Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Identification of Source Terms in the Ginzburg-Landau Equation from Final Data

Roberto Morales, Javier-Ramírez-Ganga

TL;DR

This work addresses the inverse-source problem for the complex Ginzburg-Landau equation from final-time data by formulating a Tikhonov-regularized variational problem on a weak-solution framework. It derives an explicit adjoint-based gradient, proves its Lipschitz continuity, and establishes existence (and under a positivity condition, uniqueness) of quasi-solutions, providing a solid theoretical foundation for reconstruction. Numerically, it implements a Crank-Nicolson–based discretization with a conjugate-gradient scheme using adjoint gradients to successfully recover diverse space-time sources in 2D, even under moderate noise. The results highlight the practical viability of adjoint-based regularization for dissipative-dispersive PDEs and outline open challenges, including nonlinear GL extensions, stability estimates, and data-driven enhancements for large-scale problems.

Abstract

In this article, we study an inverse problem consisting in the identification of a space-time dependent source term in the Ginzburg-Landau equation from final-time observations. We adopt a weak-solution framework and analyze Tikhonov's functional, deriving an explicit gradient formula via an adjoint system and proving its Lipschitz continuity. We then establish existence and uniqueness results for quasi-solutions, and validate the theory with numerical experiments based on iterative methods.

Identification of Source Terms in the Ginzburg-Landau Equation from Final Data

TL;DR

This work addresses the inverse-source problem for the complex Ginzburg-Landau equation from final-time data by formulating a Tikhonov-regularized variational problem on a weak-solution framework. It derives an explicit adjoint-based gradient, proves its Lipschitz continuity, and establishes existence (and under a positivity condition, uniqueness) of quasi-solutions, providing a solid theoretical foundation for reconstruction. Numerically, it implements a Crank-Nicolson–based discretization with a conjugate-gradient scheme using adjoint gradients to successfully recover diverse space-time sources in 2D, even under moderate noise. The results highlight the practical viability of adjoint-based regularization for dissipative-dispersive PDEs and outline open challenges, including nonlinear GL extensions, stability estimates, and data-driven enhancements for large-scale problems.

Abstract

In this article, we study an inverse problem consisting in the identification of a space-time dependent source term in the Ginzburg-Landau equation from final-time observations. We adopt a weak-solution framework and analyze Tikhonov's functional, deriving an explicit gradient formula via an adjoint system and proving its Lipschitz continuity. We then establish existence and uniqueness results for quasi-solutions, and validate the theory with numerical experiments based on iterative methods.

Paper Structure

This paper contains 15 sections, 8 theorems, 57 equations, 16 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature
  3. The input-output operator
  4. Outline
  5. Fréchet differentiability and gradient formula
  6. Existence and uniqueness of the solution to (ISP)
  7. Numerical results
  8. Discretization and numerical implementation
  9. Numerical experiments
  10. Example 1 (Smooth real-valued source)
  11. Example 2 (Localized imaginary Gaussian source)
  12. Example 3 (Complex source with mixed frequencies)
  13. Example 4 (Space–time forcing)
  14. Conclusions and open problems
  15. Existence and uniqueness results for the linearized GL equation

Key Result

Proposition 1.1

The input-output operator $\Psi:L^2(0,T;L^2(\Omega))\to L^2(\Omega)$ is compact.

Figures (16)

  • Figure 1: Final-time comparison: real part. Parameters $(N_x,N_y,N_t)=(100,100,70)$, $\\tau = 10^{-5}$, $\epsilon=10^{-5}$.
  • Figure 2: Final-time comparison: imaginary part. Parameters $(N_x,N_y,N_t)=(100,100,70)$, $\\tau = 10^{-5}$, $\epsilon=10^{-5}$.
  • Figure 3: Recovered vs. true source (real part). Parameters $(N_x,N_y,N_t)=(100,100,70)$, $\\tau = 10^{-5}$, $\epsilon=10^{-5}$.
  • Figure 4: Recovered vs. true source (imaginary part). Parameters $(N_x,N_y,N_t)=(100,100,70)$, $\\tau = 10^{-5}$, $\epsilon=10^{-5}$.
  • Figure 5: Final-time comparison: real part. Parameters $(N_x,N_y,N_t)=(100,100,70)$, $\tau = 10^{-5}$, $\varepsilon=10^{-5}$.
  • ...and 11 more figures

Theorems & Definitions (17)

  • Proposition 1.1
  • proof
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • Lemma 3.3
  • ...and 7 more