Model Reduction for Parametrized Optimal Control Problems in Environmental Marine Sciences and Engineering

We propose reduced order methods as a suitable approach to face parametrized optimal control problems governed by partial differential equations, with applications in en- vironmental marine sciences and engineering. Environmental parametrized optimal control problems are usually studied for different configurations described by several physical and/or geometrical parameters representing different phenomena and structures. The solution of parametrized problems requires a demanding computational effort. In order to save com- putational time, we rely on reduced basis techniques as a reliable and rapid tool to solve parametrized problems. We introduce general parametrized linear quadratic optimal control problems, and the saddle-point structure of their optimality system. Then, we propose a POD-Galerkin reduction of the optimality system. Finally, we test the resulting method on two environmental applications: a pollutant control in the Gulf of Trieste, Italy and a solution tracking governed by quasi-geostrophic equations, in its linear and nonlinear version, describing North Atlantic Ocean dynamic. The two experiments underline how reduced order methods are a reliable and convenient tool to manage several environmental optimal control problems, for different mathematical models, geographical scale as well as physical meaning.

[1]  Gianluigi Rozza,et al.  RBniCS - reduced order modelling in FEniCS , 2015 .

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

[3]  Fredi Tröltzsch,et al.  Optimal Control of the Stationary Navier--Stokes Equations with Mixed Control-State Constraints , 2007, SIAM J. Control. Optim..

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

[5]  James A. Carton,et al.  A Simple Ocean Data Assimilation Analysis of the Global Upper Ocean 1950–95. Part I: Methodology , 2000 .

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

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

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

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

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

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

[12]  Max Gunzburger,et al.  Perspectives in flow control and optimization , 1987 .

[13]  Gianluigi Rozza,et al.  Stabilized Weighted Reduced Basis Methods for Parametrized Advection Dominated Problems with Random Inputs , 2017, SIAM/ASA J. Uncertain. Quantification.

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

[15]  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.

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

[17]  Joachim Schöberl,et al.  Symmetric Indefinite Preconditioners for Saddle Point Problems with Applications to PDE-Constrained Optimization Problems , 2007, SIAM J. Matrix Anal. Appl..

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

[19]  Jiping Liu,et al.  Intensification and poleward shift of subtropical western boundary currents in a warming climate , 2016 .

[20]  S. Taasan One shot methods for optimal control of distributed parameter systems 1: Finite dimensional control , 1991 .

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

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

[23]  Raino A. E. Mäkinen,et al.  Introduction to shape optimization - theory, approximation, and computation , 2003, Advances in design and control.

[24]  Luca Dedè,et al.  Optimal flow control for Navier–Stokes equations: drag minimization , 2007 .

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

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

[27]  Christopher K. Wikle,et al.  Atmospheric Modeling, Data Assimilation, and Predictability , 2005, Technometrics.

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

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

[30]  Ming Ji,et al.  An Improved Coupled Model for ENSO Prediction and Implications for Ocean Initialization. Part I: The Ocean Data Assimilation System , 1998 .

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

[32]  K. Kunisch,et al.  Control of the Burgers Equation by a Reduced-Order Approach Using Proper Orthogonal Decomposition , 1999 .

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

[34]  J. Carton,et al.  A Reanalysis of Ocean Climate Using Simple Ocean Data Assimilation (SODA) , 2008 .

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

[36]  M. Delfour,et al.  Shapes and Geometries: Analysis, Differential Calculus, and Optimization , 1987 .

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

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

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

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

[41]  Luca Dedè Reduced Basis Method and Error Estimation for Parametrized Optimal Control Problems with Control Constraints , 2012, J. Sci. Comput..

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

[43]  Gianluigi Rozza,et al.  Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants , 2013, Numerische Mathematik.

[44]  Traian Iliescu,et al.  A stabilized proper orthogonal decomposition reduced-order model for large scale quasigeostrophic ocean circulation , 2014, Adv. Comput. Math..

[45]  Traian Iliescu,et al.  B-spline based finite-element method for the stationary quasi-geostrophic equations of the ocean , 2015 .

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

[47]  Tomás Roubícek,et al.  Optimal control of Navier-Stokes equations by Oseen approximation , 2007, Comput. Math. Appl..

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

[49]  Olivier Pironneau,et al.  Applied Shape Optimization for Fluids, Second Edition , 2009, Numerical mathematics and scientific computation.

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

[51]  Volker Schulz,et al.  One-Shot Methods for Aerodynamic Shape Optimization , 2009 .

[52]  Gianluigi Rozza,et al.  Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations , 2016, Numerische Mathematik.

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

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

[55]  Federico Negri,et al.  Reduced basis method for parametrized optimal control problems governed by PDEs , 2011 .