论文标题
基于$ \ MATHCAL {H}^{ - S} $ SOBOLEV NORM的ANDERSON加速
Anderson Acceleration Based on the $\mathcal{H}^{-s}$ Sobolev Norm for Contractive and Noncontractive Fixed-Point Operators
论文作者
论文摘要
安德森加速度（AA）是一种加速定点迭代的收敛性的技术。在本文中，我们将AA应用于一系列函数，并将其内部优化问题中的规范修改为$ \ Mathcal {h}^{ - s} $ norm，对于某些积极的整数$ s $，以将其偏向于残留物中的低频光谱。我们通过量化对PICARD迭代的改进来分析AA的收敛性。我们发现，基于$ \ Mathcal {H}^{ - 2} $ NORM的AA非常适合求解固定点运算符，这些固定点运算符衍生自二阶椭圆差分运算符，包括Helmholtz方程。
Anderson acceleration (AA) is a technique for accelerating the convergence of fixed-point iterations. In this paper, we apply AA to a sequence of functions and modify the norm in its internal optimization problem to the $\mathcal{H}^{-s}$ norm, for some positive integer $s$, to bias it towards low-frequency spectral content in the residual. We analyze the convergence of AA by quantifying its improvement over Picard iteration. We find that AA based on the $\mathcal{H}^{-2}$ norm is well-suited to solve fixed-point operators derived from second-order elliptic differential operators, including the Helmholtz equation.
