New POD Error Expressions, Error Bounds, and Asymptotic Results for Reduced Order Models of Parabolic PDEs

The derivations of existing error bounds for reduced order models of time varying partial differential equations (PDEs) constructed using proper orthogonal decomposition (POD) have relied on bounding the error between the POD data and various POD projections of that data. Furthermore, the asymptotic behavior of the model reduction error bounds depends on the asymptotic behavior of the POD data approximation error bounds. We consider time varying data taking values in two different Hilbert spaces $ H $ and $ V $, with $ V \subset H $, and prove exact expressions for the POD data approximation errors considering four different POD projections and the two different Hilbert space error norms. Furthermore, the exact error expressions can be computed using only the POD eigenvalues and modes, and we prove the errors converge to zero as the number of POD modes increases. We consider the POD error estimation approaches of Kunisch and Volkwein [SIAM J. Numer. Anal., 40 (2002), pp. 492--515] and Chapelle, Gariah, an...

[1]  J. G. Heywood.  An error estimate uniform in time for spectral Galerkin approximations of the Navier-Stokes problem. , 1982 .

[2]  C. Fletcher.  Computational techniques for fluid dynamics , 1992 .

[3]  Zhu Wang,et al.  Are the Snapshot Difference Quotients Needed in the Proper Orthogonal Decomposition? , 2013, SIAM J. Sci. Comput..

[4]  Stefan Volkwein.  Optimal Control of a Phase‐Field Model Using Proper Orthogonal Decomposition , 2001 .

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

[6]  Ping Sun,et al.  Finite element formulation based on proper orthogonal decomposition for parabolic equations , 2009 .

[7]  Yanjie Zhou,et al.  A reduced finite difference scheme based on singular value decomposition and proper orthogonal decomposition for Burgers equation , 2009 .

[8]  Zhenghui Xie,et al.  A reduced finite volume element formulation and numerical simulations based on POD for parabolic problems , 2011, J. Comput. Appl. Math..

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

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

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

[12]  Stefan Volkwein,et al.  Control of laser surface hardening by a reduced-order approach using proper orthogonal decomposition , 2003 .

[13]  C.A.J. Fletcher,et al.  The group finite element formulation , 1983 .

[14]  Zhu Wang,et al.  Artificial viscosity proper orthogonal decomposition , 2011, Math. Comput. Model..

[15]  J. Aubin,et al.  APPLIED FUNCTIONAL ANALYSIS , 1981, The Mathematical Gazette.

[16]  R. Rannacher,et al.  Finite element approximation of the nonstationary Navier-Stokes problem. I : Regularity of solutions and second-order error estimates for spatial discretization , 1982 .

[17]  A. R. Mitchell,et al.  Product Approximation for Non-linear Problems in the Finite Element Method , 1981 .

[18]  Pedro Galán del Sastre,et al.  Error estimates of proper orthogonal decomposition eigenvectors and Galerkin projection for a general dynamical system arising in fluid models , 2008, Numerische Mathematik.

[19]  L. Sirovich.  TURBULENCE AND THE DYNAMICS OF COHERENT STRUCTURES PART I : COHERENT STRUCTURES , 2016 .

[20]  Rolf Rannacher,et al.  On the finite element approximation of the nonstationary Navier-Stokes problem , 1980 .

[21]  Sivaguru S. Ravindran,et al.  Error analysis for Galerkin POD approximation of the nonstationary Boussinesq equations , 2011 .

[22]  Rolf Rannacher,et al.  Finite element approximation of the nonstationary Navier-Stokes problem, part II: Stability of solutions and error estimates uniform in time , 1986 .

[23]  Jing Chen,et al.  Mixed Finite Element Formulation and Error Estimates Based on Proper Orthogonal Decomposition for the Nonstationary Navier-Stokes Equations , 2008, SIAM J. Numer. Anal..

[24]  Zhu Wang,et al.  Variational multiscale proper orthogonal decomposition: Convection-dominated convection-diffusion-reaction equations , 2013, Math. Comput..

[25]  Matthew F. Barone,et al.  On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far‐field boundary treatment , 2010 .

[26]  Danny C. Sorensen,et al.  A State Space Error Estimate for POD-DEIM Nonlinear Model Reduction , 2012, SIAM J. Numer. Anal..

[27]  William Layton,et al.  Introduction to the Numerical Analysis of Incompressible Viscous Flows , 2008 .

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

[29]  Linda R. Petzold,et al.  Error Estimation for Reduced-Order Models of Dynamical Systems , 2005, SIAM J. Numer. Anal..

[30]  Richard E. Mortensen,et al.  Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam) , 1991, SIAM Rev..

[31]  D. Chapelle,et al.  Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples , 2012 .