AI 中文总结
本文研究原始-对偶活动集策略（PDAS）在离散化无限维问题中的迭代次数增长问题，通过数值与理论分析，解释其在障碍问题中指数增长、Signorini型问题中线性增长的现象及收敛特性。
AI 中文摘要
原始-对偶活动集策略（PDAS）是混合互补问题的常用迭代求解器，适用于带逐点不等式约束的约束优化问题，例如PETSc中包含的降维活动集算法vinewtonrsls。当应用于离散化的无限维问题时，PDAS因与半光滑牛顿法（SSN）等价而具有局部超线性收敛性。然而，对于许多问题类，随着网格加密，PDAS达到收敛所需的迭代次数会无限制增长。本文对障碍问题、Signorini问题及相关模型在均匀加密网格上的PDAS迭代次数进行数值研究：当网格尺寸趋于零时，应用于Signorini型问题的PDAS失去局部超线性收敛性，导致迭代次数呈线性增长（每次加密增加若干次迭代）；对于障碍问题，PDAS出现停滞，导致迭代次数呈指数增长（每次加密渐近翻倍）。本文通过三方面解释这些现象：（i）证明对于障碍问题，节点自由度在去激活阶段仅逐层从障碍处脱离；（ii）推导PDAS的一般全局收敛率，其依赖于对偶可行性违反的程度；（iii）说明在无限维情形下，这一特性为何使PDAS对部分问题成为定义明确的求解器，但无局部超线性收敛性，甚至对另一些问题会发散。
英文摘要
Primal-dual active set strategies (PDAS) are popular iterative solvers for mixed complementarity problems such as constrained optimization problems with pointwise inequality constraints. Examples include the reduced-space active set algorithm vinewtonrsls found in PETSc. When applied to discretized infinite-dimensional problems, PDAS exhibit local superlinear convergence thanks to their equivalence to a semismooth Newton method (SSN). However, for many problem classes the number of iterations, to reach convergence, grows without bound under mesh refinement. In this paper we numerically study PDAS iteration counts on uniformly refined meshes for obstacle problems, Signorini problems, and related models. As the mesh size tends to zero, PDAS applied to Signorini-type problems lose their local superlinear convergence, resulting in linear growth of the iteration count (adding some iterations with each refinement). For obstacle problems, PDAS stagnates, leading to exponential iteration growth (asymptotically doubling with each refinement). We explain these phenomena by (i) proving that, for obstacle problems, nodal degrees of freedom only peel away from the obstacle layer-by-layer during the deactivation phase, (ii) deriving a general global convergence rate for PDAS that depends on the magnitude of dual feasibility violation, and (iii) demonstrating why, in the infinite-dimensional setting, this leads to a well-defined solver, but without local superlinear convergence, for some problems yet divergence for others.
