Efficient solution of large-scale algebraic Riccati equations associated with index-2 DAEs via the inexact low-rank Newton-ADI method
Efficient solution of large-scale algebraic Riccati equations associated with index-2 DAEs via the inexact low-rank Newton-ADI method
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DOI：
10.1016/j.apnum.2019.11.016
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发表时间：
2018-04
影响因子：
2.8
通讯作者：
P. Benner;M. Heinkenschloss;J. Saak;H. Weichelt
中科院分区：
文献类型：
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作者：
P. Benner;M. Heinkenschloss;J. Saak;H. Weichelt
本文将Benner et al.(2016)[10]的算法扩展到与Hessenberg指数-2微分代数方程（DAE）系统相关的Riccati方程。例如，这种DAE系统产生于半离散的、线性化的（围绕稳态的）Navier-Stokes方程。相关Riccati方程的解是重要的，例如，计算稳定Navier-Stokes方程的反馈律。Riccati方程数值解的挑战来自于基础系统的大规模和DAE系统中的代数约束。这些挑战是通过对DAE系统的不精确低秩牛顿- adi方法的仔细扩展来解决的。在DAE情况下的推广的一个主要成分是由代数约束描述的流形上的投影。在该算法中，方程从未显式投影，但投影仅在需要时应用。数值经验表明，控制不精确性和直线搜索的算法选择有助于避免矩阵仅为边缘稳定的子问题。该算法的性能在一个与圆柱周围Navier-Stokes流稳定化相关的大规模Riccati方程上得到了说明。
This paper extends the algorithm of Benner et al. (2016) [10] to Riccati equations associated with Hessenberg index-2 Differential Algebratic Equation (DAE) systems. Such DAE systems arise, e.g., from semi-discretized, linearized (around steady state) Navier-Stokes equations. The solution of the associated Riccati equation is important, e.g., to compute feedback laws that stabilize the Navier-Stokes equations. Challenges in the numerical solution of the Riccati equation arise from the large-scale of the underlying systems and the algebraic constraint in the DAE system. These challenges are met by a careful extension of the inexact low-rank Newton-ADI method to the case of DAE systems. A main ingredient in the extension to the DAE case is the projection onto the manifold described by the algebraic constraints. In the algorithm, the equations are never explicitly projected, but the projection is only applied as needed. Numerical experience indicates that the algorithmic choices for the control of inexactness and line-search can help avoid subproblems with matrices that are only marginally stable. The performance of the algorithm is illustrated on a large-scale Riccati equation associated with the stabilization of Navier-Stokes flow around a cylinder.