Preconditioning Navier–Stokes control using multilevel sequentially semiseparable matrix computations. Issue 2 (12th November 2020)

• Record Type: Journal Article
• Authors: Qiu, Yue; van Gijzen, Martin B.; van Wingerden, Jan‐Willem; Verhaegen, Michel; Vuik, Cornelis
• Is Part Of: Numerical linear algebra with applications. Volume 28:Issue 2(2021)
• Page Start: n/a
• Page End: n/a
• Publication Date: 2020-11-12
• Subjects: generalized saddle‐point systems -- MSSS preconditioners -- Navier–Stokes control; Algebras, Linear -- Periodicals; 512.5
• DOI: 10.1002/nla.2349
• Languages: English
• ISSNs: 1070-5325

Abstract

Abstract: In this article, we study preconditioning techniques for the control of the Navier–Stokes equation, where the control only acts on a few parts of the domain. Optimization, discretization, and linearization of the control problem results in a generalized linear saddle‐point system. The Schur complement for the generalized saddle‐point system is very difficult or even impossible to approximate, which prohibits satisfactory performance of the standard block preconditioners. We apply the multilevel sequentially semiseparable (MSSS) preconditioner to the underlying system. Compared with standard block preconditioning techniques, the MSSS preconditioner computes an approximate factorization of the global generalized saddle‐point matrix up to a prescribed accuracy in linear computational complexity. This in turn gives parameter independent convergence for MSSS preconditioned Krylov solvers. We use a simplified wind farm control example to illustrate the performance of the MSSS preconditioner. We also compare the performance of the MSSS preconditioner with the performance of the state‐of‐the‐art preconditioning techniques. Our results show the superiority of the MSSS preconditioning techniques to standard block preconditioning techniques for the control of the Navier–Stokes equation.