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Inverse scattering for Schrödinger equation in the frequency domain via data-driven reduced order modeling

Andreas Tataris, Tristan van Leeuwen, Alexander V. Mamonov

TL;DR

This work addresses the inverse scattering problem for the Schrödinger potential in the frequency domain by combining data-driven reduced order models with nonlinear ROM misfit minimization. A data-driven ROM is constructed from boundary measurements by projecting the Schrödinger operator onto a space spanned by solution snapshots at multiple wavenumbers, and its stiffness/mass structure is recovered from boundary data. Two inversion schemes are developed: one directly fitting the stiffness ROM and another transforming it to a block-tridiagonal form via block-Lanczos before misfit minimization, both solved with an adaptive regularized Gauss-Newton method. Numerical experiments in 2D demonstrate that ROM-based misfit minimization yields higher-quality reconstructions than conventional frequency-domain full waveform inversion, with the S-variant showing the best performance under noise. The approach promises improved convergence and robustness for frequency-domain inverse problems and can be extended to Helmholtz-type problems in future work.

Abstract

In this paper we develop a numerical method for solving an inverse scattering problem of estimating the scattering potential in a Schrödinger equation from frequency domain measurements based on reduced order models (ROM). The ROM is a projection of Schrödinger operator onto a subspace spanned by its solution snapshots at certain wavenumbers. Provided the measurements are performed at these wavenumbers, the ROM can be constructed in a data-driven manner from the measurements on a surface surrounding the scatterers. Once the ROM is computed, the scattering potential can be estimated using non-linear optimization that minimizes the ROM misfit. Such an approach typically outperforms the conventional methods based on data misfit minimization. We develop two variants of ROM-based algorithms for inverse scattering and test them on a synthetic example in two spatial dimensions.

Inverse scattering for Schrödinger equation in the frequency domain via data-driven reduced order modeling

TL;DR

This work addresses the inverse scattering problem for the Schrödinger potential in the frequency domain by combining data-driven reduced order models with nonlinear ROM misfit minimization. A data-driven ROM is constructed from boundary measurements by projecting the Schrödinger operator onto a space spanned by solution snapshots at multiple wavenumbers, and its stiffness/mass structure is recovered from boundary data. Two inversion schemes are developed: one directly fitting the stiffness ROM and another transforming it to a block-tridiagonal form via block-Lanczos before misfit minimization, both solved with an adaptive regularized Gauss-Newton method. Numerical experiments in 2D demonstrate that ROM-based misfit minimization yields higher-quality reconstructions than conventional frequency-domain full waveform inversion, with the S-variant showing the best performance under noise. The approach promises improved convergence and robustness for frequency-domain inverse problems and can be extended to Helmholtz-type problems in future work.

Abstract

In this paper we develop a numerical method for solving an inverse scattering problem of estimating the scattering potential in a Schrödinger equation from frequency domain measurements based on reduced order models (ROM). The ROM is a projection of Schrödinger operator onto a subspace spanned by its solution snapshots at certain wavenumbers. Provided the measurements are performed at these wavenumbers, the ROM can be constructed in a data-driven manner from the measurements on a surface surrounding the scatterers. Once the ROM is computed, the scattering potential can be estimated using non-linear optimization that minimizes the ROM misfit. Such an approach typically outperforms the conventional methods based on data misfit minimization. We develop two variants of ROM-based algorithms for inverse scattering and test them on a synthetic example in two spatial dimensions.

Paper Structure

This paper contains 16 sections, 8 theorems, 137 equations, 2 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and problem formulation
  3. Forward model
  4. Operator form and Fréchet differentiation
  5. Measurements and the inverse scattering problem
  6. Projection reduced order model
  7. Main results and the method
  8. Data-driven ROM
  9. Data-driven stiffness and mass matrix computaion
  10. Orthogonal projection ROM via block-Lanczos algorithm
  11. Numerical solution of ISP via ROM misfit minimization
  12. Numerical Results
  13. Conclusions and future work
  14. Proofs
  15. Proof of Theorem \ref{['thm:uderiv']}
  16. ...and 1 more sections

Key Result

Theorem 2.1

Given $k \in \mathbb{R}_+:=(0,\infty)$, $q\in L^\infty_+(\Omega)$ with $\overline{\text{supp(q)}}\subset \subset \Omega$, and boundary source $p_s \in {H^{1/2}(\partial \Omega)}$, the problem eqn:weaksom admits the unique weak solution $u^{(s)}(\;\cdot\; ; k) \in H^1(\Omega)$. \newlabelthm:fwdpde0

Figures (2)

  • Figure 1: Model used for testing Algorithm \ref{['alg:gn']} (top row) and estimated potentials (bottom row). The dashed yellow lines in top left plot indicate the location of vertical slices shown in quality control plots in Figure \ref{['fig:rectschrodoptbslc']}.
  • Figure 2: Potential estimate quality control: vertical slices of the target potential $q^{\text{\scriptsize 2inc}}$ (solid blue lines) and its estimates (dashed red lines) for three different values of $x_1$. Top row: $q^{\text{\scriptsize est}}_{\mathbf{S}}$; middle row: $q^{\text{\scriptsize est}}_{\mathbf{T}}$; bottom row: $q^{\text{\scriptsize est}}_{\text{\scriptsize FWI}}$.

Theorems & Definitions (16)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 3.1
  • Proof 1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • ...and 6 more