论文标题
安德森加速度具有近似计算：科学计算的应用
Anderson acceleration with approximate calculations: applications to scientific computing
论文作者
论文摘要
我们为安德森加速度（AA）提供严格的理论界限，该界限允许在解决线性问题时进行近似计算。我们表明，当近似计算满足提供的误差界限时，在可以减少计算时间的同时，保持AA的收敛性。我们还提供可计算的启发式量，在理论误差界的指导下，可以在执行近似计算时自动化精度调整。对于线性问题，使用启发式方法来监视通过近似计算引入的误差，并结合对残差单调性的检查，可确保在规定的残差公差内数值方案的收敛性。在理论研究的激励下，我们提出了一个减少的AA变体，该变体包括投射用于计算Anderson混合的最小二乘，以降低尺寸的子空间。该子空间的维度在每次迭代中都会通过可计算的启发式数量进行动态调整。我们通过数值显示和评估AA的性能，通过以下近似计算：（i）由Richardson的计划进行线性确定性的固定点迭代，该方案求解具有开源基准矩阵的线性系统，并具有各种预处理和（ii）非线性确定性定点迭代，由非线性的确定性迭代引起。
We provide rigorous theoretical bounds for Anderson acceleration (AA) that allow for approximate calculations when applied to solve linear problems. We show that, when the approximate calculations satisfy the provided error bounds, the convergence of AA is maintained while the computational time could be reduced. We also provide computable heuristic quantities, guided by the theoretical error bounds, which can be used to automate the tuning of accuracy while performing approximate calculations. For linear problems, the use of heuristics to monitor the error introduced by approximate calculations, combined with the check on monotonicity of the residual, ensures the convergence of the numerical scheme within a prescribed residual tolerance. Motivated by the theoretical studies, we propose a reduced variant of AA, which consists in projecting the least-squares used to compute the Anderson mixing onto a subspace of reduced dimension. The dimensionality of this subspace adapts dynamically at each iteration as prescribed by the computable heuristic quantities. We numerically show and assess the performance of AA with approximate calculations on: (i) linear deterministic fixed-point iterations arising from the Richardson's scheme to solve linear systems with open-source benchmark matrices with various preconditioners and (ii) non-linear deterministic fixed-point iterations arising from non-linear time-dependent Boltzmann equations.
