Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples

We propose a numerical analysis of proper orthogonal decomposition (POD) model reductions in which a priori error estimates are expressed in terms of the projection errors that are controlled in the construction of POD bases. These error estimates are derived for generic parabolic evolution PDEs, including with non-linear Lipschitz right-hand sides, and for wave-like equations. A specific projection continuity norm appears in the estimates and – whereas a general uniform continuity bound seems out of reach – we prove that such a bound holds in a variety of Galerkin bases choices. Furthermore, we directly numerically assess this bound – and the effectiveness of the POD approach altogether – for test problems of the type considered in the numerical analysis, and also for more complex equations. Namely, the numerical assessment includes a parabolic equation with super-linear reaction terms, inspired from the FitzHugh-Nagumo electrophysiology model, and a 3D biomechanical heart model. This shows that the effectiveness established for the simpler models is also achieved in the reduced-order simulation of these highly complex systems.

[1]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[2]  T. M. Flett.  Differential Analysis: Appendix , 1980 .

[3]  J. M. Thomas,et al.  Introduction à l'analyse numérique des équations aux dérivées partielles , 1983 .

[4]  C. DeWitt-Morette,et al.  Mathematical Analysis and Numerical Methods for Science and Technology , 1990 .

[5]  K. Bathe.  Finite Element Procedures , 1995 .

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

[7]  B. Feeny,et al.  On the physical interpretation of proper orthogonal modes in vibrations , 1998 .

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

[9]  Anthony T. Patera,et al.  A Priori Convergence Theory for Reduced-Basis Approximations of Single-Parameter Elliptic Partial Differential Equations , 2002, J. Sci. Comput..

[10]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

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

[12]  J. Peraire,et al.  Balanced Model Reduction via the Proper Orthogonal Decomposition , 2002 .

[13]  Anthony T. Patera,et al.  A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations , 2002 .

[14]  A. Patera,et al.  Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds , 2003 .

[15]  Jacob K. White,et al.  A multiparameter moment-matching model-reduction approach for generating geometrically parameterized interconnect performance models , 2004, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[16]  A. Antoulas,et al.  A Survey of Model Reduction by Balanced Truncation and Some New Results , 2004 .

[17]  M. Hinze,et al.  Proper Orthogonal Decomposition Surrogate Models for Nonlinear Dynamical Systems: Error Estimates and Suboptimal Control , 2005 .

[18]  Tatjana Stykel,et al.  Balanced truncation model reduction for semidiscretized Stokes equation , 2006 .

[19]  Dimitrios V. Rovas,et al.  Reduced-basis output bound methods for parabolic problems , 2006 .

[20]  D. Chapelle,et al.  MODELING AND ESTIMATION OF THE CARDIAC ELECTROMECHANICAL ACTIVITY , 2006 .

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

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

[23]  Stefan Volkwein,et al.  Galerkin proper orthogonal decomposition methods for parameter dependent elliptic systems , 2007 .

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

[25]  C. Farhat,et al.  Interpolation Method for Adapting Reduced-Order Models and Application to Aeroelasticity , 2008 .

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

[27]  D. Chapelle,et al.  Validation of a biomechanical heart model using animal data with acute myocardial infarction , 2009 .

[28]  Alessandro Astolfi,et al.  Model Reduction by Moment Matching for Linear and Nonlinear Systems , 2010, IEEE Transactions on Automatic Control.