Stability of the multifrequency inverse medium problem in two and three dimensions

Bao and Triki (J. Differential Equations 269 (2020) 7106-7128) proved that the one-dimensional multifrequency inverse medium problem - recovering a refractive index from reflection data over a band of frequencies - satisfies Hölder stability with an exponent that grows with the band, and Lipschitz stability for an 'observable part' of the medium defined by a truncated trace formula, and they posed the extension to dimensions two and three as an open problem, obstructed by strong high-frequency nonlinearity, the meromorphic resonance structure of the scattering data, and trapped rays. The core increasing-stability question has been substantially resolved for the fixed-potential Schrödinger formulation: Zhai and Zhao (arXiv:2306.10211), published with Bao as Bao, Zhai and Zhao (SIAM J. Math. Anal., 2026), proved the first increasing-stability estimates in two and three dimensions from multi-wavenumber near-field or far-field data, with extensions to magnetic, nonlinear, random, and elastic settings. A precise open core survives: increasing-stability and observable-part estimates for the acoustic refractive-index medium itself in dimensions two and three, where the coefficient couples frequency to the unknown and no reduction to a fixed potential exists.

Background

In inverse scattering, an unknown inhomogeneity is probed with waves and reconstructed from measurements of the scattered field. In the acoustic inverse medium problem the unknown is the refractive index $n(x)=\sqrt{1+q(x)}$ of a compactly supported medium: for each wavenumber (frequency) $k>0$ an incident wave $e^{ikx\cdot d}$ propagates according to the Helmholtz equation

$$\Delta u + k^{2}\,n^{2}(x)\,u = 0,$$

and one measures the scattered field either on a surface enclosing the medium (near-field data) or through its asymptotic amplitude at infinity (the far-field pattern). At a single frequency the problem is severely ill-posed: classical results establish only logarithmic-type stability in dimensions two and three - reconstruction error bounded by a power of the logarithm of the data error - and this is essentially optimal, so that tiny measurement errors can be amplified enormously. The multifrequency variant, measuring at all wavenumbers in a band $(0,K]$, is motivated by the experimental and numerical observation that widening the band dramatically improves reconstructions; recursive linearization algorithms exploit this by marching from low to high frequency, but lacked a rigorous stability theory (a topical review is Bao, Li, Lin and Triki, Inverse Problems 31 (2015) 093001).

Bao and Triki (2020) made the phenomenon rigorous in one dimension. For $\phi''+k^{2}(1+q(x))\phi=0$ on $[0,1]$ with $q\in C^{m+1}_{0}$, $m\geq 4$, and $1+q$ bounded away from zero, they measured the reflection coefficient $\mu(k)$ - the complex amplitude of the wave reflected by the medium when a wave of frequency $k$ is incident - over a band $(0,k_{0})$ and proved: (i) Hölder stability $\|q-\tilde q\|_{L^{\infty}}\lesssim\varepsilon^{\gamma(k_{0})}$, where $\varepsilon$ is the data-error level and the exponent $\gamma$ is an increasing function of the largest measured frequency $k_{0}$ (with logarithmic-type stability for narrow bands and a crossover at a critical band width depending on the noise level and the a priori class of media); and (ii) Lipschitz stability - error bounded proportionally to the data error - for the observable part $q_{k_{0}}$ of the medium, the function obtained by truncating a trace formula, an identity expressing the medium through frequency integrals of the impedance functions $p_{\pm}=\pm\phi'_{\pm}/(ik\phi_{\pm})$ (logarithmic derivatives of the total field, which solve Riccati equations, i.e. nonlinear first-order ODEs quadratic in the unknown). The proof rests on a one-dimensional reduction, in the spirit of the Gel'fand-Levitan inverse spectral theory, that converts the Helmholtz equation into a Schrödinger equation in which frequency and medium separate, followed by analytic continuation of the scattering data in the complex frequency plane. Bao and Triki state that in higher dimensions the corresponding problem is still open: the reduction has no analogue there, the nonlinearity of the data-to-medium map strengthens at high frequency, the scattering data possess a complicated meromorphic (resonance) structure - poles of the analytic continuation whose positions depend on the unknown medium - and trapped rays, ray trajectories confined by the medium, obstruct the analytic-continuation arguments.

Substantial progress came for the fixed-potential formulation. Writing the model as the Schrödinger equation $-\Delta u-\kappa^{2}u+V(x)u=0$ with an unknown compactly supported potential $V$ (in one dimension this is equivalent to the acoustic problem via a Liouville change of variables, but in dimensions two and three it is a genuinely different model, since the acoustic coefficient $k^{2}n^{2}(x)$ couples the frequency to the unknown), Zhai and Zhao - published with Bao as Bao, Zhai and Zhao (2026) - proved the first increasing-stability estimates in two and three dimensions from multi-wavenumber near-field or far-field data: for real $V\in H^{s}\cap L^{\infty}$ supported in a ball,

$$\|V_{1}-V_{2}\|^{2}_{L^{2}} \lesssim K^{\alpha}\varepsilon^{2}+\frac{1}{K^{\beta}(\ln|\ln\varepsilon|)^{\beta}},$$

with $\alpha,\beta>0$ depending on the smoothness $s$ - a Lipschitz-type data discrepancy plus a high-wavenumber tail that shrinks as the band widens; the same scheme covers the magnetic Schrödinger equation in three dimensions. The method replaces the complex geometric optics (CGO) solutions - special high-frequency solutions with exponentially growing phases used in earlier increasing-stability results for Schrödinger-type inverse boundary value problems with Dirichlet-to-Neumann data (the map from boundary values of solutions to their normal derivatives), which do not extend to two dimensions - by scattering-theoretic estimates for the resolvent (the operator inverse $(-\Delta+V-\lambda)^{-1}$) in an analytic region of the complex spectral plane, followed by quantitative analytic continuation. Related extensions exist for nonlinear Schrödinger models, elasticity, and random one-dimensional potentials, and a parallel multifrequency literature covers inverse source problems, where the unknown source does not feed back into the propagation.

