An Augmented Lagrangian Method-Based Framework in the Adjoint Space for Sparse Reconstruction of Acoustic Sources

TL;DR

This paper tackles the sparse inverse source reconstruction problem in inverse acoustic scattering, where boundary data are linked to a volumetric source via the Helmholtz-based volume potential. The authors develop a semismooth Newton–based augmented Lagrangian method that operates in the adjoint/measurement space of dimension $M$, with the primal source recovered from Fenchel–Rockafellar duality. They establish global convergence and a local linear convergence rate for the dual-space ALM and demonstrate substantial computational speedups, particularly when $M\ll N$, across 2D and 3D tests. A comparative study with a first-order primal-dual method and a dedicated SSN approach shows the proposed method outperforms alternative strategies, especially in three dimensions. The work suggests that the adjoint-space framework and regularized sparse inversion can be extended to multifrequency data and related inverse-wave problems in acoustics and electromagnetics.

Abstract

We propose a semismooth Newton-based augmented Lagrangian framework for reconstructing sparse sources in inverse acoustic scattering problems. Rather than working in the unknown source space, our semismooth Newton updates operate in the measurement (adjoint) space, which is especially efficient when the number of measurements is much smaller than the discretized source dimension. The source is then recovered via Fenchel-Rockafellar duality. Our approach substantially accelerates computation and reduces costs. Numerical experiments in two and three dimensions demonstrate the high efficiency of the proposed method.

Figures (9)

• Figure 1: Illustration of the acoustic source scattering: the red squares represent the receivers for recording the scattered wave, and the trapezoid labeled $A$ represents the acoustic sources.
• Figure 2: Reconstruction of sparse sources with multiple peaks in homogeneous media with $k=6$. The images in the leftmost column are the original acoustic sources, including one, four, six, and eight peaks. The images in the second, the third from the left, and the rightmost columns are reconstructed results of ALM, SSN, and PDA, respectively. The images in the first, second, third, and fourth rows are the sources with one, four, six, and eight peaks, respectively.
• Figure 3: Reconstruction of sparse sources with multiple peaks in inhomogeneous media with $k=6$. The information of the acoustic sources and the corresponding reconstruction algorithms are the same as in Figure \ref{['fig:big_figure1']}.
• Figure 4: Reconstructions of strip-shaped sparse sources in homogeneous media with $k=4$, noise level 0.1%, and $\alpha$ =2e-6, $\alpha_0$= 2e-8. The figures in the leftmost column are the original acoustic sources. The images in the second, the third from the left, and the rightmost columns are reconstructed results of ALM, SSN, and PDA, respectively. The images in the first and second rows are the sources with "skew diagonal" (the first row) and "diagonal" (the second row) types of strips, respectively.
• Figure 5: Reconstruction of strip-shaped sparse sources in inhomogeneous media with $k=4$, noise level 0.1%, and $\alpha$ =2e-6, $\alpha_0$= 2e-8. The information of the acoustic sources and the corresponding reconstruction algorithms are the same as in Figure \ref{['fig:big_figure3']}.

Theorems & Definitions (11)

• Proposition 1
• Theorem 1
• proof
• Lemma 1
• proof
• Remark 1
• Definition 1: Metric Subregularity DR
• Theorem 2
• proof
• Theorem 3