论文标题
椭圆形最佳控制问题的两种强大多机方法的平滑分析
Smoothing analysis of two robust multigrid methods for elliptic optimal control problems
论文作者
论文摘要
在本文中，我们研究和比较了两个多族松弛方案与两个，三个和四个，以解决椭圆形稀疏的最佳控制问题，并与控制约束。首先，我们对众所周知的集体雅各比弛豫（CJR）方案进行详细的局部傅立叶分析（LFA），其中得出了最佳的平滑因子。这种有见地的分析表明，最佳放松参数取决于网格的大小和正则化参数，这在文献中没有研究。其次，我们提出并分析了一种新的基于质量的盲目乳糖浆松弛（BSR）方案，事实证明，当$α\ ge ch^4 $用于小型常数$ c $时，该方案比CJR方案提供了较小的平滑因子。这里$α$是正规化参数，$ h $是空间网状步长。这些方案通过半平滑的牛顿方法成功扩展到控制受限的案例。用BSR将三个或四个块化的竞争力竞争性，而将两个则浓缩。提出了数值示例以验证我们的理论结果。提出的不精确BSR（IBSR）方案，其中将两个预处理的共轭梯度迭代应用于Schur补体系统，其计算效率比CJR方案更高。
In this paper we study and compare two multigrid relaxation schemes with coarsening by two, three, and four for solving elliptic sparse optimal control problems with control constraints. First, we perform a detailed local Fourier analysis (LFA) of a well-known collective Jacobi relaxation (CJR) scheme, where the optimal smoothing factors are derived. This insightful analysis reveals that the optimal relaxation parameters depend on mesh size and regularization parameters, which was not investigated in literature. Second, we propose and analyze a new mass-based Braess-Sarazin relaxation (BSR) scheme, which is proven to provide smaller smoothing factors than the CJR scheme when $α\ge ch^4$ for a small constant $c$. Here $α$ is the regularization parameter and $h$ is the spatial mesh step size. These schemes are successfully extended to control-constrained cases through the semi-smooth Newton method. Coarsening by three or four with BSR is competitive with coarsening by two. Numerical examples are presented to validate our theoretical outcomes. The proposed inexact BSR (IBSR) scheme, where two preconditioned conjugate gradients iterations are applied to the Schur complement system, yields better computational efficiency than the CJR scheme.
