In this work, a subspace method is proposed for efficient solution of parametric linear systems with a symmetric and positive definite coefficient matrix of the form I−A(σ). The motivation is to use the method for solution of linear systems appearing when solving parameter dependent elliptic PDEs using the finite element method (FEM). In the proposed method, one first computes a method subspace and then uses it to approximately solve the linear system for any parameter vector. The method subspace is designed in such a way that it contains the j+1-term truncated Neumann series approximation of the solution to desired accuracy for any admissible parameter vector. This allows us to use the best approximation property of subspace methods to show that the subspace solution is at least as accurate as the truncated Neumann series approximation. The performance of the method is demonstrated by numerical examples with the parametric diffusion equation. In these examples, the method yields much smaller errors than anticipated by the Neumann series based error analysis. We study this phenomenon in some special cases.

中文翻译：

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一种基于 Neumann 级数的子空间方法，用于参数线性系统的求解


在这项工作中，提出了一种子空间方法，用于有效求解具有 I-A（σ） 形式的对称和正定系数矩阵的参数线性系统。其动机是使用有限元法 （FEM） 求解参数相关椭圆偏微分方程时出现的线性方程组的方法。在所提出的方法中，首先计算一个方法子空间，然后使用它来近似求解任何参数向量的线性系统。方法子空间的设计方式使其包含解的 j+1 项截断诺伊曼级数近似，对于任何可接受的参数向量，它都可以达到所需的精度。这使我们能够使用子空间方法的最佳近似特性来证明子空间解至少与截断的诺伊曼级数近似一样准确。该方法的性能通过参数扩散方程的数值示例来证明。在这些示例中，该方法产生的误差比基于 Neumann 级数的误差分析预期的要小得多。我们在一些特殊情况下研究这种现象。