AI 中文总结
研究参数化线性系统求解问题，利用紧凑有理Krylov（CORK）框架，先线性化系统，制定左右预处理有理Krylov GMRES方法，展示其加速作用及相关应用，如通过边界元法对亥姆霍兹散射问题进行频率扫描。
AI 中文摘要
在参数化线性系统$\mathsf{P}(\mu)\mathsf{x}=\mathsf{b}$中，系统矩阵$\mathsf{P}$非线性依赖于参数$\mu$，需为该参数的多个值求解。本文表明，最初用于解决非线性特征值问题的紧凑有理Krylov（CORK）框架，可用于一次为该参数的多个值高效生成此类系统的近似解。该方法先将参数化系统线性化，得到大型移位线性系统$(\boldsymbol{\mathsf{A}}-\mu\boldsymbol{\mathsf{B}})\boldsymbol{\mathsf{y}}=\boldsymbol{\mathsf{d}}$。本文为移位线性系统制定了左右预处理有理Krylov GMRES方法，并展示了CORK框架如何在参数化线性系统中加速这些方法。此外，还展示了如何纳入也依赖于参数的右侧$\mathsf{b}(\mu)$，如何选择移位以控制收敛，以及如何在整个迭代过程中允许在这些移位处进行不精确求解。作为应用，通过边界元法（BEM）考虑亥姆霍兹散射问题的“频率扫描”，这通过密集但数据稀疏的波数相关系统矩阵的有效表示得以实现。
英文摘要
In parametrized linear systems $\mathsf{P}(μ)\mathsf{x}=\mathsf{b}$ the system matrix $\mathsf{P}$ depends nonlinearly on a parameter $μ$ and solutions are sought for many values of this parameter. We show that the compact rational Krylov (CORK) framework, originally introduced to solve nonlinear eigenvalue problems, can be used to efficiently produce approximate solutions to such a system for many values of the parameter at once. In this approach, the parametrized system is first linearized, resulting in a large shifted linear system $(\boldsymbol{\mathsf{A}}-μ\boldsymbol{\mathsf{B}})\boldsymbol{\mathsf{y}}=\boldsymbol{\mathsf{d}}$. We formulate a left- and right-preconditioned rational Krylov GMRES method for shifted linear systems. In the setting of parametrized linear systems, these can exploit the structure in the linearization, and in combination with the CORK framework, computational and memory complexity mainly depend on the problem size, less the degree of the linearization. Additionally, we show how to incorporate a right-hand side $\mathsf{b}(μ)$ that also depends on the parameter, how to choose the shifts to steer convergence and how to allow for inexact solves at these shifts throughout the iterations. As an application we consider the 'frequency sweeping' of Helmholtz scattering problems through the Boundary Element Method (BEM), enabled via an efficient representation of the dense but data-sparse wavenumber-dependent system matrix.
