A Laplace transform certified reduced basis method; application to the heat equation and wave equation

Abstract We present a certified reduced basis (RB) method for the heat equation and wave equation. The critical ingredients are certified RB approximation of the Laplace transform; the inverse Laplace transform to develop the time-domain RB output approximation and rigorous error bound; a (Butterworth) filter in time to effect the necessary “modal” truncation; RB eigenfunction decomposition and contour integration for Offline–Online decomposition. We present numerical results to demonstrate the accuracy and efficiency of the approach.

[1]  Boris Lohmann,et al.  Parametric Model Order Reduction by Matrix Interpolation (Parametrische Ordnungsreduktion mittels Matrixinterpolation). , 2010 .

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

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

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

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

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

[7]  Giorgio Pini,et al.  PARALLEL FINITE ELEMENT LAPLACE TRANSFORM METHOD FOR THE NON-EQUILIBRIUM GROUNDWATER TRANSPORT EQUATION , 1997 .

[8]  Boris Lohmann,et al.  Parametric Model Order Reduction by Matrix Interpolation , 2010, Autom..

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

[10]  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.

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

[12]  E. A. Sudicky,et al.  The Laplace Transform Galerkin Technique: A time‐continuous finite element theory and application to mass transport in groundwater , 1989 .

[13]  Anthony T. Patera,et al.  Reduced Basis Method for 2nd Order Wave Equation: Application to One-Dimensional Seismic Problem , 2007 .

[14]  Ahmed Ali Mohammed,et al.  Integral transforms and their applications , 2009 .

[15]  J. Béliveau.  Introduction to finite element vibration analysis , 1992 .