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On the Use of ANOVA Expansions in Reduced Basis Methods for Parametric Partial Differential Equations

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Abstract

We propose two different improvements of reduced basis (RB) methods to enable the efficient and accurate evaluation of an output functional based on the numerical solution of parametrized partial differential equations with a possibly high-dimensional parameter space. The element that combines these two techniques is that they both utilize analysis of variance (ANOVA) expansions to enable the improvements. The first method is a three-step RB–ANOVA–RB method, which aims to use a combination of RB methods and ANOVA expansions to effectively compress the parameter space with minimal impact on the accuracy of the output of interest under the assumption that only a selection of parameters are very important for the problem. This is achieved by first building a low-accuracy reduced model for the full high-dimensional parametric problem. This is used to recover an approximate ANOVA expansion for the output functional at marginal cost, allowing the estimation of the sensitivity of the output functional to parameter variation and enabling a subsequent compression of the parameter space. A new accurate reduced model can then be constructed for the compressed parametric problem at a substantially reduced computational cost as for the full problem. In the second approach we explore the ANOVA expansion to drive an hp RB method. This is initiated considering a RB as accurate as can be afforded during the online stage. If the offline greedy procedure for a given parameter domain converges with equal or less than the maximum basis functions, the offline algorithm stops. Otherwise, an approximate ANOVA expansion is performed for the output functional. The parameter domain is decomposed into several subdomains where the most important parameters according to the ANOVA expansion are split. The offline greedy algorithms are performed in these parameter subdomains. The algorithm is applied recursively until the offline greedy algorithm converges across all parameter subdomains. We demonstrate the accuracy, efficiency, and generality of these two approaches through a number of test cases.

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References

  1. 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, 1457–1472 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Buffa, A., Maday, Y., Patera, A.T., Prud’homme, C., Turinici, G.: A priori convergence of the Greedy algorithm for the parametrized reduced basis method. Math. Model. Numer. Anal. 46(03), 595–603 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bui-Thanh, T., Willcox, K., Ghattas, O.: Model reduction for large-scale systems with high-dimensional parametric input space. SIAM J. Sci. Comput. 30(6), 3270–3288 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dörfler, W.: A convergent adaptive algorithm for Poissons equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Eftang, J.L., Patera, A.T., Ronquist, E.M.: An “\(hp\)” certified reduced basis method for parametrized elliptic partial differential equations. SIAM J. Sci. Comput. 32(6), 3170–3200 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Eftang, J.L., Stamm, B.: Parameter multi-domain “hp” empirical interpolation. Int. J. Numer. Methods Eng. 90, 412–428 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fares, B., Hesthaven, J.S., Maday, Y., Stamm, B.: The reduced basis method for the electric field integral equation. J. Comput. Phys. 230(14), 5532–5555 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gao, Z., Hesthaven, J.S.: On ANOVA expansions and strategies for choosing the anchor point. Appl. Math. Comp. 217, 3274–3285 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gao, Z., Hesthaven, J.S.: Efficient solution of ordinary differential equations with high-dimensional parametrized uncertainty. Commun. Comput. Phys. 10, 253–278 (2011)

    Article  MathSciNet  Google Scholar 

  10. Gerstner, T., Griebel, M.: Numerical integration using sparse grid. Numer. Algorithm 18, 209–232 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Grepl, M.A., Maday, Y., Nguyen, N.C., Patera, A.T.: Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. Math. Model. Numer. Anal. 41, 575–605 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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. Mathe. Comput. Modell. Dynam. Syst. Methods Tools Appl. Eng. Relat. Sci. 17(4), 423 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hesthaven, J.S., Stamm, B., Zhang, S.: New greedy algorithms for the empirical interpolation and reduced basis methods: with applications to high dimensional parameter spaces. Math. Model. Numer. Anal. 48, 259–283 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Knuth, D.: The Art of Computer Programming: Fundamental Algorithms, vol. 1, 3rd edn. Addison-Wesley, Reading, Massachusetts (1997)

    MATH  Google Scholar 

  15. Lieberman, C., Willcox, K., Ghattas, O.: Parameter and state model reduction for large-scale statistical inverse problems. SIAM J. Sci. Comput. 32(5), 2523–2542 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu, M., Gao, Z., Hesthaven, J.S.: Adaptive sparse grid algorithms with applications to electromagnetic scattering under uncertainty. Appl. Numer. Math. 61, 24–37 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Maday, Y., Patera, A.T., Turinici, G.: A priori convergence theory for reduced-basis approximations of single-parametric elliptic partial differential equations. J. Sci. Comput. 17, 437–446 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Maday, Y., Stamm, B.: Locally adaptive greedy approximations for anisotropic parameter reduced basis spaces. SIAM J. Sci. Comput. 35(6), A2417–A2441 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Patera, A.T., Rozza, G.: Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations. Version 1.0, Copyright MIT, (2006) to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering

  20. Prud’homme, C., Rovas, D., Veroy, K., Maday, Y., Patera, A., Turinici, G.: Reliable real-time solution of parametrized partial differential equations: reduced-basis output bounds methods. J. Fluids Eng. 124, 70–80 (2002)

    Article  Google Scholar 

  21. Quarteroni, A., Rozza, G., Manzoni, A.: Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Indust. 1, 3 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Reduced Basis at MIT. (http://augustine.mit.edu/methodology.html)

  23. 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)

    Article  MathSciNet  MATH  Google Scholar 

  24. Smolyak, S.A.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Math. Dokl. 4, 240–243 (1963)

  25. Sobol’, I.M.: Sensitivity estimates for nonlinear mathematical models. Math. Model. Comput. Exp. 1, 407–414 (1993)

    MathSciNet  MATH  Google Scholar 

  26. Sobol’, I.M.: Theorems and examples on high dimensional model representation. Reliab. Eng. Syst. Saf. 79, 187–193 (2003)

    Article  Google Scholar 

  27. Stroud, A.: Remarks on the disposition of points in numerical integration formulas. Math. Comput. 11, 257–261 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  28. Udawalpola, R., Berggren, M.: Optimization of an acoustic horn with respect to efficiency and directivity. Internat J. Numer. Methods Eng. 73(11), 1571–1606 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. 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, Paper 2003-3847 (2003)

  30. Zhang, S.: Efficient greedy algorithms for successive constraints methods with high-dimensional parameters. Brown Division of Applied Math Scientific Computing Tech Report (2011-23)

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Acknowledgments

The authors acknowledge partial support by OSD/AFOSR FA9550-09-1-0613. The second author is also supported in part by Research Grants Council of the Hong Kong SAR, China under the GRF Grant Project No. 11303914, CityU 9042090.

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Authors and Affiliations

  1. Chair of Computational Mathematics and Simulation Science (MCSS), Mathematics Institute of Computational Science and Engineering (MATHICSE), Ecole Polytechnique Federale de Lausanne (EPFL), 1015, Lausanne, Switzerland

    Jan S. Hesthaven

  2. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    Jan S. Hesthaven

  3. Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong SAR, China

    Shun Zhang

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  1. Jan S. Hesthaven
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  2. Shun Zhang
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Correspondence to Shun Zhang.

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Hesthaven, J.S., Zhang, S. On the Use of ANOVA Expansions in Reduced Basis Methods for Parametric Partial Differential Equations. J Sci Comput 69, 292–313 (2016). https://doi.org/10.1007/s10915-016-0194-9

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  • DOI: https://doi.org/10.1007/s10915-016-0194-9

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