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Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM

  • Received: 01 August 2020 Revised: 01 September 2020 Published: 14 December 2020
  • Primary: 76D55, 49M25, 49M41, 68T07; Secondary: 65M60

  • An efficient computing method for a target velocity tracking problem of fluid flows is considered. We first adopts the Lagrange multipliers method to obtain the optimality system, and then designs a simple and effective feedback control law based on the relationship between the control $ {{\boldsymbol f}} $ and the adjoint variable $ {{\boldsymbol w}} $ in the optimality system. We consider a reduced order modeling (ROM) of this problem for real-time computing. In order to improve the existing ROM method, the deep learning technique, which is currently being actively researched, is applied. We review previous research results and some computational results are presented.

    Citation: Hyung-Chun Lee. Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM[J]. Electronic Research Archive, 2021, 29(3): 2533-2552. doi: 10.3934/era.2020128

    Related Papers:

  • An efficient computing method for a target velocity tracking problem of fluid flows is considered. We first adopts the Lagrange multipliers method to obtain the optimality system, and then designs a simple and effective feedback control law based on the relationship between the control $ {{\boldsymbol f}} $ and the adjoint variable $ {{\boldsymbol w}} $ in the optimality system. We consider a reduced order modeling (ROM) of this problem for real-time computing. In order to improve the existing ROM method, the deep learning technique, which is currently being actively researched, is applied. We review previous research results and some computational results are presented.



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    [1] On some control problems in fluid mechanics. Theoret. Comput. Fluid Dynamics (1990) 1: 303-325.
    [2] (1975) Sobolev Spaces.Academic Press.
    [3] S. E. Ahmed, O. San, A. Rasheed and T. Iliescu, A long short-term memory embedding for hybrid uplifted reduced order models, Phys. D, 409 (2020), 132471, 16 pp. doi: 10.1016/j.physd.2020.132471
    [4] D. Amsallem, Interpolation on Manifolds of CFD-Based Fluid and Finite Element-Based Structural Reduced-Order Models for On-Line Aeroelastic Predictions, Ph. D. Thesis, Stanford University, 2010.
    [5] A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. (2015) 57: 483-531.
    [6] The proper orthogonal decomposition in the analysis of turbulent flows. Annual review of fluid mechanics (1993) 25: 539-575.
    [7] Machine Learning for Fluid Mechanics. Annu. Rev. Fluid Mech. (2020) 52: 477-508.
    [8] Centroidal voronoi tessellation-based reduced-order modeling of complex systems. SIAM J. Sci. Comput. (2006) 28: 459-484.
    [9] POD and CVT-based reduced-order modeling of Navier-Stokes flows. Comput. Methods Appl. Mech. Engrg. (2006) 196: 337-355.
    [10] Theory-guided data science for climate change. Computer (2014) 47: 74-78.
    [11] Active control of vortex shedding. J. Fluids Struct. (1989) S3: 115-122.
    [12] V. Girault and P. Raviart, Navier-Stokes Equations, North-Hollan, Amsterdam, 1979.
    [13] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5
    [14] G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins University, Baltimore, 1996.
    [15] Finite-dimensional approximation of a class of constrained nonlinear optimal control problems. SIAM J. Control Optim. (1996) 34: 1001-1043.
    [16] Active control of vortex shedding. J. Appl. Mech. (1996) 63: 828-835.
    [17] Analysis and approximation of the velocity tracking problem for Navier-Stokes flows with distributed control. SIAM J. Numer. Anal. (2000) 37: 1481-1512.
    [18] The velocity tracking problem for Navier-Stokes flows with boundary control. SIAM J. Control Optim. (2000) 39: 594-634.
    [19] Analysis and approximation for linear feedback control for tracking the velocity in Navier-Stokes flows. Comput. Methods Appl. Mech. Engrg. (2000) 189: 803-823.
    [20] New development in FreeFem++. J. Numer. Math. (2012) 20: 251-265.
    [21] Dynamics for controlled Navier-Stokes systems with distributed controls. SIAM J. Control Optim. (1997) 35: 654-677.
    [22] Dynamics and approximations of a velocity tracking problem for the Navier-Stokes flows with piecewise distributed controls. SIAM J. Control Optim. (1997) 35: 1847-1885.
    [23] Theory-guided data science: A new paradigm for scientific discovery from data. IEEE Trans. Knowl. Data Eng. (2017) 29: 2318-2331.
    [24] Strategies for reduced-order models for predicting the statistical responses and uncertainty quantification in complex turbulent dynamical systems. SIAM Rev. (2018) 60: 491-549.
    [25] S. Pawar, S. Ahmed, O. San and A. Rasheed, An evolve-then-correct reduced order model for hidden fluid dynamics. Mathematics, Mathematics, 8 (2020), 570.
    [26] S. Pawar, S. E. Ahmed, O. San and A. Rasheed, Data-driven recovery of hidden physics in reduced order modeling of fluid flows, preprint, arXiv: 1910.13909 doi: 10.1063/5.0002051
    [27] M. Rahman, S. Pawar, O. San, A. Rasheed and T. Iliescu, A non-intrusive reduced order modeling framework for quasi-geostrophic turbulence, preprint, arXiv: 1906.11617
    [28] A new look at proper orthogonal decomposition. SIAM J. Numer. Anal. (2003) 41: 1893-1925.
    [29] L. Scarpa, Analysis and optimal velocity control of a stochastic convective Cahn-Hilliard equation, preprint, arXiv: 2007.14735
    [30] Turbulence and the dynamics of coherent structures, part ⅰ: Coherent structures; part ⅱ: symmetries and transformations; part ⅲ: Dynamics and scaling. Quart. Appl. Math. (1987) 45: 561-590.
    [31] M. Strazzullo, Z. Zainib, F. Ballarin and G. Rozza, Reduced order methods for parametrized non-linear and time dependent optimal flow control problems, towards applications in biomedical and environmental sciences, preprint, arXiv: 1912.07886
    [32] Kailai Xu, Bella Shi and Shuyi Yin, Deep Learning for Partial Differential Equations, Stanford University, 2018.
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