We consider the problem of approximating the solution to $A(\mu) x(\mu) = b$ for many different values of the parameter $\mu$. Here we assume $A(\mu)$ is large, sparse, and nonsingular with a nonlinear dependence on $\mu$. Our method is based on a companion linearization derived from an accurate Chebyshev interpolation of $A(\mu)$ on the interval $[-a,a]$, $a \in \mathbb{R}$. The solution to the linearization is approximated in a preconditioned BiCG setting for shifted systems, where the Krylov basis matrix is formed once. This process leads to a short-term recurrence method, where one execution of the algorithm produces the approximation to $x(\mu)$ for many different values of the parameter $\mu \in [-a,a]$ simultaneously. In particular, this work proposes one algorithm which applies a shift-and-invert preconditioner exactly as well as an algorithm which applies the preconditioner inexactly. The competitiveness of the algorithms are illustrated with large-scale problems arising from a finite element discretization of a Helmholtz equation with parameterized material coefficient. The software used in the simulations is publicly available online, and thus all our experiments are reproducible.

翻译：我们考虑求解多参数值下 $A(\mu) x(\mu) = b$ 的近似解问题。其中假设 $A(\mu)$ 是大型稀疏非奇异矩阵，且对参数 $\mu$ 具有非线性依赖性。我们的方法基于一种伴随线性化技术，该技术源于 $A(\mu)$ 在区间 $[-a,a]$（$a \in \mathbb{R}$）上的精确切比雪夫插值。该线性化问题的解在位移系统的预处理双共轭梯度（BiCG）框架下近似求解，其中Krylov基矩阵仅构建一次。该过程形成短递推方法，算法执行一次即可同时生成区间 $\mu \in [-a,a]$ 内多个参数值对应的 $x(\mu)$ 近似解。具体而言，本文提出两种算法：一种精确应用位移-求逆预处理器，另一种非精确应用该预处理器。通过参数化材料系数的亥姆霍兹方程有限元离散形成的大规模问题，验证了所提算法的竞争力。仿真所用软件已公开可获取，因此所有实验均可复现。