论文标题
参数化线性系统的预处理无限GMRE
Preconditioned infinite GMRES for parameterized linear systems
论文作者
论文摘要
我们有兴趣获得$ a（μ）x（μ）x（μ）= b $的参数化线性系统的近似解决方案。这里$ a（μ）$很大，稀疏且非语言，对$μ$的非线性分析依赖性。我们的方法基于用于参数化线性系统的伴随线性化。伴随矩阵与无限Arnoldi方法中的操作员相似，我们使用它来调整柔性GMRES设置。通过这种方式，我们的方法返回一个函数$ \ tilde {x}（μ）$，该$廉价用于不同$μ$，并且仅应用大约应用预处理。这种新颖的方法导致增加了不推翻操作的行动的自由，从而在无限制的方法上提供了性能改善，而通常不会损失准确性。我们表明，我们的方法的误差是根据参数$μ$的大小，预处理的不确定性和线性伴随矩阵的频谱估算的。具有参数化材料系数的Helmholtz方程的有限元离散化的数值示例说明了我们方法的竞争力。模拟是可重现的，可以在线公开。
We are interested in obtaining approximate solutions to parameterized linear systems of the form $A(μ) x(μ) = b$ for many values of the parameter $μ$. Here $A(μ)$ is large, sparse, and nonsingular, with a nonlinear analytic dependence on $μ$. Our approach is based on a companion linearization for parameterized linear systems. The companion matrix is similar to the operator in the infinite Arnoldi method, and we use this to adapt the flexible GMRES setting. In this way, our method returns a function $\tilde{x}(μ)$ which is cheap to evaluate for different $μ$, and the preconditioner is applied only approximately. This novel approach leads to increased freedom to carry out the action of the operation inexactly, which provides performance improvement over the method infinite GMRES, without a loss of accuracy in general. We show that the error of our method is estimated based on the magnitude of the parameter $μ$, the inexactness of the preconditioning, and the spectrum of the linear companion matrix. Numerical examples from a finite element discretization of a Helmholtz equation with a parameterized material coefficient illustrate the competitiveness of our approach. The simulations are reproducible and publicly available online.
