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Offline-enhanced reduced basis method through adaptive construction of the surrogate training set. (English) Zbl 1381.65004

Summary: The reduced basis method (RBM) is a popular certified model reduction approach for solving parametrized partial differential equations. One critical stage of the offline portion of the algorithm is a greedy algorithm, requiring maximization of an error estimate over parameter space. In practice this maximization is usually performed by replacing the parameter domain continuum with a discrete “training” set. When the dimension of parameter space is large, it is necessary to significantly increase the size of this training set in order to effectively search parameter space. Large training sets diminish the attractiveness of RBM algorithms since this proportionally increases the cost of the offline phase. In this work we propose novel strategies for offline RBM algorithms that mitigate the computational difficulty of maximizing error estimates over a training set. The main idea is to identify a subset of the training set, a “surrogate training set” (STS), on which to perform greedy algorithms. The STS we construct is much smaller in size than the full training set, yet our examples suggest that it is accurate enough to induce the solution manifold of interest at the current offline RBM iteration. We propose two algorithms to construct the STS: our first algorithm, the successive maximization method, is inspired by inverse transform sampling for non-standard univariate probability distributions. The second constructs an STS by identifying pivots in the Cholesky decomposition of an approximate error correlation matrix. We demonstrate the algorithm through numerical experiments, showing that it is capable of accelerating offline RBM procedures without degrading accuracy, assuming that the solution manifold has rapidly decaying Kolmogorov width.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
35J25 Boundary value problems for second-order elliptic equations
65N15 Error bounds for boundary value problems involving PDEs
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations

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[1] Babuška, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Rev. 52(2), 317-355 (2010) · Zbl 1226.65004 · doi:10.1137/100786356
[2] Barrault, M., Maday, Y., Nguyen, N.C., Patera, A.T.: An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. Acad. Sci. Paris 339(9), 667-672 (2004) · Zbl 1061.65118 · doi:10.1016/j.crma.2004.08.006
[3] Berkooz, G., Holmes, P., Lumley, J.L.: The proper orthogonal decomposition in the analysis of turbulent flows. In Ann. Rev. Fluid Mech. 25, 539-575. Annual Reviews, Palo Alto, CA, (1993) · Zbl 0586.65040
[4] Binev, P., Cohen, A., Dahmen, W., DeVore, R., Petrova, G., Wojtaszczyk, P.: Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43(3), 1457-1472 (2011) · Zbl 1229.65193 · doi:10.1137/100795772
[5] Chen, P., Quarteroni, A.: Accurate and efficient evaluation of failure probability for partial different equations with random input data. Comput. Methods Appl. Mech. Eng. 267, 233-260 (2013) · Zbl 1286.65156 · doi:10.1016/j.cma.2013.08.016
[6] Chen, P., Quarteroni, A., Rozza, G.: A weighted reduced basis method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 51(6), 3163-3185 (2013) · Zbl 1288.65007 · doi:10.1137/130905253
[7] Chen, P., Quarteroni, A., Rozza, G.: Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations. Numer. Math. 133(1), 67-102 (2016) · Zbl 1344.93109 · doi:10.1007/s00211-015-0743-4
[8] Chen, Y.: A certified natural-norm successive constraint method for parametric inf-sup lower bounds. Appl. Numer. Math. 99, 98-108 (2016) · Zbl 1329.65269 · doi:10.1016/j.apnum.2015.09.003
[9] Chen, Y., Hesthaven, J.S., Maday, Y., Rodríguez, J.: Certified reduced basis methods and output bounds for the harmonic Maxwell’s equations. SIAM J. Sci. Comput. 32(2), 970-996 (2010) · Zbl 1213.78011 · doi:10.1137/09075250X
[10] Feldmann, P., Freund, R.W.: Efficient linear circuit analysis by padé approximation via the lanczos process. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 14(5), 639-649 (1995) · doi:10.1109/43.384428
[11] Grepl, M.A., Maday, Y., Nguyen, N.C., Patera, A.T.: Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. M2AN. Math. Model. Numer. Anal. 41(3), 575-605 (2007) · Zbl 1142.65078 · doi:10.1051/m2an:2007031
[12] Grepl, M.A., Patera, A.T.: A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM Math. Model. Numer. Anal. 39(1), 157-181 (2005) · Zbl 1079.65096 · doi:10.1051/m2an:2005006
