We present a polynomial preconditioner for solving large systems of linear equations. The polynomial is derived from the minimum residual polynomial (the GMRES polynomial) and is more straightforward to compute and implement than many previous polynomial preconditioners. Our current implementation of this polynomial using its roots is naturally more stable than previous methods of computing the same polynomial. We implement further stability control using added roots, and this allows for high degree polynomials. We discuss the effectiveness and challenges of root-adding and give an additional check for stability. In this paper, we study the polynomial preconditioner applied to GMRES; however it could be used with any Krylov solver. This polynomial preconditioning algorithm can dramatically improve convergence for some problems, especially for difficult problems, and can reduce dot products by an even greater margin.
翻译:我们提出了一个解决大型线性方程式的多面性先决条件。 多面性来自最低残留多面性( GMRES 多面性), 并且比以往许多多面性先决条件者更直截了当地计算和执行。 我们目前使用其根部的多面性比以往计算同一多面性方程式的方法更加稳定。 我们使用增加的根进一步实施稳定控制, 允许高度多面性。 我们讨论根添加的效果和挑战, 并对稳定性进行额外的检查。 在本文中, 我们研究适用于 GMRES 的多面性先决条件; 但是, 它可以与任何 Krylov 解决方案一起使用。 这种多面性先决条件性算法可以极大地改善一些问题的趋同性, 特别是困难的问题, 并且可以将点性产品减少更大的空间 。