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An Adaptive ANOVA Stochastic Galerkin Method for Partial Differential Equations with High-dimensional Random Inputs

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Abstract

It is known that standard stochastic Galerkin methods encounter challenges when solving partial differential equations with high-dimensional random inputs, which are typically caused by the large number of stochastic basis functions required. It becomes crucial to properly choose effective basis functions, such that the dimension of the stochastic approximation space can be reduced. In this work, we focus on the stochastic Galerkin approximation associated with generalized polynomial chaos (gPC), and explore the gPC expansion based on the analysis of variance (ANOVA) decomposition. A concise form of the gPC expansion is presented for each component function of the ANOVA expansion, and an adaptive ANOVA procedure is proposed to construct the overall stochastic Galerkin system. Numerical results demonstrate the efficiency of our proposed adaptive ANOVA stochastic Galerkin method for both diffusion and Helmholtz problems.

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References

  1. Agarwal, N., Aluru, N.R.: A domain adaptive stochastic collocation approach for analysis of MEMS under uncertainties. J. Comput. Phys. 228(20), 7662–7688 (2009). https://doi.org/10.1016/j.jcp.2009.07.014

    Article  MathSciNet  Google Scholar 

  2. Askey, R.: Orthogonal Polynomials and Special Functions. SIAM, Philadelphia (1975)

    Book  Google Scholar 

  3. Babuška, I., Tempone, R.I., Zouraris, G.E.: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42(2), 800–825 (2004). https://doi.org/10.1137/S0036142902418680

    Article  MathSciNet  Google Scholar 

  4. Berenger, J.P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114(2), 185–200 (1994). https://doi.org/10.1006/jcph.1994.1159

    Article  MathSciNet  Google Scholar 

  5. Caflisch, R.E.: Monte Carlo and quasi-Monte Carlo methods. Acta Numer. 7, 1–49 (1998). https://doi.org/10.1017/S0962492900002804

  6. Cheng, M., Hou, T.Y., Zhang, Z.: A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations I: derivation and algorithms. J. Comput. Phys. 242, 843–868 (2013). https://doi.org/10.1016/j.jcp.2013.02.033

    Article  MathSciNet  Google Scholar 

  7. Cheng, M., Hou, T.Y., Zhang, Z.: A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations II: adaptivity and generalizations. J. Comput. Phys. 242, 753–776 (2013). https://doi.org/10.1016/j.jcp.2013.02.020

    Article  MathSciNet  Google Scholar 

  8. Cho, H., Elman, H.C.: An adaptive reduced basis collocation method based on PCM ANOVA decomposition for anisotropic stochastic PDEs. Int. J. Uncertain. Quantif. (2018). https://doi.org/10.1615/Int.J.UncertaintyQuantification.2018024436

    Article  MathSciNet  Google Scholar 

  9. Elman, H., Furnival, D.: Solving the stochastic steady-state diffusion problem using multigrid. IMA J. Numer. Anal. 27(4), 675–688 (2007). https://doi.org/10.1093/imanum/drm006

    Article  MathSciNet  Google Scholar 

  10. Elman, H., Liao, Q.: Reduced basis collocation methods for partial differential equations with random coefficients. SIAM/ASA J. Uncertain. Quantif. 1, 192–217 (2013). https://doi.org/10.1137/120881841

    Article  MathSciNet  Google Scholar 

  11. Elman, H.C., Ernst, O.G., O’Leary, D.P., Stewart, M.: Efficient iterative algorithms for the stochastic finite element method with application to acoustic scattering. Comput. Methods Appl. Mech. Eng. 194, 1037–1055 (2005). https://doi.org/10.1016/j.cma.2004.06.028

    Article  MathSciNet  Google Scholar 

  12. Feng, X., Lin, J., Lorton, C.: An efficient numerical method for acoustic wave scattering in random media. SIAM/ASA J. Uncertain. Quantif. 3(1), 790–822 (2015). https://doi.org/10.1137/140958232

