发表机构
VIT-AP University（维特-阿普大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出新型并行迭代格式求解障碍与自由边值问题，结合SPIKE算法、方向分裂法等，成功求解高维问题并验证其在图像去模糊中的有效性。
AI 中文摘要
本文提出一类新型高效的并行直接与间接迭代格式，用于求解一般障碍问题和自由边值问题。在模型问题生成M矩阵的假设下，证明了直接并行方法解的唯一性；间接方法的收敛性分析基于在线性近似过程的每个迭代阶段最小化离散能量泛函。在上述设定下，针对Ferris和Mangasarian定义的凸集上的光滑与非光滑能量泛函，建立了理论收敛结果。对于一维障碍问题的数值计算，采用基于SPIKE算法的直接广义并行方法；为加速收敛并降低计算复杂度，将该格式的快速递归版本推广至非光滑情形。对于二维和三维障碍问题，采用方向分裂法，将每个方向子问题视为一维障碍极小化问题处理；通过将能量泛函极小化构造为抛物型时变问题，并在解更新过程中采用Armijo时间步长规则，成功得到高维障碍问题的求解结果。此外，本研究还探索了将该算法扩展为约束二次规划优化求解器的可能性，并在上述设定下针对图像去模糊现象对该框架进行评估，成功恢复出原始图像，最后通过数值算例验证了理论结果。
英文摘要
This paper introduces a novel, efficient class of parallel direct and indirect iterative schemes to solve general obstacle and free boundary value problems. The uniqueness of the solution for the direct parallel method is established under the assumption that the model problem yields an $M$-matrix. The convergence analysis of the indirect approach is predicated on minimizing the discrete energy functional at each iterative stage of the linear approximation process. Under the aforementioned setting, we establish theoretical convergence results for both smooth and nonsmooth energy functionals defined over a convex set, in the sense of Ferris and Mangasarian \cite{ferris1994parallel}. For the numerical computation of the one-dimensional obstacle problem, we adopt a direct generalized parallel approach based on the SPIKE algorithm \cite{polizzi2006parallel}. To accelerate convergence and reduce computational complexity, a fast recursive version of the scheme is generalized to the nonsmooth case. For two- and three-dimensional obstacle problems, we employ a directional splitting method that treats each directional subproblem as a one-dimensional obstacle minimization problem. By framing the energy functional minimization as a parabolic time-dependent problem and utilizing an Armijo time-stepping rule during the solution update process, we successfully obtain results for higher-dimensional obstacles. Additionally, this study explores the possibility of extending the algorithm into a constrained quadratic programming optimization solver. We also evaluate the framework on image deblurring phenomena under the aforementioned setting, successfully recovering the original images. Finally, numerical illustrations are provided to validate the theoretical results.
