOI 10.1016/j.matpur.2024.05.014; arXiv:2210.10359, https://arxiv.org/abs/2210.10359
  4. Yi Wang, Junliang Lv, Shuxin Li, Numerical method for inverse scattering by random penetrable periodic structures, Inverse Problems and Imaging (2024), DOI 10.3934/ipi.2024038, https://doi.org/10.3934/ipi.2024038
  5. Hao Gu, Xiang Xu, Liang Yan, Inverse elastic scattering by random periodic structures, Journal of Computational Physics 501 (2024) 112785, DOI 10.1016/j.jcp.2024.112785, https://doi.org/10.1016/j.jcp.2024.112785
  6. Yi Wang, Lei Lin, Junliang Lv, Numerical method for the inverse scattering problem of acoustic-elastic interaction by random periodic structures, Journal of Computational Physics 562 (2026) 115028, DOI 10.1016/j.jcp.2026.115028; arXiv:2504.18356, https://arxiv.org/abs/2504.18356
  7. Zhiqi Sun, Yiwen Lin, An inverse random diffraction grating problem for the Helmholtz equation, arXiv:2507.19744 (2025), https://arxiv.org/abs/2507.19744
  8. Zhiqi Sun, Xiang Xu, Yiwen Lin, Inverse acoustic scattering for random obstacles with multi-frequency data, arXiv:2601.22560 (2026), https://arxiv.org/abs/2601.22560
  9. Zhiqi Sun, Yiwen Lin, Inverse scattering for three-dimensional random obstacles with multi-frequency data, arXiv:2607.13473 (2026), https://arxiv.org/abs/2607.13473
  10. Tianjiao Wang, Yiwen Lin, Xiang Xu, A Priori Bounds for Elastic Scattering by Deterministic and Random Unbounded Rough Surfaces, CSIAM Transactions on Applied Mathematics 4 (2023) 696-720, DOI 10.4208/csiam-am.so-2023-0001, https://doi.org/10.4208/csiam-am.so-2023-0001
  11. Guanghui Hu, Tianjiao Wang, Xiang Xu, Yue Zhao, Time-Harmonic Elastic Scattering by Unbounded Deterministic and Random Rough Surfaces in Three Dimensions, SIAM Journal on Mathematical Analysis 57(2) (2025) 1964-1989, DOI 10.1137/24m1640306, https://doi.org/10.1137/24m1640306