发表机构
Tufts University; University of Houston（塔夫茨大学; 休斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出统一两网格框架，将Anderson加速与非线性GMRES统一，并据此设计新变体，通过扩大粗空间提升性能，数值实验验证其优越性。
AI 中文摘要
本文中，我们将两种广泛用于不动点迭代的加速方法——Anderson加速（AA）和非线性广义最小残差法（NGMRES）——重新表述为两网格方法。通过显式推导AA和NGMRES在线性问题上的误差传播矩阵，我们表明这两种方法及其若干现有变体，均可重构为带单次预平滑或后平滑步骤、以及由相对于适当内积的投影给出的粗网格校正的两网格方法。这一重构不仅将AA、NGMRES及其变体统一在单一算法框架内，还使得通过独立调整两网格方法的组件（如粗空间和平滑步骤）来设计新变体成为可能。特别地，我们提出了一种新变体，通过纳入最近更新的残差来扩大粗空间，这一改进由两网格公式自然揭示，但在AA和NGMRES算法的标准公式中却是隐藏的。在SPD Poisson问题及一族非SPD对流-扩散问题上的数值实验表明，该新变体通常在性能上优于AA和NGMRES及其变体。
英文摘要
In this work, we recast two widely used acceleration methods for fixed-point iterations, Anderson acceleration (AA) and the nonlinear generalized minimal residual method (NGMRES), as two-grid methods. By explicitly deriving the error propagation matrices for AA and NGMRES on linear problems, we show that both methods, together with several of their existing variants, can be reformulated as two-grid methods with a single pre- or postsmoothing step and a coarse-grid correction given by a projection with respect to a suitable inner product. This reformulation not only unifies AA, NGMRES, and their variants within a single algorithmic framework, but also enables the design of new variants by independently adjusting the components of the two-grid method, such as the coarse space and the smoothing steps. In particular, we propose a new variant that enlarges the coarse space by incorporating the most recently updated residual, an improvement that is naturally revealed by the two-grid formulation but is hidden in the standard formulations of the AA and NGMRES algorithms. Numerical experiments on both an SPD Poisson problem and a family of non-SPD convection-diffusion problems show that this new variant often outperforms both AA and NGMRES, as well as their variants.
