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A weighted POD method for elliptic PDEs with random inputs. (English) Zbl 1462.65201
Summary: In this work we propose and analyze a weighted proper orthogonal decomposition method to solve elliptic partial differential equations depending on random input data, for stochastic problems that can be transformed into parametric systems. The algorithm is introduced alongside the weighted greedy method. Our proposed method aims to minimize the error in a $$L^2$$ norm and, in contrast to the weighted greedy approach, it does not require the availability of an error bound. Moreover, we consider sparse discretization of the input space in the construction of the reduced model; for high-dimensional problems, provided the sampling is done accordingly to the parameters distribution, this enables a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We provide many numerical tests to assess the performance of the proposed method compared to an equivalent reduced order model without weighting, as well as to the weighted greedy approach, in both low and high dimensional problems.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
##### Software:
RBniCS; SyFi; FEniCS
Full Text:
##### References:
 [1] Bäck, J.; Nobile, F.; Tamellini, L.; Tempone, R.; Hesthaven, JS (ed.); Rønquist, EM (ed.), Stochastic spectral galerkin and collocation methods for pdes with random coefficients: a numerical comparison, 43-62 (2011), Berlin · Zbl 1216.65004 [2] Ballarin, F., Sartori, A., Rozza, G.: RBniCS—reduced order modelling in FEniCS. http://mathlab.sissa.it/rbnics (2015) [3] Barrault, M., Maday, Y., Nguyen, N., Patera, A.: An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Mathematique 339(9), 667-672 (2004) · Zbl 1061.65118 [4] Bathelmann, V., Novak, E., Ritter, K.: High dimensional polynomial interpolation on sparse grids. Adv. Comput. Math. 4(12), 273-288 (2000) · Zbl 0944.41001 [5] Benner, P., Cohen, A., Ohlberger, M., Willcox, K.: Model Reduction and Approximation: Theory and Algorithms, vol. 15. SIAM (2017) · Zbl 1378.65010 [6] Benner, P., Gugercin, S., Willcox, K.: A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57(4), 483-531 (2015) · Zbl 1339.37089 [7] Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K.: Model Reduction of Parametrized Systems. Springer, Berlin (2017) · Zbl 1381.65001 [8] Bistrian, D., Susan-Resiga, R.: Weighted proper orthogonal decomposition of the swirling flow exiting the hydraulic turbine runner. Appl. Math. Model. 40(5-6), 4057-4078 (2016) · Zbl 1459.76077 [9] Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2010) · Zbl 1220.46002 [10] Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737-2764 (2010) · Zbl 1217.65169 [11] Chen, P.: Model order reduction techniques for uncertainty quantification problems. Ph.D. thesis, École polytechnique fédérale de Lausanne EPFL (2014) [12] 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 [13] Chen, P., Quarteroni, A., Rozza, G.: Comparison between reduced basis and stochastic collocation methods for elliptic problems. J. Sci. Comput. 1(59), 187-216 (2014) · Zbl 1301.65007 [14] Chen, P., Quarteroni, A., Rozza, G.: A weighted empirical interpolation method: a priori convergence analysis and applications. SESAIM. Math. Model. Numer. Anal. 4(48), 943-953 (2014) · Zbl 1304.65097 [15] Chen, P., Quarteroni, A., Rozza, G.: Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by stokes equations. Numerische Mathematik 1(133), 67-102 (2016) · Zbl 1344.93109 [16] Chen, P., Quarteroni, A., Rozza, G.: Reduced basis methods for uncertainty quantification. SIAM/ASA J. Uncertain. Quantif. 