Comparison of the reduced-basis and POD a posteriori error estimators for an elliptic linear-quadratic optimal control problem

In this article, a linear-quadratic optimal control problem governed by the Helmholtz equation is considered. For the computation of suboptimal solutions, two different model reduction techniques are compared: the reduced-basis method and proper orthogonal decomposition. By an a posteriori error estimator for the optimal control problem, the accuracy of the suboptimal solutions is ensured. The efficiency of both model reduction approaches is illustrated by a numerical example for the stationary Helmholtz equation.

[1]  Stefan Volkwein,et al.  Admittance Identification from Point-wise Sound Pressure Measurements Using Reduced-order Modelling , 2010, J. Optim. Theory Appl..

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

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

[4]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems , 2005, Advances in Design and Control.

[5]  Stefan Volkwein,et al.  Impedance Identification by POD Model Reduction Techniques (Impedanz-Identifikation mittels POD Modellreduktion) , 2008, Autom..

[6]  M. Reed.  Methods of Modern Mathematical Physics. I: Functional Analysis , 1972 .

[7]  B. Haasdonk,et al.  REDUCED BASIS METHOD FOR FINITE VOLUME APPROXIMATIONS OF PARAMETRIZED LINEAR EVOLUTION EQUATIONS , 2008 .

[8]  Athanasios C. Antoulas,et al.  Approximation of Large-Scale Dynamical Systems (Advances in Design and Control) (Advances in Design and Control) , 2005 .

[9]  P. Holmes,et al.  Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 1996 .

[10]  A. Patera,et al.  A Successive Constraint Linear Optimization Method for Lower Bounds of Parametric Coercivity and Inf-Sup Stability Constants , 2007 .

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

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

[13]  Pablo Gamallo,et al.  Finite Element Methods in Local Active Control of Sound , 2004, SIAM J. Control. Optim..

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

[15]  Stefan Volkwein,et al.  POD A-POSTERIORI ERROR BASED INEXACT SQP METHOD FOR BILINEAR ELLIPTIC OPTIMAL CONTROL PROBLEMS , 2010 .

[16]  Enrique S. Quintana-Ortí,et al.  Model Reduction Based on Spectral Projection Methods , 2005 .