Lippmann-Schwinger-Lanczos algorithm for inverse scattering problems with unknown reflectivity and loss distributions: One-dimensional Case

TL;DR

This work extends the Lippmann-Schwinger-Lanczos framework to one-dimensional inverse scattering in attenuating media with unknown reflectivity $r(T)$ and loss $κ(T)$. It develops a data-driven internal-field approximation using a complex-symmetric Lanczos reduction of a rational ROM, enabling a linearized Lippmann-Schwinger equation that preserves a port-Hamiltonian structure and enforces physically plausible constraints. Two internal-solution constructions are proposed: a spectral-measure ROM and a data-driven adaptive ROM built from frequency-domain data, both yielding more accurate reconstructions than the Born approximation and requiring fewer forward solves. Numerical experiments demonstrate robust, high-fidelity reconstructions even with significant loss and measurement noise, indicating potential for scalable 1D and higher-dimensional imaging in lossy media such as ground-penetrating radar.

Abstract

We consider one-dimensional inverse scattering in attenuating media where both the reflectivity and loss distributions are unknown. Mathematically, this corresponds to recovering the coefficients of a damped wave operator, or equivalently, a quadratic operator pencil in the frequency domain. The Lippmann-Schwinger equation maps the unknown reflectivity and loss distribution to the measured scattered data. This mapping is nonlinear, as it requires knowledge of the internal wavefield, which itself depends on the reflectivity and loss distribution. The Lippmann-Schwinger-Lanczos method addresses this nonlinearity by approximating the internal solutions through the lifting of states from a reduced-order model constructed directly from the measured data. In this work, we extend the method to dissipative problems, enabling the approximation of internal partial differential equation (PDE) solutions in media with both reflectivity and loss distributions. We present two complementary constructions of such internal solutions: one based on spectral data and another on frequency-domain measurements over a finite interval. This development establishes a direct link between data-driven reduced-order models for inverse problems and port-Hamiltonian dynamical systems, with reduced models obtained either from the associated spectral measure or via rational approximation. Compared to the Born approximation, which replaces the internal field with the background field, our approach yields more accurate internal reconstructions and enables faster and more robust recovery of the contrast as evidenced by our numerical experiments.

Figures (3)

• Figure 1: Internal solutions for $n=40$ at frequency $s = 4i$. The top row shows the primary wave $w$, and the bottom row shows the dual wave $\widehat{w}$. The left column displays the imaginary part, while the right column displays the real part. Note that the chosen forcing function is equivalent to enforcing the condition $\lim_{T\to 0^+} \widehat{w}(T) = 1$. The wave of the background $w_0$, $\widehat{w}_0$ is the Born approximation frequently used in the Lippmann--Schwinger equations.
• Figure 2: Inversion results after a single iteration of the LSL algorithm for $n=10,25,$ and $40$, with Tikhonov regularization. Left: recovery of a large attenuation profile $r(T)$. Right: First order potential $\kappa(T)$.
• Figure 3: Numerical examples with losses and impedance profiles given by Gaussians (black curves). In the noiseless case (top plots), the LSL approach (red curves) allows to produce qualitatively better reconstructions compared to the Born method (blue curves). Bottom plots show that LSL remains stable even with 20% noise whereas the Born method fails.

Theorems & Definitions (1)

• Remark 1