We provide rigorous theoretical bounds for Anderson acceleration (AA) that allow for efficient approximate residual calculations in AA, which in turn reduce computational time and memory storage while maintaining convergence. Specifically, 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 computable heuristic quantities guided by rigorous theoretical error bounds. The use of heuristics to monitor the error introduced by approximate calculations, combined with the check on monotonicity of the convergence, ensures the convergence of the numerical scheme within a prescribed tolerance threshold on the residual. 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.

翻译：本文为允许在Anderson加速(AA)中进行高效近似残差计算的算法提供了严格的理论界，该方法在保持收敛性的同时降低了计算时间和存储开销。具体而言，我们提出了一种AA的降阶变体，其核心思想是将用于计算Anderson混合的最小二乘问题投影到降维子空间上。该子空间的维数依据严格理论误差界导出的可计算启发式量，在每次迭代中动态自适应调整。结合收敛单调性检验，采用启发式方法监控近似计算引入的误差，确保了数值方案能在预设残差容限阈值内收敛。我们通过以下两类问题数值验证并评估了近似计算AA的性能：(i) 使用开源基准矩阵配合各类预条件子求解线性系统时，由Richardson迭代产生的线性确定性不动点迭代；(ii) 求解非线性时间依赖Boltzmann方程时产生的非线性确定性不动点迭代。