[13] Gugercin, S., Antoulas, A.C.: A survey of model reduction by balanced truncation and some new results. Int. J. Control 77(8), 748-766 (2004) · Zbl 1061.93022 · doi:10.1080/00207170410001713448
[14] Haasdonk, B.: Reduced basis methods for parametrized pdes-a tutorial introduction for stationary and instationary problems. Reduced Order Modelling, Luminy Book Series (2014)
[15] Haasdonk, B.: RBmatlab 2016) · Zbl 1226.65004
[16] Haasdonk, B., Dihlmann, M., Ohlberger, M.: A training set and multiple bases generation approach for parameterized model reduction based on adaptive grids in parameter space. Math. Comput. Model. Dyn. Syst. 17(4), 423-442 (2011) · Zbl 1302.65221 · doi:10.1080/13873954.2011.547674
[17] Harbrecht, H., Peters, M., Schneider, R.: On the low-rank approximation by the pivoted Cholesky decomposition. Appl. Numer. Math. 62(4), 428-440 (2012) · Zbl 1244.65042 · doi:10.1016/j.apnum.2011.10.001
[18] Hesthaven, J.S., Stamm, B., Zhang, S.: Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods? ESAIM Math. Model. Numer. Anal. 48(1), 259-283 (2014) · Zbl 1292.41001 · doi:10.1051/m2an/2013100
[19] 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) · Zbl 1231.76208 · doi:10.1016/j.cma.2010.02.011
[20] 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. Acad. Sci. Paris 345(8), 473-478 (2007) · Zbl 1127.65086 · doi:10.1016/j.crma.2007.09.019
[21] Jiang, J., Chen, Y., Narayan, A.: A goal-oriented reduced basis methods-accelerated generalized polynomial Chaos algorithm. SIAM/ASA J. Uncertain. Quant. 4(1), 1398-1420 (2016) · Zbl 1352.65012 · doi:10.1137/16M1055736
[22] Lassila, T., Quarteroni, A., Rozza, G.: A reduced basis model with parametric coupling for fluid-structure interaction problems. SIAM J. Sci. Comput. 34(2), A1187-A1213 (2012) · Zbl 1390.74053 · doi:10.1137/110819950
[23] Maday, Y., Patera, A.T., Turinici, G.: Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations. C. R. Math. Acad. Sci. Paris 335(3), 289-294 (2002) · Zbl 1009.65066 · doi:10.1016/S1631-073X(02)02466-4
[24] Moore, B.C.: Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control 26(1), 17-32 (1981) · Zbl 0464.93022 · doi:10.1109/TAC.1981.1102568
[25] Noor, A.K.: Recent advances in reduction methods for nonlinear problems. Comput. Struct. 13(1-3), 31-44 (1981) · Zbl 0455.73080 · doi:10.1016/0045-7949(81)90106-1
[26] Noor, A.K., Peters, J.M.: Reduced basis technique for nonlinear analysis of structures. AIAA J. 18(4), 455-462 (1980) · doi:10.2514/3.50778
[27] Patera, A., Rozza, G.: Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations. Copyright MIT (2007) · Zbl 1304.65251
[28] Pinkus, A.: \[n\] n-widths in approximation theory, volume 7 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin (1985) · Zbl 0551.41001
[29] Porsching, T.A.: Estimation of the error in the reduced basis method solution of nonlinear equations. Math. Comput. 45(172), 487-496 (1985) · Zbl 0586.65040 · doi:10.1090/S0025-5718-1985-0804937-0
[30] Porsching, T.A., Lee, M.-Y.L.: The reduced basis method for initial value problems. SIAM J. Numer. Anal. 24(6), 1277-1287 (1987) · Zbl 0639.65039 · doi:10.1137/0724083
[31] Quarteroni, A., Rozza, G.: Numerical solution of parametrized Navier-Stokes equations by reduced basis methods. Numer. Methods Partial Differ. Equ. 23(4), 923-948 (2007) · Zbl 1178.76238 · doi:10.1002/num.20249
[32] Quarteroni, A., Rozza, G., Manzoni, A.: Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Ind. 1: Art. 3, 44 (2011) · Zbl 1273.65148
[33] 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: application to transport and continuum mechanics. Arch. Comput. Methods Eng. 15(3), 229-275 (2008) · Zbl 1304.65251 · doi:10.1007/s11831-008-9019-9
[34] Saad, Y.: Iterative methods for sparse linear systems. Society for Industrial and Applied Mathematics, 2nd edn. (2003) · Zbl 1031.65046
[35] Sen, S.: Reduced-basis approximation and a posteriori error estimation for many-parameter heat conduction problems. Numer. Heat Transf. Part B Fundam. 54(5), 369-389 (2008) · doi:10.1080/10407790802424204
[36] Urban, K., Volkwein, S., Zeeb, O.: Greedy sampling using nonlinear optimization. In: Reduced Order Methods for Modeling and Computational Reduction, pp. 137-157. Springer (2014) · Zbl 1322.65070
[37] Veroy, K., Patera, A.T.: Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds. Internat. J. Numer. Methods Fluids 47(8-9), 773-788 (2005) · Zbl 1134.76326 · doi:10.1002/fld.867
[38] Willcox, K., Peraire, J.: Balanced model reduction via the proper orthogonal decomposition. AIAA J. 40(11), 2323-2330 (2002) · doi:10.2514/2.1570
[39] Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118-1139 (2005) · Zbl 1091.65006 · doi:10.1137/040615201
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