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Reconstruction of Boundary Data in the Helmholtz Equation Using Particle Swarm Optimization

Jamal Daoudi, Chakir Tajani

TL;DR

This work addresses an ill-posed data completion problem for the Helmholtz equation by recovering unknown boundary data on an inaccessible boundary from Cauchy data on an accessible boundary. It combines Particle Swarm Optimization with Tikhonov regularization and uses a Finite Element forward solver to evaluate the objective, formulating the inversion as a regularized least-squares problem $\mathcal{J}_{DR}(\phi_D) = \tfrac{1}{2}\|u(\phi_D,g)-f\|^2_{L^2(\Gamma_c)} + \tfrac{\alpha}{2}\|\phi_D\|^2_{H^{1/2}(\Gamma_i)}$, with $\alpha>0$. The PSO framework searches for boundary traces $\phi_D$ (and $\phi_N$ when reconstructing the Neumann data) by minimizing the data-misfit while the FEM solves the direct problem $u(\phi_D,g)$; this approach is tested on unit square and unit disc domains with synthetic noise. Numerical results show rapid convergence of the objective functions, stability under noise up to $3\%$, and accurate reconstructions, with Dirichlet data generally recovered more reliably than Neumann data. The proposed PSO-based method offers a flexible and effective alternative for ill-posed inverse boundary value problems, and its use on irregular domains suggests broad applicability to similar PDE data completion tasks.

Abstract

This paper tackles the data completion problem related to the Helmholtz equation. The goal is to identify unknown boundary conditions on parts of the boundary that cannot be accessed directly, by making use of measurements collected from accessible regions. Such inverse problems are known to be ill-posed in the Hadamard sense, which makes finding stable and dependable solutions particularly difficult. To address these challenges, we propose a bio-inspired method that combines Particle Swarm Optimization with Tikhonov regularization. The results of our numerical experiments suggest that this approach can yield solutions that are both accurate and stable, converging reliably. Overall, this method provides a promising way to handle the inherent instability and sensitivity of these types of inverse problems.

Reconstruction of Boundary Data in the Helmholtz Equation Using Particle Swarm Optimization

TL;DR

This work addresses an ill-posed data completion problem for the Helmholtz equation by recovering unknown boundary data on an inaccessible boundary from Cauchy data on an accessible boundary. It combines Particle Swarm Optimization with Tikhonov regularization and uses a Finite Element forward solver to evaluate the objective, formulating the inversion as a regularized least-squares problem , with . The PSO framework searches for boundary traces (and when reconstructing the Neumann data) by minimizing the data-misfit while the FEM solves the direct problem ; this approach is tested on unit square and unit disc domains with synthetic noise. Numerical results show rapid convergence of the objective functions, stability under noise up to , and accurate reconstructions, with Dirichlet data generally recovered more reliably than Neumann data. The proposed PSO-based method offers a flexible and effective alternative for ill-posed inverse boundary value problems, and its use on irregular domains suggests broad applicability to similar PDE data completion tasks.

Abstract

This paper tackles the data completion problem related to the Helmholtz equation. The goal is to identify unknown boundary conditions on parts of the boundary that cannot be accessed directly, by making use of measurements collected from accessible regions. Such inverse problems are known to be ill-posed in the Hadamard sense, which makes finding stable and dependable solutions particularly difficult. To address these challenges, we propose a bio-inspired method that combines Particle Swarm Optimization with Tikhonov regularization. The results of our numerical experiments suggest that this approach can yield solutions that are both accurate and stable, converging reliably. Overall, this method provides a promising way to handle the inherent instability and sensitivity of these types of inverse problems.

Paper Structure

This paper contains 16 sections, 9 equations, 10 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical Formulation
  3. Optimization Problem
  4. Solving the Cauchy Problem for the Helmholtz Equation Using a Stochastic Approach
  5. Particle Swarm Optimization
  6. Algorithm for Solving the Cauchy Problem for the Helmholtz Equation
  7. Numerical Results and Discussion
  8. Unit Square
  9. Choice of Regularization Parameter
  10. Convergence of the Algorithm
  11. Stability of the Proposed Method
  12. Unit Disc
  13. Regularization Parameter Choice
  14. Convergence of the Algorithm
  15. Stability of the Proposed Method
  16. ...and 1 more sections

Figures (10)

  • Figure 1: Configuration of the Unit Square Domain
  • Figure 2: Configuration of the Unit Disc Domain
  • Figure 3: The objective function $\mathcal{J_{DR}}(\phi_D)$ and the trace of $u$ on $\Gamma_i$ for various iterations with the best regularization parameter $\alpha=1e-08$.
  • Figure 4: The objective function $\mathcal{J_{NR}}(\phi_N)$ and the derivative of $u$$(\partial_n u)$ on $\Gamma_i$ for various iterations with the best regularization parameter $\beta=1e-8$.
  • Figure 5: The objective function $\mathcal{J_{DR}}(\phi_D)$ and the trace of $u$ on $\Gamma_i$ for various noise levels.
  • ...and 5 more figures