We consider approximating solutions to parameterized linear systems of the form $A(\mu_1,\mu_2) x(\mu_1,\mu_2) = b$, where $(\mu_1, \mu_2) \in \mathbb{R}^2$. Here the matrix $A(\mu_1,\mu_2) \in \mathbb{R}^{n \times n}$ is nonsingular, large, and sparse and depends nonlinearly on the parameters $\mu_1$ and $\mu_2$. Specifically, the system arises from a discretization of a partial differential equation and $x(\mu_1,\mu_2) \in \mathbb{R}^n$, $b \in \mathbb{R}^n$. This work combines companion linearization with the Krylov subspace method preconditioned bi-conjugate gradient (BiCG) and a decomposition of a tensor matrix of precomputed solutions, called snapshots. As a result, a reduced order model of $x(\mu_1,\mu_2)$ is constructed, and this model can be evaluated in a cheap way for many values of the parameters. Tensor decompositions performed on a set of snapshots can fail to reach a certain level of accuracy, and it is not known a priori if a decomposition will be successful. Moreover, the selection of snapshots can affect both the quality of the produced model and the computation time required for its construction. This new method offers a way to generate a new set of solutions on the same parameter space at little additional cost. An interpolation of the model is used to produce approximations on the entire parameter space, and this method can be used to solve a parameter estimation problem. Numerical examples of a parameterized Helmholtz equation show the competitiveness of our approach. The simulations are reproducible, and the software is available online.

翻译：本文考虑求解形如$A(\mu_1,\mu_2) x(\mu_1,\mu_2) = b$的参数化线性系统的近似解，其中$(\mu_1, \mu_2) \in \mathbb{R}^2$。此处矩阵$A(\mu_1,\mu_2) \in \mathbb{R}^{n \times n}$非奇异、大规模且稀疏，并关于参数$\mu_1$和$\mu_2$非线性依赖。具体而言，该系统源自偏微分方程的离散化，且$x(\mu_1,\mu_2) \in \mathbb{R}^n$，$b \in \mathbb{R}^n$。本文将伴随线性化与Krylov子空间方法——预处理双共轭梯度法(BiCG)以及预计算解的张量矩阵分解（称为快照）相结合。由此构造出$x(\mu_1,\mu_2)$的降阶模型，该模型可在参数取多值时以较低代价进行评估。对一组快照进行张量分解可能无法达到特定精度，且分解能否成功在事前无法预知。此外，快照的选择会影响所生成模型的质量和构造所需的计算时间。新方法提供了一种在相同参数空间中生成新解集的途径，且额外成本较低。通过模型插值可在整个参数空间上生成近似解，该方法可用于求解参数估计问题。针对参数化亥姆霍兹方程的数值算例展示了本方法的竞争力。该模拟结果可复现，相关软件已在线上公开。