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：我们考虑对多个不同参数 $\mu$ 值逼近求解 $A(\mu) x(\mu) = b$ 的问题。假设 $A(\mu)$ 是大规模、稀疏且非奇异的矩阵，其对参数 $\mu$ 的依赖是非线性的。本方法基于从区间 $[-a,a]$（$a \in \mathbb{R}$）上 $A(\mu)$ 的精确切比雪夫插值导出的伴随线性化。在移位系统的预处理双共轭梯度（BiCG）框架下逼近线性化的解，其中Krylov基矩阵仅需构建一次。该过程产生一种短递推方法，算法单次执行即可同时为多个不同的 $\mu \in [-a,a]$ 值生成 $x(\mu)$ 的近似解。具体而言，本研究提出两种算法：一种精确应用移位-求逆预处理子，另一种非精确应用该预处理子。通过参数化材料系数的亥姆霍兹方程有限元离散产生的大规模问题，验证了所提算法的竞争力。仿真所用软件已公开获取，因此所有实验均可复现。