In this work, we propose and study a preconditioned framework with a graphic Ginzburg-Landau functional for image segmentation and data clustering by parallel computing. Solving nonlocal models is usually challenging due to the huge computation burden. For the nonconvex and nonlocal variational functional, we propose several damped Jacobi and generalized Richardson preconditioners for the large-scale linear systems within a difference of convex functions algorithms framework. They are efficient for parallel computing with GPU and can leverage the computational cost. Our framework also provides flexible step sizes with a global convergence guarantee. Numerical experiments show the proposed algorithms are very competitive compared to the singular value decomposition based spectral method.

翻译：本文提出并研究了一种基于图Ginzburg-Landau泛函的预处理框架，用于通过并行计算实现图像分割与数据聚类。求解非局部模型通常因计算量巨大而颇具挑战。针对非凸非局部变分泛函，我们在凸函数差分算法框架内，为大规模线性系统提出了几种阻尼Jacobi预条件子与广义Richardson预条件子。这些预条件子在GPU上并行计算高效，并能有效降低计算成本。我们的框架还提供了具有全局收敛保证的灵活步长。数值实验表明，与基于奇异值分解的谱方法相比，所提算法极具竞争力。