A Posteriori Error Bounds for Reduced Basis Approximations of Nonaffine and Nonlinear Parabolic Partial Differential Equations

We present a posteriori error bounds for reduced basis approximations of parabolic partial differential equations involving (i) a nonaffine dependence on the parameter and (ii) a nonlinear dependence on the field variable. The method employs the Empirical Interpolation Method in order to construct “affine” coefficient-function approximations of the “nonaffine” (or nonlinear) parametrized functions. Our a posteriori error bounds take both error contributions explicitly into account — the error introduced by the reduced basis approximation and the error induced by the coefficient function interpolation. We show that these bounds are rigorous upper bounds for the approximation error under certain conditions on the function interpolation, thus addressing the demand for certainty of the approximation. As regards efficiency, we develop an efficient offline-online computational procedure for the calculation of the reduced basis approximation and associated error bound. The method is thus ideally suited for the many-query or real-time contexts. We also introduce a new sampling approach to generate the collateral reduced basis space for functions with a nonlinear dependence on the field variable. Numerical results are presented to confirm and test our approach.

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

[2]  Anthony T. Patera,et al.  A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations : “convex inverse” bound conditioners , 2002 .

[3]  Z. Bai.  Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems , 2002 .

[4]  Alfio Quarteroni,et al.  Numerical Mathematics (Texts in Applied Mathematics) , 2006 .

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

[6]  Ahmed K. Noor,et al.  Reduced Basis Technique for Nonlinear Analysis of Structures , 1979 .

[7]  Werner C. Rheinboldt,et al.  On the theory and error estimation of the reduced basis method for multi-parameter problems , 1993 .

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

[9]  Marcus Meyer,et al.  Efficient model reduction in non-linear dynamics using the Karhunen-Loève expansion and dual-weighted-residual methods , 2003 .

[10]  N. Nguyen,et al.  A general multipurpose interpolation procedure: the magic points , 2008 .

[11]  Claudio Canuto,et al.  A Posteriori Error Analysis of the Reduced Basis Method for Nonaffine Parametrized Nonlinear PDEs , 2009, SIAM J. Numer. Anal..

[12]  G. Reddien,et al.  On the reduced basis method , 1995 .

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

[14]  Ngoc Cuong Nguyen,et al.  A posteriori error estimation and basis adaptivity for reduced-basis approximation of nonaffine-parametrized linear elliptic partial differential equations , 2007, J. Comput. Phys..

[15]  K. Gallivan,et al.  Reduced-order modeling of multiscreen frequency-selective surfaces using Krylov-based rational interpolation , 2001 .

[16]  J. Marsden,et al.  A subspace approach to balanced truncation for model reduction of nonlinear control systems , 2002 .

[17]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

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

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

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

[21]  Werner C. Rheinboldt,et al.  On the Error Behavior of the Reduced Basis Technique for Nonlinear Finite Element Approximations , 1983 .

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

[23]  Sung-Mo Kang,et al.  Model-order reduction of nonlinear MEMS devices through arclength-based Karhunen-Loeve decomposition , 2001, ISCAS 2001. The 2001 IEEE International Symposium on Circuits and Systems (Cat. No.01CH37196).

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

[25]  Yong P. Chen,et al.  A Quadratic Method for Nonlinear Model Order Reduction , 2000 .

[26]  Jens Nørkær Sørensen,et al.  Evaluation of Proper Orthogonal Decomposition-Based Decomposition Techniques Applied to Parameter-Dependent Nonturbulent Flows , 1999, SIAM J. Sci. Comput..

[27]  S. Ravindran,et al.  A Reduced-Order Method for Simulation and Control of Fluid Flows , 1998 .

[28]  N. Nguyen,et al.  EFFICIENT REDUCED-BASIS TREATMENT OF NONAFFINE AND NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS , 2007 .

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

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

[31]  D. Rovas,et al.  Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems , 2000 .

[32]  J. M. A. Scherpen,et al.  Balancing for nonlinear systems , 1993 .

[33]  Etienne Balmes,et al.  PARAMETRIC FAMILIES OF REDUCED FINITE ELEMENT MODELS. THEORY AND APPLICATIONS , 1996 .

[34]  Anthony T. Patera,et al.  A Posteriori Error Bounds for the Empirical Interpolation Method , 2010 .

[35]  N. Nguyen,et al.  An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations , 2004 .

[36]  Anthony T. Patera,et al.  A Certified Reduced Basis Method for the Fokker--Planck Equation of Dilute Polymeric Fluids: FENE Dumbbells in Extensional Flow , 2010, SIAM J. Sci. Comput..

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

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

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

[40]  Jacob K. White,et al.  A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices , 2001, IEEE/ACM International Conference on Computer Aided Design. ICCAD 2001. IEEE/ACM Digest of Technical Papers (Cat. No.01CH37281).

[41]  D. Rovas,et al.  A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations , 2003 .