参数化线性系的无穷GMRES
Infinite GMRES for parameterized linear systems
摘要：
我们考虑线性参数依赖系统$A(μ) x(μ) = b$为许多不同的$μ$，其中$A$是大的和稀疏的，并依赖于$μ$非线性。为每个$μ$单独求解此类系统将需要大量的计算工作。在这项工作中，我们建议计算部分参数化$\tilde{x} \approx x(μ)$，其中$\tilde{x}(μ)$是便宜的计算许多不同的$μ$。我们的方法是基于这样的观察，即可以形成伴随线性化，其中对$μ$的依赖性仅是线性的。特别是，我们开发的方法相结合，完善的Krylov子空间方法的线性系，GMRES，与算法的非线性本征值问题（NEPs）生成的基础Krylov子空间。在这种新方法中，基矩阵以三种不同的方式构造，使用张量结构并利用某些问题具有低秩性质。我们显示的收敛因子界获得类似的方法GMRES线性系。更具体地，基于参数$μ$的大小和线性友矩阵的谱来获得界限，其对应于对应NEP的倒数解。数值实验说明了我们的方法对大规模问题的竞争力。这些模拟是可重复的，并可在网上公开获得。
英文摘要：
We consider linear parameter-dependent systems $A(μ) x(μ) = b$ for many different $μ$, where $A$ is large and sparse, and depends nonlinearly on $μ$. Solving such systems individually for each $μ$ would require great computational effort. In this work we propose to compute a partial parameterization $\tilde{x} \approx x(μ)$ where $\tilde{x}(μ)$ is cheap to compute for many different $μ$. Our methods are based on the observation that a companion linearization can be formed where the dependence on $μ$ is only linear. In particular, we develop methods which combine the well-established Krylov subspace method for linear systems, GMRES, with algorithms for nonlinear eigenvalue problems (NEPs) to generate a basis for the Krylov subspace. Within this new approach, the basis matrix is constructed in three different ways, using a tensor structure and exploiting that certain problems have low-rank properties. We show convergence factor bounds obtained similarly to those for the method GMRES for linear systems. More specifically, a bound is obtained based on the magnitude of the parameter $μ$ and the spectrum of the linear companion matrix, which corresponds to the reciprocal solutions to the corresponding NEP. Numerical experiments illustrate the competitiveness of our methods for large-scale problems. The simulations are reproducible and publicly available online.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