    Article  MathSciNet  Google Scholar 

  13. Fishman, G.: Monte Carlo: Concepts, Algorithms, and Applications. Springer, New York (2013)

    Google Scholar 

  14. Gao, Z., Hesthaven, J.S.: On ANOVA expansions and strategies for choosing the anchor point. Appl. Math. Comput. 217(7), 3274–3285 (2010). https://doi.org/10.1016/j.amc.2010.08.061

    Article  MathSciNet  Google Scholar 

  15. Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Courier Corporation, North Chelmsford (2003)

    Google Scholar 

  16. Guo, L., Narayan, A., Zhou, T.: Constructing least-squares polynomial approximations. SIAM Rev. 62(2), 483–508 (2020). https://doi.org/10.1137/18M1234151

    Article  MathSciNet  Google Scholar 

  17. Jakeman, J.D., Narayan, A., Zhou, T.: A generalized sampling and preconditioning scheme for sparse approximation of polynomial chaos expansions. SIAM J. Sci. Comput. 39(3), A1114–A1144 (2017). https://doi.org/10.1137/16M1063885

    Article  MathSciNet  Google Scholar 

  18. Kämmerer, L., Potts, D., Taubert, F.: The uniform sparse FFT with application to PDEs with random coefficients. In: Sampling Theory, Signal Processing, and Data Analysis, vol. 20, no. 19 (2021). https://doi.org/10.1007/s43670-022-00037-3

  19. Lee, K., Elman, H.C.: A preconditioned low-rank projection method with a rank-reduction scheme for stochastic partial differential equations. SIAM J. Sci. Comput. 39(5), S828–S850 (2017). https://doi.org/10.1137/16M1075582

    Article  MathSciNet  Google Scholar 

  20. Lee, K., Elman, H.C., Sousedik, B.: A low-rank solver for the Navier-Stokes equations with uncertain viscosity. SIAM/ASA J. Uncertain. Quantif. 7(4), 1275–1300 (2019). https://doi.org/10.1137/17M1151912

    Article  MathSciNet  Google Scholar 

  21. Liao, Q., Lin, G.: Reduced basis ANOVA methods for partial differential equations with high-dimensional random inputs. J. Comput. Phys. 317, 148–164 (2016). https://doi.org/10.1016/j.jcp.2016.04.029

    Article  MathSciNet  Google Scholar 

  22. Liu, F., Ying, L.: Additive sweeping preconditioner for the Helmholtz equation. Multiscale Model. Simul. 14(2), 799–822 (2016). https://doi.org/10.1137/15M1017144

    Article  MathSciNet  Google Scholar 

  23. Ma, X., Zabaras, N.: An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations. J. Comput. Phys. 229(10), 3884–3915 (2010). https://doi.org/10.1016/j.jcp.2010.01.033

    Article  MathSciNet  Google Scholar 

  24. Musharbash, E., Nobile, F., Zhou, T.: Error analysis of the dynamically orthogonal approximation of time dependent random PDEs. SIAM J. Sci. Comput. 37(2), A776–A810 (2015). https://doi.org/10.1137/140967787

    Article  MathSciNet  Google Scholar 

  25. Potts, D., Schmischke, M.: Approximation of high-dimensional periodic functions with Fourier-based methods. SIAM J. Numer. Anal. 59(5), 2393–2429 (2021). https://doi.org/10.1137/20M1354921

    Article  MathSciNet  Google Scholar 

  26. Powell, C.E., Elman, H.C.: Block-diagonal preconditioning for spectral stochastic finite element systems. IMA J. Numer. Anal. 29(2), 350–375 (2009). https://doi.org/10.1093/imanum/drn014

    Article  MathSciNet  Google Scholar 

  27. Powell, C.E., Silvester, D., Simoncini, V.: An efficient reduced basis solver for stochastic Galerkin matrix equations. SIAM J. Sci. Comput. 39(1), A141–A163 (2017). https://doi.org/10.1137/15M1032399

    Article  MathSciNet  Google Scholar 

  28. Sobol’, I.M.: Theorems and examples on high dimensional model representation. Reliab. Eng. Syst. Saf. 79(2), 187–193 (2003). https://doi.org/10.1016/S0951-8320(02)00229-6

    Article  Google Scholar 

  29. Sudret, B.: Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf. 93(7), 964–979 (2008). https://doi.org/10.1016/j.ress.2007.04.002

