Anderson acceleration for contractive and noncontractive operators
Anderson acceleration for contractive and noncontractive operators
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DOI：
10.1093/imanum/draa095
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发表时间：
2019-09
期刊：
影响因子：
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通讯作者：
Sara N. Pollock;L. Rebholz
中科院分区：
文献类型：
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作者：
Sara N. Pollock;L. Rebholz
给出了一般算法深度下安德森加速度的一步分析。在收缩和非收缩设置内得到的残差界限揭示了高阶项和低阶项的贡献之间的平衡，这两者都取决于在算法的每个步骤中解决的优化问题的成功。新的残差界限显示，由外推法引入的附加项产生的项比以前理解的项具有更高的阶数。在收缩设置这些界限锐化以前的收敛和加速的结果。边界依赖于连续残差之间的差异的足够的线性独立性，而不是假设的优化系数的有界性，允许引入一个理论上合理的保障策略。几个数值试验说明了分析主要是在非收缩设置，并展示了使用的方法，保障战略和理论为基础的指导动态选择的算法深度，对p-Laplace方程，非线性Helmholtz方程和稳定的Navier-Stokes方程与高雷诺数在三维空间。
A one-step analysis of Anderson acceleration with general algorithmic depths is presented. The resulting residual bounds within both contractive and noncontractive settings reveal the balance between the contributions from the higher and lower order terms, which are both dependent on the success of the optimization problem solved at each step of the algorithm. The new residual bounds show the additional terms introduced by the extrapolation produce terms that are of a higher order than was previously understood. In the contractive setting these bounds sharpen previous convergence and acceleration results. The bounds rely on sufficient linear independence of the differences between consecutive residuals, rather than assumptions on the boundedness of the optimization coefficients, allowing the introduction of a theoretically sound safeguarding strategy. Several numerical tests illustrate the analysis primarily in the noncontractive setting, and demonstrate the use of the method, the safeguarding strategy and theory-based guidance on dynamic selection of the algorithmic depth, on a p-Laplace equation, a nonlinear Helmholtz equation and the steady Navier–Stokes equations with high Reynolds number in three spatial dimensions.