This paper proposes and tests the first-ever reduced basis warm-start iterative method for the parametrized linear systems, exemplified by those discretizing the parametric partial differential equations. Traditional iterative methods are usually used to obtain the high-fidelity solutions of these linear systems. However, they typically come with a significant computational cost which becomes challenging if not entirely untenable when the parametrized systems need to be solved a large number of times (e.g. corresponding to different parameter values or time steps). Classical techniques for mitigating this cost mainly include acceleration approaches such as preconditioning. This paper advocates for the generation of an initial prediction with controllable fidelity as an alternative approach to achieve the same goal. The proposed reduced basis warm-start iterative method leverages the mathematically rigorous and efficient reduced basis method to generate a high-quality initial guess thereby decreasing the number of iterative steps. Via comparison with the iterative method initialized with a zero solution and the RBM preconditioned and initialized iterative method tested on two 3D steady-state diffusion equations, we establish the efficacy of the proposed reduced basis warm-start approach.

翻译：本文提出并首次测试了一种针对参数化线性系统（以参数化偏微分方程离散化系统为例）的降基暖启动迭代方法。传统迭代方法通常用于获取这些线性系统的高保真解，但其计算成本显著，尤其在需要大量求解参数化系统（如对应不同参数值或时间步长）时，可能变得极具挑战性甚至完全不可行。降低此类成本的经典技术主要包括预条件等加速方法。本文提倡生成可控保真度的初始预测作为实现相同目标的替代方案。所提出的降基暖启动迭代方法利用数学上严谨且高效的降基方法生成高质量初始猜测，从而减少迭代步数。通过与零解初始化的迭代方法及基于降基预条件初始化的迭代方法进行对比，并基于两个三维稳态扩散方程的数值实验，验证了所提降基暖启动方法的有效性。