Generalized Reduced Basis Methods and n-Width Estimates for the Approximation of the Solution Manifold of Parametric PDEs

The set of solutions of a parameter-dependent linear partial differential equation with smooth coefficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold.

[1]  S. D. Chatterji.  Proceedings of the International Congress of Mathematicians , 1995 .

[2]  R. DeVore,et al.  Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs , 2010 .

[3]  Charles A. Micchelli,et al.  SPLINE SPACES ARE OPTIMAL FOR L 2 n-WIDTH , 1978 .

[4]  Gianluigi Rozza,et al.  Certified reduced basis approximation for parametrized partial differential equations and applications , 2011 .

[5]  A. Noor.  Recent advances in reduction methods for nonlinear problems. [in structural mechanics , 1981 .

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

[7]  G. Rozza,et al.  ON THE APPROXIMATION OF STABILITY FACTORS FOR GENERAL PARAMETRIZED PARTIAL DIFFERENTIAL EQUATIONS WITH A TWO-LEVEL AFFINE DECOMPOSITION , 2012 .

[8]  Jens Markus Melenk,et al.  On n-widths for elliptic problems , 2000 .

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

[10]  Yvon Maday,et al.  Reduced basis method for the rapid and reliable solution of partial differential equations , 2006 .

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

[12]  Wolfgang Dahmen,et al.  Convergence Rates for Greedy Algorithms in Reduced Basis Methods , 2010, SIAM J. Math. Anal..

[13]  A. Hay,et al.  Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition , 2009, Journal of Fluid Mechanics.

[14]  Dominique Pelletier,et al.  Reduced-order models for parameter dependent geometries based on shape sensitivity analysis , 2010, J. Comput. Phys..

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

[16]  Jens L. Eftang,et al.  An hp certified reduced basis method for parametrized parabolic partial differential equations , 2011 .

[17]  Timo Tonn,et al.  Comparison of the reduced-basis and POD a posteriori error estimators for an elliptic linear-quadratic optimal control problem , 2011 .

[18]  Anthony T. Patera,et al.  Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations , 2002 .

[19]  Anthony T. Patera,et al.  A natural-norm Successive Constraint Method for inf-sup lower bounds , 2010 .

[20]  Anthony T. Patera,et al.  An "hp" Certified Reduced Basis Method for Parametrized Elliptic Partial Differential Equations , 2010, SIAM J. Sci. Comput..

[21]  George Adomian,et al.  Solving Frontier Problems of Physics: The Decomposition Method , 1993 .

[22]  W. A. Light.  n -WIDTHS IN APPROXIMATION THEORY (Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, Band 7) , 1985 .

[23]  Simone Deparis,et al.  Reduced Basis Error Bound Computation of Parameter-Dependent Navier-Stokes Equations by the Natural Norm Approach , 2008, SIAM J. Numer. Anal..

[24]  A. Patera,et al.  A PRIORI CONVERGENCE OF THE GREEDY ALGORITHM FOR THE PARAMETRIZED REDUCED BASIS METHOD , 2012 .

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

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

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

[28]  C. Farhat,et al.  A low‐cost, goal‐oriented ‘compact proper orthogonal decomposition’ basis for model reduction of static systems , 2011 .

[29]  Thomas J. R. Hughes,et al.  n-Widths, sup–infs, and optimality ratios for the k-version of the isogeometric finite element method , 2009 .

[30]  S. Fischler.  Formes linéaires en polyzêtas et intégrales multiples , 2002 .

[31]  Z. Pammer,et al.  The p–version of the finite–element method , 2014 .

[32]  Janet S. Peterson,et al.  The Reduced Basis Method for Incompressible Viscous Flow Calculations , 1989 .

[33]  R. DeVore,et al.  ANALYTIC REGULARITY AND POLYNOMIAL APPROXIMATION OF PARAMETRIC AND STOCHASTIC ELLIPTIC PDE'S , 2011 .

[34]  A. Pinkus.  n-Widths in Approximation Theory , 1985 .