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. The decomposition is performed efficiently using the sparse grid based higher-order proper generalized decomposition (HOPGD), and the snapshots are generated as one variable functions of $\mu_1$ or of $\mu_2$. Tensor decompositions performed on a set of snapshots can fail to reach a certain level of accuracy, and it is not possible to know a priori if the decomposition will be successful. This 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：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 $(\mu_1, \mu_2) \in \mathbb{R}^2$。这里矩阵 $A(\mu_1,\mu_2) \in \mathbb{R}^{n \times n}$ 为大规模、稀疏且非奇异，其元素非线性地依赖于参数 $\mu_1$ 和 $\mu_2$。具体而言，该系统源于偏微分方程的离散化，其中 $x(\mu_1,\mu_2) \in \mathbb{R}^n$，$b \in \mathbb{R}^n$。本文将伴随线性化方法与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。由此构建出 $x(\mu_1,\mu_2)$ 的降阶模型，该模型可对大量参数值进行低成本评估。分解过程采用基于稀疏网格的高阶本征正交分解（HOPGD）高效完成，而快照则生成为 $\mu_1$ 或 $\mu_2$ 的单变量函数。对快照集进行张量分解可能无法达到特定精度要求，且无法预先判断分解能否成功。本方法能够在几乎不增加额外计算成本的情况下，在同参数空间上生成一组新解。通过模型插值可在整个参数空间上获得近似解，该方法还可用于求解参数估计问题。参数化亥姆霍兹方程的数值算例验证了本方法的竞争力。所有仿真结果均可复现，相关软件已公开提供。