5(1), 813-869 (2017) · Zbl 1400.65010 [17] Cliffe, K.A., Giles, M.B., Scheichl, R., Teckentrup, A.L.: Multilevel monte carlo methods and applications to elliptic pdes with random coefficients. Comput. Visual. Sci. 14(1), 3 (2011) · Zbl 1241.65012 [18] Dahmen, W.: How to best sample a solution manifold? In: G.E. Pfander (ed.) Sampling Theory, a Renaissance: Compressive Sensing and Other Developments, pp. 403-435. Springer, Berlin (2015) · Zbl 1336.65188 [19] Fan, Z., Liu, E., Xu, B.: Weighted principal component analysis. In: International Conference on Artificial Intelligence and Computational Intelligence, pp. 569-574. Springer, Berlin (2011) [20] Gabriel, K.R., Zamir, S.: Lower rank approximation of matrices by least squares with any choice of weights. Technometrics 21(4), 489-498 (1979) · Zbl 0471.62004 [21] Gerstner, T., Griebel, M.: Numerical integration using sparse grids. Numer. Algorithms 3-4(18), 209-232 (1998) · Zbl 0921.65022 [22] Haber, M., Gabriel, K.: Weighted Least Squares Approximation of Matrices and Its Application to Canonical Correlations and Biplot Display. Tech. rep. Department of Statistics, University of Rochester, Rochester (1976) [23] Hesthaven, J., Rozza, G., Stamm, B.: Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer, Berlin (2016) · Zbl 1329.65203 [24] Holtz, M.: Sparse Grid Quadrature in High Dimensions with Applications in Finance and Insurance. Springer, Belrin (2010) · Zbl 1221.65080 [25] Lang, J., Scheichl, R.: Adaptive multilevel stochastic collocation method for randomized elliptic PDEs. Preprint 2718 (2017) [26] Loève, M.: Probability Theory. Springer, Berlin (1978) · Zbl 0385.60001 [27] Logg, A., Mardal, K.A., Wells, G.: Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, vol. 84. Springer, Berlin (2012) · Zbl 1247.65105 [28] Nobile, F., Tempone, R., Webster, C.G.: A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 5(46), 2309-2345 (2008) · Zbl 1176.65137 [29] Novak, E., Ritter, K.: High dimensional integration of smooth functions over cubes. Numerische Mathematik 1(75), 79-97 (1996) · Zbl 0883.65016 [30] Oksendal, B.: Stochastic Differential Equations. An Introduction with Applications. Springer, Berlin (1998) · Zbl 0897.60056 [31] Owen, A.B.: Monte Carlo theory, methods and examples (2013). http://statweb.stanford.edu/ owen/mc/ [32] Peherstorfer, B., Cui, T., Marzouk, Y., Willcox, K.: Multifidelity importance sampling. Comput. Methods Appl. Mech. Eng. 300, 490-509 (2016) · Zbl 1426.62087 [33] Quarteroni, A., Rozza, G., Manzoni, A.: Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math Ind. 1(1), 3 (2011) · Zbl 1273.65148 [34] Spannring, C.: Weighted reduced basis methods for parabolic PDEs with random input data. Ph.D. thesis, Graduate School CE, Technische Universität Darmstadt (2018) · Zbl 1423.35005 [35] Spannring, C., Ullmann, S., Lang, J.: A weighted reduced basis method for parabolic PDEs with random data. arXiv preprint arXiv:1712.07393 (2017) [36] Sullivan, T.: Introduction to Uncertainty Quantification. Springer, Berlin (2015) · Zbl 1336.60002 [37] Tamuz, O., Mazeh, T., Zucker, S.: Correcting systematic effects in a large set of photometric light curves. Mon. Notices R. Astron. Soc. 356(4), 1466-1470 (2005) [38] Teckentrup, A.L., Jantsch, P., Webster, C.G., Gunzburger, M.: A multilevel stochastic collocation method for partial differential equations with random input data. SIAM/ASA J. Uncertain. Quantif. 3(1), 1046-1074 (2015) · Zbl 1327.65014 [39] Torlo, D., Ballarin, F., Rozza, G.: Stabilized weighted reduced basis methods for parametrized advection dominated problems with random inputs. arXiv preprint arXiv:1711.11275 (2017) · Zbl 1408.35021 [40] Wasilkowski, G.W.: Explicit cost bounds of algorithms for multivariate tensor product problems. J. Complex. 1(11), 1-56 (1995) · Zbl 0819.65082 [41] Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 3(27), 1118-1139 (2006) · Zbl 1091.65006 [42] Xiu, D., Karniadakis, G.E.: The Wiener-Askey polynomial chaos for stochastic differential equations. S. J. Sci. Comput. 24(2), 619-644 (2002) · Zbl 1014.65004
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