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Calibri 83ffff̙̙3f3fff3f3f33333f33333.Y>TU Delft Repositoryg r2uuidrepository linktitleauthorcontributorpublication yearabstract
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departmentresearch group programmeprojectcoordinates)uuid:af5ff77f8bf541068ed8bf33ead02a26Dhttp://resolver.tudelft.nl/uuid:af5ff77f8bf541068ed8bf33ead02a26Preconditioning optimal indomain control of navierstokes equation using multilevel sequentially semiseparable matrix computationsKQiu, Y.; Van Gijzen, M.B.; Van Wingerden, J.W.; Verhaegen, M.H.G.; Vuik, C.In this manuscript, we study preconditioning techniques for optimal indomain control of the NavierStokes equation, where the control only acts on a few parts of the domain. Optimization and linearization of the optimal indomain control problem results in a generalized linear saddlepoint system. The Schur complement for the generalized saddlepoint system is very difficult or even impossible to approximate. This 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 saddlepoint 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 simple wind farm control example to illustrate the performance of the MSSS preconditioner. We also compare with the performance of the stateoftheart preconditioning techniques. Our results show the superiority of the MSSS preconditioning techniques to standard block preconditioning techniques for optimal indomain control of the NavierStokes equation.dindomain flow control; MSSS preconditioners; generalized saddlepoint system; NavierStokes controlenreportEWI Dept. Applied Mathematics8Electrical Engineering, Mathematics and Computer ScienceApplied Mathematics)uuid:9cd6cf6f004b4393b2729704d5f34754Dhttp://resolver.tudelft.nl/uuid:9cd6cf6f004b4393b2729704d5f34754mPreconditioning Optimal Flow Control Problems Using Multilevel Sequentially Semiseparable Matrix ComputationsQiu, Y.JVuik, C. (promotor); Van Gijzen, M.B. (promotor); Verhaegen, M. (promotor)Optimal flow control problems are important for applications in science and engineering. Solving such problems usually requires the solution of a large linear generalized saddlepoint system. This linear system is sparse and highly indefinite. In order to solve such systems using Krylov subspace methods, efficient preconditioners are necessary to enhance their robustness and accelerate the convergence. Standard block preconditioning techniques for the generalized saddlepoint systems require an efficient approximation of the Schur complement. This is a big challenge since the Schur complement is large and dense, and therefore computationally expensive to approximate. For some problems, it is even impossible to approximate the Schur complement efficiently. In this dissertation, we propose a new class of preconditioners for optimal flow control problems using multilevel sequentially semiseparable (MSSS) matrix computations. In contrast to standard block preconditioners, MSSS preconditioners do not approximate the Schur complement of the generalized saddlepoint system but compute an approximate factorization of the global block system in linear computational complexity. This is a big advantage over block preconditioners. The key to this global factorization is that the Schur complements in this factorization usually have low offdiagonal rank. Therefore, these Schur complements can be approximated by matrices with a low rank offdiagonal structure. For this, MSSS matrix computations are very well suited. Theoretical analysis shows that MSSS preconditioners yield a spectrum of the preconditioned system matrix that is contained in a circle centered at $(1, 0)$. This radiu<bs can be controlled arbitrarily by properly choosing a parameter in the MSSS preconditioner computations. This in turn implies that the convergence of MSSS preconditioned systems can be independent of the mesh size and regularization parameter for PDEconstrained optimization problems while for computational fluid dynamics problems, the convergence is independent of the mesh size and Reynolds number. Mesh size independent and wave number independent convergence can be also obtained when applying the MSSS preconditioning technique to the Helmholtz problem. Numerical results verify the convergence property. In this dissertation, we also studied the problem of optimal indomain control of the NavierStokes equation. We use a simplified wind farm control example to formulate such a problem. Compared with standard PDEconstrained optimization problems where the controls are distributed throughout the whole domain, this indomain control problem is even more difficult to solve since the control only acts on a few parts of the domain. This in turn gives a linear system of the generalized saddlepoint type. Block preconditioners cannot give satisfactory performance for such problem because the Schur complement for such system is very difficult or even impossible to approximate efficiently. Applying MSSS preconditioners to this problem gives superior performance compared to block preconditioning techniques.preconditioning,; Krylov methods,; multilevel sequentially semiseparable matrices,; convergence analysis,; generalized saddlepoint systems,; wind farm control,doctoral thesis&Delft Institute of Applied Mathematics)uuid:0705c999d9d14c3c92ae45b958c883a7Dhttp://resolver.tudelft.nl/uuid:0705c999d9d14c3c92ae45b958c883a7Preconditioning Optimal InDomain Control of NavierStokes Equation Using Multilevel Sequentially Semiseparable Matrix ComputationsGQiu, Y.; Van Gijzen, M.B.; Van Wingerden, J.W.; Verhaegen, M.; Vuik, C.In this manuscript, we study preconditioning techniques for optimal indomain control of the NavierStokes equation, where the control only acts on a few parts of the domain. Optimization and linearization of the optimal indomain control problem results in a generalized linear saddlepoint system. The Schur complement for the generalized saddlepoint system is very difficult or even impossible to approximate. This 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 saddlepoint 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 simple wind farm control example to illustrate the performance of the MSSS preconditioner. We also ompare with the performance of the stateoftheart preconditioning techniques. Our results show the superiority of the MSSS preconditioning techniques to standard block preconditioning techniques for optimal indomain control of the NavierStokes equation.Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft Institute of Applied Mathematics
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