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Bi-level regularization via iterative mesh refinement for aeroacoustics

Christian Aarset, Tram Thi Ngoc Nguyen

TL;DR

The paper addresses an ill-posed inverse aeroacoustic source problem for the Helmholtz equation, where an unknown source $\phi$ supported on $\Omega_0$ must be recovered from measurements on $\Omega_1$. It introduces a bi-level regularization framework that couples an upper-level source update with a lower-level PDE solve and leverages adaptive mesh refinement to maintain regularization while accelerating convergence, with the adjoint $F^*$ explicitly derived. A discrepancy-based stopping rule and a multi-grid refinement strategy tie the discretization error to data noise, enabling efficient, robust reconstructions. Numerical experiments show that bi-level Landweber with mesh refinement converges faster and yields smaller residuals than direct Landweber, especially at higher noise levels, demonstrating the practical benefits of adaptive discretization in PDE-constrained inverse problems. The approach has potential implications for optimal experimental design and extensions to nonlinear settings are discussed as future work.

Abstract

In this work, we illustrate the connection between adaptive mesh refinement for finite element discretized PDEs and the recently developed \emph{bi-level regularization algorithm}. By adaptive mesh refinement according to data noise, regularization effect and convergence are immediate consequences. We moreover demonstrate its numerical advantages to the classical Landweber algorithm in term of time and reconstruction quality for the example of the Helmholtz equation in an aeroacoustic setting.

Bi-level regularization via iterative mesh refinement for aeroacoustics

TL;DR

The paper addresses an ill-posed inverse aeroacoustic source problem for the Helmholtz equation, where an unknown source supported on must be recovered from measurements on . It introduces a bi-level regularization framework that couples an upper-level source update with a lower-level PDE solve and leverages adaptive mesh refinement to maintain regularization while accelerating convergence, with the adjoint explicitly derived. A discrepancy-based stopping rule and a multi-grid refinement strategy tie the discretization error to data noise, enabling efficient, robust reconstructions. Numerical experiments show that bi-level Landweber with mesh refinement converges faster and yields smaller residuals than direct Landweber, especially at higher noise levels, demonstrating the practical benefits of adaptive discretization in PDE-constrained inverse problems. The approach has potential implications for optimal experimental design and extensions to nonlinear settings are discussed as future work.

Abstract

In this work, we illustrate the connection between adaptive mesh refinement for finite element discretized PDEs and the recently developed \emph{bi-level regularization algorithm}. By adaptive mesh refinement according to data noise, regularization effect and convergence are immediate consequences. We moreover demonstrate its numerical advantages to the classical Landweber algorithm in term of time and reconstruction quality for the example of the Helmholtz equation in an aeroacoustic setting.
Paper Structure (4 sections, 8 equations, 5 figures, 1 algorithm)

This paper contains 4 sections, 8 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Inverse aeroacoustic source with Helmholtz equation
  2. Inversion algorithm with mesh iterative refinement
  3. Numerical experiments
  4. Conclusions

Key Result

lemma 1

The Hilbert space adjoint $F^*:L^2(\Omega_1,\mathbb{C}) \mapsto L^2(\Omega_0)$ is where $v$ is implicitly extended by zero outside of $\Omega_1$.

Figures (5)

  • Figure 1: Iterative reconstruction on each mesh in the bi-level algorithm, prior to refinement. Noise level $1\%$. $h\approx 0.531$, $0.265$, $0.139$, $0.08$.
  • Figure 2: Iterative reconstruction on each mesh in the bi-level algorithm, prior to refinement. Noise level $10\%$. $h\approx 0.531$, $0.265$.
  • Figure 3: True source $\phi$. Mesh not depicted.
  • Figure 4: Error plots of reconstructions. Left to right: Bi-level with $1\%$ noise, direct Landweber with $1\%$ noise, bi-level with $10\%$ noise, direct Landweber with $10\%$ noise. Mesh not depicted.
  • Figure 5: Error plots over time. Blue background denotes which mesh refinement the bi-level algorithm is currently on.

Theorems & Definitions (2)

  • proof
  • remark 1: Multi grid size