POD-Galerkin model order reduction for parametrized nonlinear time-dependent optimal flow control: an application to shallow water equations

Abstract In the present paper we propose reduced order methods as a reliable strategy to efficiently solve parametrized optimal control problems governed by shallow waters equations in a solution tracking setting. The physical parametrized model we deal with is nonlinear and time dependent: this leads to very time consuming simulations which can be unbearable, e.g., in a marine environmental monitoring plan application. Our aim is to show how reduced order modelling could help in studying different configurations and phenomena in a fast way. After building the optimality system, we rely on a POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space. The presented theoretical framework is actually suited to general nonlinear time dependent optimal control problems. The proposed methodology is finally tested with a numerical experiment: the reduced optimal control problem governed by shallow waters equations reproduces the desired velocity and height profiles faster than the standard model, still remaining accurate.

[1]  I. Babuska.  Error-bounds for finite element method , 1971 .

[2]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[3]  Claes Johnson,et al.  Error estimates and automatic time step control for nonlinear parabolic problems, I , 1987 .

[4]  W. Thacker,et al.  An Optimal-Control/Adjoint-Equations Approach to Studying the Oceanic General Circulation , 1989 .

[5]  M. Ghil,et al.  Data assimilation in meteorology and oceanography , 1991 .

[6]  K. Hoffmann,et al.  Optimal Control of Partial Differential Equations , 1991 .

[7]  Alfio Quarteroni,et al.  Recent developments in the numerical simulation of shallow water equations I: boundary conditions , 1992 .

[8]  A. Quarteroni,et al.  Mathematical and numerical modelling of shallow water flow , 1993 .

[9]  C. Vreugdenhil.  Numerical methods for shallow-water flow , 1994 .

[10]  A. Quarteroni,et al.  RECENT DEVELOPMENTS IN THE NUMERICAL SIMULATION OF SHALLOW WATER EQUATIONS II: TEMPORAL DISCRETIZATION , 1994 .

[11]  G. Tallini,et al.  ON THE EXISTENCE OF , 1996 .

[12]  S. Ravindran,et al.  A Reduced-Order Method for Simulation and Control of Fluid Flows , 1998 .

[13]  Alfio Quarteroni,et al.  Finite element approximation of Quasi-3D shallow water equations , 1999 .

[14]  Joseph E. Pasciak,et al.  Uzawa type algorithms for nonsymmetric saddle point problems , 2000, Math. Comput..

[15]  D. Rovas,et al.  Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods , 2002 .

[16]  Eugenia Kalnay,et al.  Atmospheric Modeling, Data Assimilation and Predictability , 2002 .

[17]  E. Zuazua,et al.  Control theory: history, mathematical achievements and perspectives , 2003 .

[18]  N. Nguyen,et al.  An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations , 2004 .

[19]  F. Saleri,et al.  A new two-dimensional shallow water model including pressure effects and slow varying bottom topography , 2004 .

[20]  Annalisa Quaini,et al.  Numerical Approximation of a Control Problem for Advection-Diffusion Processes , 2005, System Modelling and Optimization.

[21]  E. Miglio,et al.  Model coupling techniques for free-surface flow problems: Part II , 2005 .

[22]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.

[23]  R. Mosetti,et al.  Innovative strategies for marine protected areas monitoring: the experience of the Istituto Nazionale di Oceanografia e di Geofisica Sperimentale in the Natural Marine Reserve of Miramare, Trieste - Italy , 2005, Proceedings of OCEANS 2005 MTS/IEEE.

[24]  Max Gunzburger,et al.  POD and CVT-based reduced-order modeling of Navier-Stokes flows , 2006 .

[25]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[26]  G. Rozza,et al.  On the stability of the reduced basis method for Stokes equations in parametrized domains , 2007 .

[27]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[28]  Fausto Saleri,et al.  An optimal control approach for 1D-2D shallow water equations coupling , 2007 .

[29]  Annalisa Quaini,et al.  Reduced basis methods for optimal control of advection-diffusion problems ∗ , 2007 .

[30]  Rémi Abgrall,et al.  Application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes , 2007, J. Comput. Phys..

[31]  E. Miglio,et al.  Geometric multiscale approach by optimal control for shallow water equations. , 2007 .

[32]  M. Hinze,et al.  A Hierarchical Space-Time Solver for Distributed Control of the Stokes Equation , 2008 .

[33]  Stefan Volkwein,et al.  Proper orthogonal decomposition for optimality systems , 2008 .

[34]  Luca Dedè,et al.  Adaptive and Reduced Basis methods for optimal control problems in environmental applications , 2008 .

[35]  A. Malej,et al.  Native and non-native ctenophores in the Gulf of Trieste, Northern Adriatic Sea , 2008 .

[36]  Pavel B. Bochev,et al.  Least-Squares Finite Element Methods , 2009, Applied mathematical sciences.

[37]  Stefan Ulbrich,et al.  Optimization with PDE Constraints , 2008, Mathematical modelling.

[38]  Mario Ricchiuto,et al.  Stabilized residual distribution for shallow water simulations , 2009, J. Comput. Phys..

[39]  Tayfun E. Tezduyar,et al.  Space–time SUPG formulation of the shallow‐water equations , 2010 .

[40]  A. Wathen,et al.  All-at-Once Solution if Time-Dependent PDE-Constrained Optimisation Problems , 2010 .

[41]  Luca Dedè,et al.  Reduced Basis Method and A Posteriori Error Estimation for Parametrized Linear-Quadratic Optimal Control Problems , 2010, SIAM J. Sci. Comput..

[42]  F. Tröltzsch.  Optimal Control of Partial Differential Equations: Theory, Methods and Applications , 2010 .

