Physics-Aware Linearized ADMM and Its Unrolling
This work addresses the computational bottleneck of PDE-based inverse problems for signal processing researchers, offering a practical algorithm with theoretical guarantees.
The authors propose a physics-aware linearized ADMM (PA-LADMM) algorithm for inverse problems with PDE-based measurement processes, achieving cost-efficient updates with theoretical convergence guarantees. Unrolling the algorithm and training parameters yields improved performance in compressed sensing and image restoration tasks.
Recently, partial differential equations (PDEs) have been used to directly model the measurement process in signal processing, although their evaluation is costly. In this paper, we propose a novel alternating direction method of multipliers (ADMM)-based algorithm called physics-aware linearized ADMM (PA-LADMM) for inverse problems from PDE-based measurement processes. The key idea is the linearization of the subproblem with PDEs, leading to a cost-efficient update rule that calls only a PDE solver and its gradient evaluation per iteration. The algorithm has a theoretical convergence guarantee under certain conditions. In addition, we combine it with deep unfolding to unroll the PA-LADMM and train its internal parameters using supervised data. Two distinct experiments, compressed sensing with optical fiber communication and image restoration from noisy anisotropic diffusion, demonstrated the effectiveness of the proposed algorithms.