Abstract
In this paper we introduce a two-step Certified Reduced Basis (RB) method. In the first step we construct from an expensive finite element “truth” discretization of dimension \({\mathcal{N}} \) an intermediate RB model of dimension \(N\ll {\mathcal{N}}\). In the second step we construct from this intermediate RB model a derived RB (DRB) model of dimension M≤N. The construction of the DRB model is effected at cost \({\mathcal{O}}(N)\) and in particular at cost independent of \({\mathcal{N}}\); subsequent evaluation of the DRB model may then be effected at cost \({\mathcal{O}}(M)\). The DRB model comprises both the DRB output and a rigorous a posteriori error bound for the error in the DRB output with respect to the truth discretization.
The new approach is of particular interest in two contexts: focus calculations and hp-RB approximations. In the former the new approach serves to reduce online cost, M≪N: the DRB model is restricted to a slice or subregion of a larger parameter domain associated with the intermediate RB model. In the latter the new approach enlarges the class of problems amenable to hp-RB treatment by a significant reduction in offline (precomputation) cost: in the development of the hp parameter domain partition and associated “local” (now derived) RB models the finite element truth is replaced by the intermediate RB model. We present numerical results to illustrate the new approach.
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Almroth, B.O., Stern, P., Brogan, F.A.: Automatic choice of global shape functions in structural analysis. AIAA J. 16, 525–528 (1978)
Barrault, M., Nguyen, N.C., Maday, Y., Patera, A.T.: An “empirical interpolation” method: application to efficient reduced-basis discretization of partial differential equations. C. R. Acad. Sci. Paris, Sér. I 339, 667–672 (2004)
Boyaval, S.: Reduced-basis approach for homogenization beyond the periodic setting. Multiscale Model. Simul. 7(1), 466–494 (2008)
Boyaval, S., Le Bris, C., Maday, Y., Nguyen, N.C., Patera, A.T.: A reduced basis approach for variational problems with stochastic parameters: application to heat conduction with variable robin coefficient. Comput. Methods Appl. Mech. Eng. 198(41–44), 3187–3206 (2009)
Canuto, C., Tonn, T., Urban, K.: A posteriori error analysis of the reduced basis method for nonaffine parametrized nonlinear PDEs. SIAM J. Numer. Anal. 47(3), 2001–2022 (2009)
Eftang, J.L., Knezevic, D.J., Patera, A.T.: An hp certified reduced basis method for parametrized parabolic partial differential equations. Math. Comput. Model. Dyn. Syst. (2011, accepted)
Eftang, J.L., Patera, A.T., Rønquist, E.M.: An “hp” certified reduced basis method for parametrized elliptic partial differential equations. SIAM J. Sci. Comput. 32(6), 3170–3200 (2010)
Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences, 3rd edn. Johns Hopkins University Press, Baltimore (1996)
Grepl, M.A., Patera, A.T.: A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. Modél. Math. Anal. Numér. 39(1), 157–181 (2005)
Grepl, M.A., Maday, Y., Nguyen, N.C., Patera, A.T.: Efficient reduced basis treatment of nonaffine and nonlinear partial differential equations. ESAIM, Math. Model. Numer. Anal. 41(3), 575–605 (2007)
Haasdonk, B., Dihlmann, M., Ohlberger, M.: A training set and multiple bases generation approach for parametrized model reduction based on adaptive Grids in parameter space. Technical report 28, SRC SimTech (2010)
Haasdonk, B., Ohlberger, M.: Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM, Math. Model. Numer. Anal. 42(2), 277–302 (2008)
Haasdonk, B., Ohlberger, M.: Efficient reduced models and a posteriori error estimation for parametrized dynamical systems by offline/online decomposition. Math. Comput. Model. Dyn. Syst. (2011). doi:10.1080/13873954.2010.514703
Huynh, D.B.P., Knezevic, D.J., Patera, A.T.: Certified reduced basis model characterization: a frequentistic uncertainty framework. Comput. Methods Appl. Mech. Eng. (submitted January 2011). http://augustine.mit.edu/methodology/papers/atp_CMAME_preprint_Jan2011.pdf
Huynh, D.B.P., Knezevic, D.J., Peterson, J.W., Patera, A.T.: High-fidelity real-time simulation on deployed platforms. Comput. Fluids (2010). doi:10.1016/j.compfluid.2010.07.007
Huynh, D.B.P., Knezevic, D.J., Chen, Y., Hesthaven, J.S., Patera, A.T.: A natural-norm successive constraint method for inf-sup lower bounds. Comput. Methods Appl. Mech. Eng. 199(29–32), 1963–1975 (2010)
Huynh, D.B.P., Rozza, G., Sen, S., Patera, A.T.: A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C. R. Math. 345(8), 473–478 (2007)
Kamon, M., Wang, F., White, J.: Generating nearly optimally compact models from Krylov-subspace based reduced-order models. IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process. 47(4), 239–248 (2000)
Kirk, B.S., Peterson, J.W., Stogner, R.H., Carey, G.F.: libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations. Eng. Comput. 22(3–4), 237–254 (2006)
Knezevic, D.J., Patera, A.T.: A certified reduced basis method for the Fokker-Planck equation of dilute polymeric fluids: Fene dumbbells in extensional flow. SIAM J. Sci. Comput. 32(2), 793–817 (2010)
Knezevic, D.J., Peterson, J.W.: A high-performance parallel implementation of the certified reduced basis method. Comput. Methods Appl. Mech. Eng. (2010, accepted), http://augustine.mit.edu/methodology/papers/djk_CMAME_preprint_December2010.pdf
Moore, B.: Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control 26(1), 17–32 (1981)
Nguyen, N.C., Rozza, G., Huynh, D.B.P., Patera, A.T.: Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Parabolic PDEs: Application to Real-Time Bayesian Parameter Estimation. Wiley, New York (2010), pp. 151–177
Noor, A.K., Peters, J.M.: Reduced basis technique for nonlinear analysis of structures. AIAA J. 18, 455–462 (1980)
Porsching, T.A.: Estimation of the error in the reduced basis method solution of nonlinear equations. Math. Comput. 45, 487–496 (1985)
Rozza, G., Huynh, D.B.P., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15(3), 229–275 (2008)
Udawalpola, R., Berggren, M.: Optimization of an acoustic horn with respect to efficiency and directivity. Int. J. Numer. Methods Eng. 73, 1571–1606 (2008)
Veroy, K., Prud’homme, C., Rovas, D.V., Patera, A.T.: A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In: Proceedings of the 16th AIAA Computational Fluid Dynamics Conference. AIAA Paper, vol. 2003-3847. (2003)
Willcox, K., Megretski, A.: Fourier series for accurate, stable, reduced-order models in large-scale linear applications. SIAM J. Sci. Comput. 26(3), 944–962 (2005) (electronic)
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Eftang, J.L., Huynh, D.B.P., Knezevic, D.J. et al. A Two-Step Certified Reduced Basis Method. J Sci Comput 51, 28–58 (2012). https://doi.org/10.1007/s10915-011-9494-2
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DOI: https://doi.org/10.1007/s10915-011-9494-2