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：针对大规模线性方程组序列（例如由含代数约束的时间依赖偏微分方程数值求解所产生的问题），我们提出了一种加速方法。 我们探讨了利用先前时间步求解所得的解历史子空间来为迭代求解器生成良好初始猜测的不同途径。具体而言，我们创新性地将降阶投影与随机线性代数技术相结合，显著减少了收敛所需的迭代次数。 当线性系统系数随时间解析变化时，我们分析了由降阶投影生成的初始猜测的精度。 通过将 Demanet 和 Townsend 的外推结果扩展到向量值形式，我们证明了随着历史数据量的增加，精度迅速提升，这一理论结果得到了我们数值实验的验证。 特别地，我们将所开发的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显著减少。