We are interested in obtaining approximate solutions to parameterized linear systems of the form $A(\mu) x(\mu) = b$ for many values of the parameter $\mu$. Here $A(\mu)$ is large, sparse, and nonsingular, with a nonlinear analytic dependence on $\mu$. 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 $\tilde{x}(\mu)$ which is cheap to evaluate for different $\mu$, 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 $\mu$, 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.
翻译:我们有兴趣获得一些近似解决方案, 将表A(\ mu) x(\ mu) x(\ mu) 的线性系统参数化线性系统参数化为 $A (\ mu) x(\ mu) = b$, 参数值为 $\ mu$ 。 这里, $A (\ mu) 美元是大、 稀少、 非单数, 并且不单数, 以非线性分析 $ umo$为主。 我们的方法基于参数的相伴线性线性化系统。 相伴矩阵类似于无限Arnoldi 方法的操作者, 我们用它来调整灵活的 GMRES 设置。 这样, 我们的方法返回一个值为 $\ mu$ 的大小, 用于评估不同 $\ xxx(\ mu) (\ mu) (\ mu) $) 的值低廉的函数值, 来评估不同的 $\ 。 和 前提是 仅大致适用 先决条件 。 这个前提性 。 这个新方法可以提高 操作的自由性,,,,, 相对于 离 离 Q 的 。