论文标题
用于参数化线性系统的预处理Chebyshev BICG
Preconditioned Chebyshev BiCG for parameterized linear systems
论文作者
论文摘要
对于许多不同值的参数$μ$，我们考虑将解决方案近似于$ a（μ）x（μ）= b $的问题。在这里，我们假设$ a（μ）$很大，稀疏且非单词，非线性依赖于$μ$。我们的方法是基于从$ [ - a，a] $，$ a \ in \ mathbb {r} $上的准确的chebyshev插值得出的伴侣线性化。线性化的解决方案在移位系统的预处理的BICG设置中近似，其中Krylov基矩阵一次形成一次。此过程导致了一种短期复发方法，其中一种算法的执行会产生近似值为$ x（μ）$，对于[-a，a] $的许多不同值的$ x（μ）$。特别是，这项工作提出了一种算法，该算法准确地应用了换档预处理，以及一种不确定地应用预处理的算法。说明了算法的竞争力，该算法是由具有参数化材料系数的Helmholtz方程的有限元离散化引起的大规模问题。模拟中使用的软件在线公开可用，因此我们所有的实验都是可重现的。
We consider the problem of approximating the solution to $A(μ) x(μ) = b$ for many different values of the parameter $μ$. Here we assume $A(μ)$ is large, sparse, and nonsingular with a nonlinear dependence on $μ$. Our method is based on a companion linearization derived from an accurate Chebyshev interpolation of $A(μ)$ 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 $x(μ)$ for many different values of the parameter $μ\in [-a,a]$ 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.