What this progress leaves open is the subject of the present problem: the frequency-coupled acoustic medium itself in dimensions two and three, and the higher-dimensional observable-part theory.

Problem Statement

Complete the quantitative stability theory of multifrequency inverse medium reconstruction in dimensions two and three, at the level of precision of the one-dimensional theory of Bao and Triki. Concretely, for the acoustic medium problem - recover the refractive index $n(x)$, equivalently the medium function $q$ with $n^{2}=1+q$, supported in a ball in $\mathbb{R}^{d}$, $d=2,3$, and belonging to a fixed a priori class (for example bounded in $H^{s}\cap L^{\infty}$), from near-field or far-field scattering data measured for all wavenumbers $k$ in a band $(0,K]$:

1. (Increasing stability for the acoustic medium.) Prove stability estimates $\|q_{1}-q_{2}\|\lesssim F(\varepsilon,K)$, where $\varepsilon$ bounds the data discrepancy and $F$ improves as the band widens - approaching Lipschitz behavior, or Hölder behavior with an exponent growing in $K$, as in the one-dimensional theory - or establish the matching obstruction showing that such an estimate provably cannot hold for the frequency-coupled coefficient $k^{2}n^{2}(x)$. The existing two- and three-dimensional results cover only the fixed potential $V$ in $-\Delta u-\kappa^{2}u+Vu=0$, which in one dimension is equivalent to the acoustic problem but in higher dimensions is not.

2. (Observable part in higher dimension.) Construct the analogue, for $d=2,3$, of the one-dimensional observable part $q_{k_{0}}$ - the component of the medium actually determined by band-limited data, defined there by truncating the trace formula - and prove that it is recovered with Lipschitz-type stability once the band is large enough, with explicit control of the truncation error.

These two components are facets of a single objective: a quantitative theory of what a finite frequency band determines about an unknown medium in dimensions two and three, and at what stability cost. The source work also discussed the convergence analysis of efficient three-dimensional reconstruction algorithms; that computational objective is not part of this problem, and the one-dimensional regularity restriction has largely been superseded by the weaker assumptions available in higher dimensions.

References

1. Gang Bao and Faouzi Triki, Stability for the multifrequency inverse medium problem, Journal of Differential Equations 269 (2020) 7106-7128, DOI: 10.1016/j.jde.2020.05.021, arXiv:2002.09242, https://arxiv.org/abs/2002.09242.
2. Jian Zhai and Yue Zhao, Increasing stability estimates for the inverse potential scattering problems, arXiv:2306.10211 (2023), https://arxiv.org/abs/2306.10211.
3. Gang Bao, Jian Zhai, and Yue Zhao, Stability for the Inverse Potential Scattering Problems with Multi-Wavenumbers, SIAM Journal on Mathematical Analysis 58 (2026) 2111-2137, DOI: 10.1137/25m1767108, https://doi.org/10.1137/25m1767108.
4. Yue Zhao, Increasing stability in the inverse scattering problem for a nonlinear Schrödinger equation with multiple frequencies, Physica D: Nonlinear Phenomena 449 (2023) 133746, DOI: 10.1016/j.physd.2023.133746, https://doi.org/10.1016/j.physd.2023.133746.
5. Rong Sun, Ganghua Yuan, and Yue Zhao, Stability estimates for inverse scattering problems in elasticity, Science China Mathematics (2026), DOI: 10.1007/s11425-024-2513-3, https://doi.org/10.1007/s11425-024-2513-3.
6. Tianjiao Wang, Xiang Xu, and Yue Zhao, Inverse random scattering for the one-dimensional Helmholtz equation, Inverse Problems 41 (2025) 065011, DOI: 10.1088/1361-6420/addde3, https://doi.org/10.1088/1361-6420/addde3.
7. Shuai Lu, Mikko Salo, and Boxi Xu, Increasing stability in the linearized inverse Schrödinger potential problem with power type nonlinearities, Inverse Problems 38 (2022) 065009, DOI: 10.1088/1361-6420/ac637a, https://doi.org/10.1088/1361-6420/ac637a.
8. Peijun Li, Jian Zhai, and Yue Zhao, Stability for the Acoustic Inverse Source Problem in Inhomogeneous Media, SIAM Journal on Applied Mathematics 80 (2020) 2547-2559, DOI: 10.1137/20m1334267, https://doi.org/10.1137/20m1334267.
9. Mirza Karamehmedović, Martin Sæbye Carøe, and Faouzi Triki, Estimates on the stability constant for the truncated Fourier transform, arXiv:2407.06656 (2024), https://arxiv.org/abs/2407.06656.
10. Gang Bao, Peijun Li, Junshan Lin, and Faouzi Triki, Inverse scattering problems with multi-frequencies, Inverse Problems 31 (2015) 093001, DOI: 10.1088/0266-5611/31/9/093001, https://doi.org/10.1088/0266-5611/31/9/093001.
11. Sei Nagayasu, Gunther Uhlmann, and Jenn-Nan Wang, Increasing stability in an inverse problem for the acoustic equation, Inverse Problems 29 (2013) 025012, DOI: 10.1088/0266-5611/29/2/025012, https://doi.org/10.1088/0266-5611/29/2/025012.
12. Xiang Xu and Yue Zhao, Inverse scattering problems of the biharmonic Schrödinger operator with a first order perturbation, Journal of Mathematical Physics 65 (2024) 062105, DOI: 10.1063/5.0202903, https://doi.org/10.1063/5.0202903.