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.

翻译：本文针对求解线性问题时允许近似计算的Anderson加速（AA）方法，提供了严格的理论误差界。我们证明，当近似计算满足所给出的误差界时，AA的收敛性得以保持，同时计算时间可能减少。基于理论误差界的指导，我们进一步提出了可计算的启发式量，用于在近似计算过程中自动调节精度。在线性问题中，结合监测近似计算引入误差的启发式方法与残差单调性检查，能够确保数值方案在指定的残差容限内收敛。受理论研究的启发，我们提出了一种AA的简化变体，其核心思想是将用于计算Anderson混合的最小二乘投影到低维子空间上。该子空间的维度可根据每次迭代中可计算的启发式量动态自适应调整。我们通过数值实验评估了近似计算下AA的性能，测试场景包括：（i）基于Richardson格式求解线性系统的线性确定性不动点迭代（采用多种预处理器与开源基准矩阵），以及（ii）由非线性时变玻尔兹曼方程导出的非线性确定性不动点迭代。