A residual based snapshot location strategy for POD in distributed optimal control of linear parabolic equations

In this paper we study the approximation of a distributed optimal control problem for linear para\-bolic PDEs with model order reduction based on Proper Orthogonal Decomposition (POD-MOR). POD-MOR is a Galerkin approach where the basis functions are obtained upon information contained in time snapshots of the parabolic PDE related to given input data. In the present work we show that for POD-MOR in optimal control of parabolic equations it is important to have knowledge about the controlled system at the right time instances. For the determination of the time instances (snapshot locations) we propose an a-posteriori error control concept which is based on a reformulation of the optimality system of the underlying optimal control problem as a second order in time and fourth order in space elliptic system which is approximated by a space-time finite element method. Finally, we present numerical tests to illustrate our approach and to show the effectiveness of the method in comparison to existing approaches.

[1]  K. Kunisch,et al.  Optimal snapshot location for computing POD basis functions , 2010 .

[2]  Stefan Volkwein,et al.  Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics , 2002, SIAM J. Numer. Anal..

[3]  Stefan Volkwein,et al.  Galerkin proper orthogonal decomposition methods for parabolic problems , 2001, Numerische Mathematik.

[4]  K. Afanasiev,et al.  Adaptive Control Of A Wake Flow Using Proper Orthogonal Decomposition1 , 2001 .

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

[7]  Michael Hinze,et al.  A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case , 2005, Comput. Optim. Appl..

[8]  Wei Gong,et al.  Space-time finite element approximation of parabolic optimal control problems , 2012, J. Num. Math..

[9]  Stephen J. Wright,et al.  Numerical Optimization (Springer Series in Operations Research and Financial Engineering) , 2000 .

[10]  S. Volkwein,et al.  Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control , 2013 .

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

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

[13]  L. Sirovich.  Turbulence and the dynamics of coherent structures. I. Coherent structures , 1987 .

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

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

[16]  R. H. W. HOPPE,et al.  Snapshot location by error equilibration in proper orthogonal decomposition for linear and semilinear parabolic partial differential equations , 2014, J. Num. Math..