Reduced Basis Method and Error Estimation for Parametrized Optimal Control Problems with Control Constraints

We propose a Reduced Basis method for the solution of parametrized optimal control problems with control constraints for which we extend the method proposed in Dedè, L. (SIAM J. Sci. Comput. 32:997, 2010) for the unconstrained problem. The case of a linear-quadratic optimal control problem is considered with the primal equation represented by a linear parabolic partial differential equation. The standard offline–online decomposition of the Reduced Basis method is employed with the Finite Element approximation as the “truth” one for the offline step. An error estimate is derived and an heuristic indicator is proposed to evaluate the Reduced Basis error on the optimal control problem at the online step; also, the indicator is used at the offline step in a Greedy algorithm to build the Reduced Basis space. We solve numerical tests in the two-dimensional case with applications to heat conduction and environmental optimal control problems.

[1]  Gianluigi Rozza,et al.  Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation , 2009 .

[2]  A. Quarteroni,et al.  Shape optimization for viscous flows by reduced basis methods and free‐form deformation , 2012 .

[3]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[4]  Luca Dedè,et al.  Reduced Basis method for parametrized elliptic advection-reaction problems , 2010 .

[5]  Belinda B. King,et al.  Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations , 2001 .

[6]  高等学校計算数学学報編輯委員会編,et al.  高等学校計算数学学報 = Numerical mathematics , 1979 .

[7]  S. Ravindran,et al.  A Reduced Basis Method for Control Problems Governed by PDEs , 1998 .

[8]  Gianluigi Rozza,et al.  Reduced basis method for linear elasticity problems with many parameters , 2008 .

[9]  A. Patera,et al.  A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations , 2005 .

[10]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[11]  P. Stern,et al.  Automatic choice of global shape functions in structural analysis , 1978 .

[12]  Sidney Yip,et al.  Handbook of Materials Modeling , 2005 .

[13]  Timo Tonn,et al.  Comparison of the reduced-basis and POD a posteriori error estimators for an elliptic linear-quadratic optimal control problem , 2011 .

[14]  Boris Vexler,et al.  Adaptive Space-Time Finite Element Methods for Parabolic Optimization Problems , 2007, SIAM J. Control. Optim..

[15]  Anthony T. Patera,et al.  10. Certified Rapid Solution of Partial Differential Equations for Real-Time Parameter Estimation and Optimization , 2007 .

[16]  Anthony T. Patera,et al.  Reduced basis approximation and a posteriori error estimation for stress intensity factors , 2007 .

[17]  Rolf Rannacher,et al.  Adaptive Finite Element Methods for Optimal Control of Partial Differential Equations: Basic Concept , 2000, SIAM J. Control. Optim..

[18]  Ahmed K. Noor,et al.  Reduced Basis Technique for Nonlinear Analysis of Structures , 1980 .

[19]  K. ITO,et al.  Reduced Basis Method for Optimal Control of Unsteady Viscous Flows , 2001 .

[20]  Stefan Volkwein,et al.  POD a-posteriori error estimates for linear-quadratic optimal control problems , 2009, Comput. Optim. Appl..

[21]  Nguyen Ngoc Cuong,et al.  Certified Real-Time Solution of Parametrized Partial Differential Equations , 2005 .

[22]  Roland Griesse,et al.  Parametric Sensitivity Analysis in Optimal Control of a Reaction Diffusion System. I. Solution Differentiability , 2004 .

[23]  Anthony T. Patera,et al.  "Natural norm" a posteriori error estimators for reduced basis approximations , 2006, J. Comput. Phys..

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

[25]  Stefan Volkwein,et al.  Error estimates for abstract linear–quadratic optimal control problems using proper orthogonal decomposition , 2008, Comput. Optim. Appl..

[26]  Bernard Haasdonk,et al.  Adaptive Basis Enrichment for the Reduced Basis Method Applied to Finite Volume Schemes , 2008 .

[27]  J. Hesthaven,et al.  Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations , 2007 .

[28]  M. Heinkenschloss,et al.  Real-Time PDE-Constrained Optimization , 2007 .

[29]  Karl Kunisch,et al.  Control and estimation of distributed parameter systems , 1998 .

[30]  S. Sen.  Reduced-Basis Approximation and A Posteriori Error Estimation for Many-Parameter Heat Conduction Problems , 2008 .

[31]  Karen Willcox,et al.  Model Reduction for Large-Scale Systems with High-Dimensional Parametric Input Space , 2008, SIAM J. Sci. Comput..

[32]  A. Patera,et al.  Certified real‐time solution of the parametrized steady incompressible Navier–Stokes equations: rigorous reduced‐basis a posteriori error bounds , 2005 .

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

[34]  H. Maurer.  First and second order sufficient optimality conditions in mathematical programming and optimal control , 1981 .

[35]  C. Farhat,et al.  A low‐cost, goal‐oriented ‘compact proper orthogonal decomposition’ basis for model reduction of static systems , 2011 .

[36]  G. Rozza,et al.  Parametric free-form shape design with PDE models and reduced basis method , 2010 .

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

[38]  Bernard Haasdonk,et al.  A training set and multiple bases generation approach for parameterized model reduction based on adaptive grids in parameter space , 2011 .

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

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

[41]  F. Kozin,et al.  System Modeling and Optimization , 1982 .

[42]  Jens L. Eftang,et al.  An hp certified reduced basis method for parametrized parabolic partial differential equations , 2011 .

[43]  S. Ravindran.  A reduced-order approach for optimal control of fluids using proper orthogonal decomposition , 2000 .

[44]  Anthony T. Patera,et al.  An "hp" Certified Reduced Basis Method for Parametrized Elliptic Partial Differential Equations , 2010, SIAM J. Sci. Comput..

[45]  Marco Fahl,et al.  Reduced Order Modelling Approaches to PDE-Constrained Optimization Based on Proper Orthogonal Decomposition , 2003 .

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

[47]  T. A. Porsching,et al.  Estimation of the error in the reduced basis method solution of nonlinear equations , 1985 .

[48]  Boris Vexler,et al.  Adaptive Finite Elements for Elliptic Optimization Problems with Control Constraints , 2008, SIAM J. Control. Optim..

[49]  Roland Griesse,et al.  Parametric sensitivity analysis in optimal control of a reaction-diffusion system – part II: practical methods and examples , 2004, Optim. Methods Softw..

[50]  Rolf Rannacher,et al.  An optimal control approach to a posteriori error estimation in finite element methods , 2001, Acta Numerica.

[51]  M. Giles,et al.  Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality , 2002, Acta Numerica.

[52]  M. Heinkenschloss,et al.  Large-Scale PDE-Constrained Optimization , 2003 .

[53]  Alfio Quarteroni,et al.  Optimal Control and Numerical Adaptivity for Advection-Diffusion Equations , 2005 .

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

[55]  J. Oden,et al.  Goal-oriented error estimation and adaptivity for the finite element method , 2001 .

[56]  Y. Marzouk,et al.  Large-Scale Inverse Problems and Quantification of Uncertainty , 1994 .

[57]  Gianluigi Rozza,et al.  Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Parabolic PDEs: Application to Real‐Time Bayesian Parameter Estimation , 2010 .

[58]  G. Burton.  Sobolev Spaces , 2013 .

[59]  M. Grepl.  Reduced-basis approximation a posteriori error estimation for parabolic partial differential equations , 2005 .

[60]  N. Nguyen,et al.  REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS , 2011 .

[61]  Endre Süli,et al.  Acta Numerica 2002: Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality , 2002 .

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