We provide rigorous theoretical bounds for Anderson acceleration (AA) that allow for efficient approximate calculations of the residual, which reduce computational time and memory storage while maintaining convergence. Specifically, we propose a reduced variant of AA, which consists in projecting the least squares 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 the theoretical error bounds. The use of the heuristic 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 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：我们为安德森加速（AA）提供了严格的理论界限，允许对残差进行高效的近似计算，从而在保持收敛性的同时减少计算时间和内存存储。具体而言，我们提出了一种AA的简化变体，其核心在于将用于计算安德森混合的最小二乘投影到降维子空间上。该子空间的维度在每次迭代中动态自适应调整，其依据由理论误差界限导出的可计算启发式量决定。利用启发式量监控近似计算引入的误差，并结合收敛单调性检查，确保了数值方案在残差预设容差阈值内的收敛性。我们在以下两类问题上对基于近似计算的AA性能进行了数值评估：（i）由理查森格式求解线性系统产生的线性确定性不动点迭代，使用了带有多种预处理器的开源基准矩阵；（ii）由非线性时变玻尔兹曼方程产生的非线性确定性不动点迭代。