[43]  Gianluigi Rozza,et al.  Reduction strategies for PDE-constrained oprimization problems in Haemodynamics , 2013 .

[44]  Anders Logg,et al.  Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book , 2012 .

[45]  Stefan Turek,et al.  A Space-Time Multigrid Method for Optimal Flow Control , 2012, Constrained Optimization and Optimal Control for Partial Differential Equations.

[46]  Karen Veroy,et al.  Certified Reduced Basis Methods for Parametrized Saddle Point Problems , 2012, SIAM J. Sci. Comput..

[47]  Fabio Cavallini,et al.  Quasi-Geostrophic Theory of Oceans and Atmosphere: Topics in the Dynamics and Thermodynamics of the Fluid Earth , 2012 .

[48]  Karsten Urban,et al.  A new error bound for reduced basis approximation of parabolic partial differential equations , 2012 .

[49]  Martin Stoll,et al.  All-at-once solution of time-dependent Stokes control , 2013, J. Comput. Phys..

[50]  Dominique Chapelle,et al.  A Galerkin strategy with Proper Orthogonal Decomposition for parameter-dependent problems – Analysis, assessments and applications to parameter estimation , 2013 .

[51]  Gianluigi Rozza,et al.  Reduced Basis Method for Parametrized Elliptic Optimal Control Problems , 2013, SIAM J. Sci. Comput..

[52]  M. Fortin,et al.  Mixed Finite Element Methods and Applications , 2013 .

[53]  A. Quarteroni,et al.  A reduced computational and geometrical framework for inverse problems in hemodynamics , 2013, International journal for numerical methods in biomedical engineering.

[54]  Razvan Stefanescu,et al.  POD/DEIM nonlinear model order reduction of an ADI implicit shallow water equations model , 2012, J. Comput. Phys..

[55]  Andreas Griewank,et al.  Trends in PDE Constrained Optimization , 2014 .

[56]  Karsten Urban,et al.  A space-time hp-interpolation-based certified reduced basis method for Burgers' equation , 2014 .

[57]  Stefan Volkwein,et al.  Model Order Reduction for PDE Constrained Optimization , 2014 .

[58]  Masayuki Yano,et al.  A Space-Time Petrov-Galerkin Certified Reduced Basis Method: Application to the Boussinesq Equations , 2014, SIAM J. Sci. Comput..

[59]  Adrian Sandu,et al.  Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations , 2014, International Journal for Numerical Methods in Fluids.

[60]  Mark Kärcher,et al.  A POSTERIORI ERROR ESTIMATION FOR REDUCED ORDER SOLUTIONS OF PARAMETRIZED PARABOLIC OPTIMAL CONTROL PROBLEMS , 2014 .

[61]  Mark Kärcher,et al.  A certified reduced basis method for parametrized elliptic optimal control problems , 2014 .

[62]  Karl Kunisch,et al.  Uniform convergence of the POD method and applications to optimal control , 2015 .

[63]  Gianluigi Rozza,et al.  Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations , 2015 .

[64]  Eduard Bader,et al.  A Certified Reduced Basis Approach for Parametrized Linear–Quadratic Optimal Control Problems with Control Constraints (two-sided) , 2015 .

[65]  Stefan Volkwein,et al.  Reduced-Order Multiobjective Optimal Control of Semilinear Parabolic Problems , 2016, ENUMATH.

[66]  Gianluigi Rozza,et al.  Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations , 2015, Comput. Math. Appl..

[67]  J. Hesthaven,et al.  Certified Reduced Basis Methods for Parametrized Partial Differential Equations , 2015 .

[68]  Karen Veroy,et al.  Certified Reduced Basis Methods for Parametrized Distributed Elliptic Optimal Control Problems with Control Constraints , 2016, SIAM J. Sci. Comput..

[69]  Eduardo Casas,et al.  Optimal Control of Partial Differential Equations , 2017 .

[70]  Karen Veroy,et al.  Certified Reduced Basis Methods for Parametrized Elliptic Optimal Control Problems with Distributed Controls , 2017, Journal of Scientific Computing.

[71]  Gianluigi Rozza,et al.  Model Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering , 2017, SIAM J. Sci. Comput..

[72]  Alfio Quarteroni,et al.  MOX-Report No . 13 / 2019 A saddle point approach to an optimal boundary control problem for steady Navier-Stokes equations , 2019 .

[73]  Gianluigi Rozza,et al.  Reduced order methods for parametrized non-linear and time dependent optimal flow control problems, towards applications in biomedical and environmental sciences , 2019, ENUMATH.

[74]  A. Quarteroni,et al.  A saddle point approach to an optimal boundary control problem for steady Navier-Stokes equations , 2019, Mathematics in Engineering.

[75]  Sebastian Grimberg,et al.  On the stability of projection-based model order reduction for convection-dominated laminar and turbulent flows , 2020, J. Comput. Phys..

[76]  Gianluigi Rozza,et al.  Reduced order methods for parametric optimal flow control in coronary bypass grafts, toward patient‐specific data assimilation , 2019, International journal for numerical methods in biomedical engineering.

[77]  G. Rozza,et al.  POD–Galerkin Model Order Reduction for Parametrized Time Dependent Linear Quadratic Optimal Control Problems in Saddle Point Formulation , 2019, J. Sci. Comput..

[78]  Gianluigi Rozza,et al.  A weighted POD-reduction approach for parametrized PDE-constrained Optimal Control Problems with random inputs and applications to environmental sciences , 2021, Comput. Math. Appl..

[79]  Lei Zhang,et al.  Space-time registration-based model reduction of parameterized one-dimensional hyperbolic PDEs , 2020, ESAIM: Mathematical Modelling and Numerical Analysis.

[80]  Atmospheric Modeling , 2022, General Aviation Aircraft Design.