用物理启发的线性化ADMM加速偏微分方程逆问题求解
Physics-Aware Linearized ADMM and Its Unrolling
- 通过线性化PDE子问题，每迭代仅需调用一次PDE求解器及梯度
- 理论保证在特定条件下收敛，计算效率显著提升
- 可与深度展开结合，适用于光学通信压缩感知和图像恢复
近年来，偏微分方程（PDE）被用于直接建模信号处理中的测量过程，尽管其求解成本较高。本文提出一种基于交替方向乘子法（ADMM）的新算法——物理感知线性化ADMM（PA-LADMM），用于基于PDE的测量过程中的逆问题求解。核心思想是将含PDE的子问题进行线性化，从而得到仅需每轮迭代调用一次PDE求解器及其梯度评估的高效更新规则。该算法在一定条件下具有理论收敛性。此外，我们将其与深度展开结合，通过监督数据训练内部参数。两个实验分别在光纤通信压缩感知和噪声各向异性扩散图像恢复中验证了所提算法的有效性。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。