    Article  Google Scholar 

  30. Tang, K., Congedo, P.M., Abgrall, R.: Sensitivity analysis using anchored ANOVA expansion and high-order moments computation. Int. J. Numer. Meth. Eng. 102(9), 1554–1584 (2015). https://doi.org/10.1002/nme.4856

    Article  MathSciNet  Google Scholar 

  31. Tang, K., Congedo, P.M., Abgrall, R.: Adaptive surrogate modeling by ANOVA and sparse polynomial dimensional decomposition for global sensitivity analysis in fluid simulation. J. Comput. Phys. 314(1), 557–589 (2016). https://doi.org/10.1016/j.jcp.2016.03.026

    Article  MathSciNet  Google Scholar 

  32. Tang, T., Zhou, T.: Convergence analysis for stochastic collocation methods to scalar hyperbolic equations with a random wave speed. Commun. Comput. Phys 8(1), 226–248 (2010). https://doi.org/10.4208/cicp.060109.130110a

    Article  MathSciNet  Google Scholar 

  33. Wan, X., Karniadakis, G.E.: An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys. 209(2), 617–642 (2005). https://doi.org/10.1016/j.jcp.2005.03.023

    Article  MathSciNet  Google Scholar 

  34. Wang, X.: On the approximation error in high dimensional model representation. In: 2008 Winter Simulation Conference, pp. 453–462. IEEE (2008). https://doi.org/10.1109/WSC.2008.4736100

  35. Williamson, K., Cho, H., Sousedík, B.: Application of adaptive ANOVA and reduced basis methods to the stochastic Stokes–Brinkman problem. Comput. Geosci. 25(3), 1191–1213 (2021). https://doi.org/10.1007/s10596-021-10048-z

    Article  MathSciNet  Google Scholar 

  36. Xiu, D.: Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, Princeton (2010)

    Book  Google Scholar 

  37. Xiu, D., Hesthaven, J.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27, 1118–1139 (2005). https://doi.org/10.1137/040615201

    Article  MathSciNet  Google Scholar 

  38. Xiu, D., Karniadakis, G.E.: Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput. Methods Appl. Mech. Eng. 191(43), 4927–4948 (2002). https://doi.org/10.1016/S0045-7825(02)00421-8

    Article  MathSciNet  Google Scholar 

  39. Xiu, D., Karniadakis, G.E.: The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002). https://doi.org/10.1137/S1064827501387826

    Article  MathSciNet  Google Scholar 

  40. Xiu, D., Karniadakis, G.E.: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187(1), 137–167 (2003). https://doi.org/10.1016/S0021-9991(03)00092-5

    Article  MathSciNet  Google Scholar 

  41. Yan, L., Zhou, T.: Adaptive multi-fidelity polynomial chaos approach to Bayesian inference in inverse problems. J. Comput. Phys. 381, 110–128 (2019). https://doi.org/10.1016/j.jcp.2018.12.025

    Article  MathSciNet  Google Scholar 

  42. Yang, X., Choi, M., Lin, G., Karniadakis, G.E.: Adaptive ANOVA decomposition of stochastic incompressible and compressible flows. J. Comput. Phys. 231(4), 1587–1614 (2012). https://doi.org/10.1016/j.jcp.2011.10.028

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Funding

This work is supported by the National Natural Science Foundation of China (No. 12071291), and the Science and Technology Commission of Shanghai Municipality (STCSM) under Grant 22DZ1101200.

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Authors and Affiliations

  1. School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai, China

    Guanjie Wang

  2. School of Mathematics and Physics, University of Portsmouth, Lion Terrace, PO1 3HF, UK

    Smita Sahu

  3. School of Information Science and Technology, ShanghaiTech University, Shanghai, China

    Qifeng Liao

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  1. Guanjie Wang
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  2. Smita Sahu
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  3. Qifeng Liao
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Correspondence to Qifeng Liao.

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Wang, G., Sahu, S. & Liao, Q. An Adaptive ANOVA Stochastic Galerkin Method for Partial Differential Equations with High-dimensional Random Inputs. J Sci Comput 98, 24 (2024). https://doi.org/10.1007/s10915-023-02417-w

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