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Numerical reconstruction of Schrödinger equations with quadratic nonlinearities

Khaoula El Maddah, Matti Lassas, Teemu Tyni

TL;DR

The paper develops a numerical framework to reconstruct a spatially varying potential in a semilinear elliptic PDE from nonlinear boundary measurements by exploiting higher order linearization to recover Fourier data of the potential. It combines a Newton-based forward solver for the nonlinear forward problem with a regularized Fourier inversion to recover $q$ from the extracted data, offering Tikhonov and Total Variation options to handle smooth and discontinuous potentials, respectively. The method is validated through 2D numerical experiments on smooth and discontinuous test potentials, showing accurate localization and shape recovery even with modest bandwidth and noise, and highlighting stability considerations via Savitzky–Golay differentiation and careful harmonic testing strategies. This work provides a practical, numerically implementable pathway for nonlinear Calderón-type inverse problems in elliptic settings, with potential applicability to imaging problems where nonlinear interactions carry information about spatial heterogeneity.

Abstract

We introduce a numerical framework for reconstructing the potential in two dimensional semilinear elliptic PDEs with power type nonlinearities from the nonlinear Dirichlet to Neumann map. By applying higher order linearization method, we compute the Fourier data of the unknown potential and then invert it to recover $q$. Numerical experiments show accurate reconstructions for both smooth and discontinuous test cases.

Numerical reconstruction of Schrödinger equations with quadratic nonlinearities

TL;DR

The paper develops a numerical framework to reconstruct a spatially varying potential in a semilinear elliptic PDE from nonlinear boundary measurements by exploiting higher order linearization to recover Fourier data of the potential. It combines a Newton-based forward solver for the nonlinear forward problem with a regularized Fourier inversion to recover from the extracted data, offering Tikhonov and Total Variation options to handle smooth and discontinuous potentials, respectively. The method is validated through 2D numerical experiments on smooth and discontinuous test potentials, showing accurate localization and shape recovery even with modest bandwidth and noise, and highlighting stability considerations via Savitzky–Golay differentiation and careful harmonic testing strategies. This work provides a practical, numerically implementable pathway for nonlinear Calderón-type inverse problems in elliptic settings, with potential applicability to imaging problems where nonlinear interactions carry information about spatial heterogeneity.

Abstract

We introduce a numerical framework for reconstructing the potential in two dimensional semilinear elliptic PDEs with power type nonlinearities from the nonlinear Dirichlet to Neumann map. By applying higher order linearization method, we compute the Fourier data of the unknown potential and then invert it to recover . Numerical experiments show accurate reconstructions for both smooth and discontinuous test cases.

Paper Structure

This paper contains 15 sections, 1 theorem, 90 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Higher order linearization method for the semilinear elliptic equation
  3. Numerical solution of the Nonlinear Schrödinger equation
  4. Implementation notes
  5. Reconstruction of the potential
  6. Choice of the harmonic functions and boundary values
  7. Numerical reconstruction of the potential from its Fourier transform
  8. Reconstruction of the potential from noisy measurements
  9. Tikhonov regularization
  10. Total Variation (TV) regularization
  11. Numerical examples
  12. Computational setup
  13. Test potentials
  14. Justification of the Newton method
  15. Savitzky-Golay regularized higher order differentiation as local polynomial regression

Key Result

Theorem 3.1

Let $F:V \to V_0'$ be the nonlinear residual associated with the boundary value problem eq:schrodinger on a smooth bounded domain $\Omega$, with $V$ and $V_0$ as defined above.

Figures (6)

  • Figure 1: Singular values of the Fourier matrix $E \in \mathbb{C}^{1800 \times 17692}$ in log-log scale ($N_r = 60$, $N_\theta = 30$, $N_x = N_y = 150$).
  • Figure 2: Example 1: Smooth, localized bump (centered).
  • Figure 3: Example 1: Smooth, localized bump (off center).
  • Figure 4: Example 2: Two bump potential $q(x,y;0.2,0.4, 0.4)$: ground truth (left), reconstruction (middle), and residual (right). The localization and width of the bumps are well reconstructed with Tikhonov regularization ($L^2$ error is $0.153$)
  • Figure 5: Example 3: Ring potential $q(x,y;0.3, 0.5)$: ground truth (left), reconstruction (middle), and residual (right). The jump discontinuities are slightly blurred, but piecewise constantness is well-captured with TV regularization ($L^2$ error is $0.229$).
  • ...and 1 more figures

Theorems & Definitions (5)

  • Theorem 3.1: Well-posedness of the Newton step and convergence
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark A.1