The generalized linear sampling method (GLSM) reconstructs the support of an unknown scatterer from multi-static scattering data by minimizing a Tikhonov-type functional whose penalty term is weighted through a factorization of the measurement operator. When the incident and observation directions cover only limited apertures, this factorization loses the Hermitian symmetry on which the original GLSM theory rests, and the corrected nonsymmetric GLSM of Audibert and Haddar (2017) recovers only asymptotic convergence guarantees: the theory prescribes no rule for the regularization parameter or for the relative weights of the multiple penalty terms. This problem asks for the missing quantitative layer of that method: a theoretically justified choice of the regularization parameter and of the relative weights of the multiple penalty terms — a priori, or a posteriori/data-driven with proof — that provably preserves the noisy-data convergence guarantees of the 2017 analysis, replacing the purely heuristic weight choices currently in use. This core remains open: later work either treats parameter choice only for the symmetric linear sampling method (Morozov-type discrepancy rules), learns parameter maps without GLSM-specific guarantees, or bypasses the nonsymmetric setting via data completion and alternative sampling indicators.

Background

Time-harmonic inverse acoustic scattering asks to recover an unknown penetrable inhomogeneity — a refractive index (contrast) nn supported in a bounded region DD — from measurements of waves it scatters. A standard data set is the far-field operator FF, which maps an incident plane wave with direction x^\hat{x} to the far-field pattern (the angular dependence of the scattered field at large distance, governed by the Sommerfeld radiation condition) observed over observation directions y^\hat{y}. The linear sampling method (LSM) is a qualitative reconstruction technique: it tests, for each sampling point zz on a grid, whether a special right-hand side ϕz\phi_z (the far field of a point source at zz) lies approximately in the range of FF; the indicator stays small inside DD and blows up outside, so its level sets image the scatterer without any a priori knowledge of boundary conditions or material parameters. The generalized linear sampling method (GLSM) of Audibert and Haddar (2014) gave this idea a rigorous variational foundation: instead of approximately solving the far-field equation Fg≈ϕzFg\approx\phi_z by plain Tikhonov regularization (whose regularization parameter choice is itself heuristic), one minimizes a cost functional Jα(φ;g)=α∣⟨Bg,g⟩∣+∥Fg−φ∥2J_\alpha(\varphi;g)=\alpha|\langle Bg,g\rangle|+\|Fg-\varphi\|^2, in which HH is the Herglotz operator (mapping a density gg on the unit sphere to the incident Herglotz wave it synthesizes) and the penalty term is tied to the factorizations F=GHF=GH and B=H∗THB=H^*TH with a middle operator TT. The justification relies on FF being factorized through a coercive middle operator, which holds when the measurement and illumination apertures are full and the contrast satisfies a sign condition near ∂D\partial D and k2k^2 is not a transmission eigenvalue (an interior resonance-type value of the wavenumber at which the associated interior transmission problem has a nontrivial solution). With full apertures this factorization is Hermitian-symmetric, and GLSM inherits exact support characterization and convergence with noisy data. In practice, however, sensors and sources often cover only limited angular apertures. Then the far-field operator is restricted to subintervals of directions, its factorization loses the symmetry F=F∗F=F^*-type structure, and the original GLSM theory breaks down. Audibert and Haddar (2017) repaired this by adding an extra penalty term that asymptotically corrects the nonsymmetry, proving in an abstract framework (with an auxiliary ambient space Y^\hat{Y} and a noise threshold δ0\delta_0) that the resulting nonsymmetric GLSM still enjoys strong convergence of the Herglotz wave to the point-source field for points inside the scatterer, under coercivity and injectivity hypotheses on the middle operator. But the correction introduces several regularization weights at once (α\alpha plus the relative scalings of the penalty terms), the theory says nothing about how to choose them — the authors test three explicit heuristics and note the choice matters far more than in the symmetric case — and the analysis yields only asymptotic boundedness statements, while the numerical experiments indicate that nonsymmetric apertures are intrinsically worse than symmetric ones, with no quantitative rates, resolution limits, or stability constants describing how much worse. Classical regularization theory for ill-posed operator equations provides parameter rules such as the Morozov discrepancy principle for single-parameter Tikhonov problems, but the nonsymmetric GLSM functional couples multiple weights with an asymptotically corrected penalty, a structure outside that theory.

Problem Statement

For the nonsymmetric generalized linear sampling method of Audibert and Haddar (2017), applied to time-harmonic inverse acoustic scattering from a contrast nn supported in DD, with incident directions restricted to an aperture Σs\Sigma_s and measurements restricted to an aperture Σm\Sigma_m (far-field or near-field), give a theoretically justified rule for choosing the regularization parameter α\alpha and the relative scalings among the multiple penalty terms of the nonsymmetric GLSM cost functional (the functional of Audibert and Haddar, SIAM J. Imaging Sci. 10(2):845–870, 2017, DOI 10.1137/16M110112X) — a priori, or a posteriori/data-driven with proof — that preserves the strong-convergence guarantees with noisy data established by the 2017 analysis, in place of the purely heuristic weight choices currently in use. The required deliverable is this choice rule for α\alpha and the relative penalty scalings; its convergence proof need not furnish rates, resolution limits, or stability constants quantifying the imaging-quality loss intrinsic to nonsymmetry.

References

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