Overlapping Domain Decomposition Methods for Ptychographic Imaging

Abstract: In ptychography experiments, redundant scanning is usually required to guarantee the stable recovery, such that a huge amount of frames are generated, and thus it poses a great demand of parallel computing in order to solve this large-scale inverse problem. In this paper, we propose the overlapping Domain Decomposition Methods(DDMs) to solve the nonconvex optimization problem in ptychographic imaging. They decouple the problem defined on the whole domain into subproblems only defined on the subdomains with synchronizing information in the overlapping regions of these subdomains,thus leading to highly parallel algorithms with good load balance. More specifically, for the nonblind recovery (with known probe in advance), by enforcing the continuity of the overlapping regions for the image (sample), the nonlinear optimization model is established based on a novel smooth-truncated amplitude-Gaussian metric (ST-AGM). Such metric allows for fast calculation of the proximal mapping with closed form, and meanwhile provides the possibility for the convergence guarantee of the first-order nonconvex optimization algorithm due to its Lipschitz smoothness. Then the Alternating Direction Method of Multipliers (ADMM) is utilized to generate an efficient Overlapping Domain Decomposition based Ptychography algorithm(OD2P) for the two-subdomain domain decomposition (DD), where all subproblems can be computed with close-form solutions.Due to the Lipschitz continuity for the gradient of the objective function with ST-AGM, the convergence of the proposed OD2P is derived under mild conditions. Moreover, it is extended to more general case including multiple-subdomain DD and blind recovery. Numerical experiments are further conducted to show the performance of proposed algorithms, demonstrating good convergence speed and robustness to the noise.