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：针对大规模线性系统序列（例如由时间相关的偏微分方程与代数约束耦合问题的数值求解所产生的系统），我们提出了一种加速方法。我们探讨了利用先前时间步计算解的历史子空间来为迭代求解器生成良好初始猜测的不同策略。特别地，我们提出了一种将降阶投影与随机线性代数技术相结合的新方法，该方法显著减少了收敛所需的迭代次数。当线性系统系数关于时间解析变化时，我们分析了由降阶投影生成的初始猜测的精度。通过将Demanet与Townsend的外推结果推广至向量值情形，我们证明随着历史数据规模的增大，精度会快速提升，这一理论结果得到了数值实验的验证。我们将所发展方法应用于聚变装置边界等离子体湍流的模拟中，结果表明求解线性系统所需的时间得到了显著缩减。