We present an acceleration method for sequences of large-scale linear systems, such as the ones arising from the numerical solution of time-dependent partial differential equations coupled with algebraic constraints. We discuss different approaches to leverage the subspace containing the history of solutions computed at previous time steps in order to generate a good initial guess for the iterative solver. In particular, we propose a novel combination of reduced-order projection with randomized linear algebra techniques, which drastically reduces the number of iterations needed for convergence. We analyze the accuracy of the initial guess produced by the reduced-order projection when the coefficients of the linear system depend analytically on time. Extending extrapolation results by Demanet and Townsend to a vector-valued setting, we show that the accuracy improves rapidly as the size of the history increases, a theoretical result confirmed by our numerical observations. In particular, we apply the developed method to the simulation of plasma turbulence in the boundary of a fusion device, showing that the time needed for solving the linear systems is significantly reduced.

翻译：我们提出了一种针对大规模线性系统序列的加速方法，此类系统通常源于时间相关偏微分方程与代数约束耦合问题的数值求解。为生成迭代求解器的优质初始猜测，我们探讨了利用历史时刻解向量构成子空间的不同策略。特别地，我们提出了一种结合降阶投影与随机线性代数技术的新型方法，该方法可大幅减少迭代收敛所需步数。当线性系统系数关于时间解析变化时，我们分析了降阶投影所得初始猜测的精度。通过将Demanet与Townsend的外推结果推广至向量值框架，我们证明精度随历史数据规模扩大而快速提升，这一理论结果与数值观测高度吻合。我们进一步将该方法应用于聚变装置边界等离子体湍流模拟，结果表明求解线性系统的时间显著缩短。