Advertisement

Journal of Scientific Computing

, Volume 66, Issue 1, pp 358–405 | Cite as

A Novel Weakly-Intrusive Non-linear Multiresolution Framework for Uncertainty Quantification in Hyperbolic Partial Differential Equations

  • Gianluca GeraciEmail author
  • Pietro Marco Congedo
  • Rémi Abgrall
  • Gianluca Iaccarino
Article

Abstract

In this paper, a novel multiresolution framework, namely the Truncate and Encode (TE) approach, previously proposed by some of the authors (Abgrall et al. in J Comput Phys 257:19–56, 2014. doi: 10.1016/j.jcp.2013.08.006), is generalized and extended for taking into account uncertainty in partial differential equations (PDEs). Innovative ingredients are given by an algorithm permitting to recover the multiresolution representation without requiring the fully resolved solution, the possibility to treat a whatever form of pdf and the use of high-order (even non-linear, i.e. data-dependent) reconstruction in the stochastic space. Moreover, the spatial-TE method is introduced, which is a weakly intrusive scheme for uncertainty quantification (UQ), that couples the physical and stochastic spaces by minimizing the computational cost for PDEs. The proposed scheme is particularly attractive when treating moving discontinuities (such as shock waves in compressible flows), even if they appear during the simulations as it is common in unsteady aerodynamics applications. The proposed method is very flexible since it can easily coupled with different deterministic schemes, even with high-resolution features. Flexibility and performances of the present method are demonstrated on various numerical test cases (algebraic functions and ordinary differential equations), including partial differential equations, both linear and non-linear, in presence of randomness. The efficiency of the proposed strategy for solving stochastic linear advection and Burgers equation is shown by comparison with some classical techniques for UQ, namely Monte Carlo or the non-intrusive polynomial chaos methods.

Keywords

Multiresolution Uncertainty quantification Adaptive grid ENO MUSCL Hyperbolic conservation laws 

Notes

Acknowledgments

R. Abgrall and G. Geraci have been supported by the ERC Advanced Grant ADDECCO N. 226316, partially and fully respectively. Moreover, part of the present work has been conceived during the participation of P.M. Congedo and G. Geraci at the 25th Summer Program 2012 of the Center for Turbulence Research at the Stanford University thanks to the financial support of the Associated Team Aquarius (Inria-Stanford). PM Congedo and G. Geraci are also grateful to Jeroen Witteveen for the very interesting discussions on the adaptive strategies for discontinuous problems in UQ.

References

  1. 1.
    Abgrall, R., Congedo, P.M.: A semi-intrusive deterministic approach to uncertainty quantifications in non-linear fluid flow problems. J. Comput. Phys. 235, 828–845 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Abgrall, R., Congedo, P.M., Corre, C., Galera, S.: A simple semi-intrusive method for uncertainty quantification of shocked flows, comparison with non-intrusive polynomial chaos method. In: Pereira, J.C.F., Sequeira, A., Pereira, J.M.C. (eds.) Proceedings of the V European Conference on Computational Fluid Dynamics ECCOMAS CFD 2010, Lisbon, Portugal, 14–17 June 2010Google Scholar
  3. 3.
    Abgrall, R., Congedo, P.M., Galéra, S., Geraci, G.: Semi-intrusive and non-intrusive stochastic methods for aerospace applications. In: 4th European Conference for Aerospace Sciences, Saint Petersburg, Russia, pp. 1–8 (2011)Google Scholar
  4. 4.
    Abgrall, R., Congedo, P.M., Geraci, G.: An adaptive multiresolution inspired scheme for solving the stochastic differential equations. In: Spitaleri (ed.) MASCOT11 11th Meeting on Applied Scientific Computing and Tools. Rome, Italy (2011)Google Scholar
  5. 5.
    Abgrall, R., Congedo, P.M., Geraci, G.: Toward a unified multiresolution scheme in the combined physical/stochastic space for stochastic differential equations. Tech. rep., INRIA (2012)Google Scholar
  6. 6.
    Abgrall, R., Congedo, P.M., Geraci, G.: A high-order adaptive semi-intrusive finite volume scheme for stochastic partial differential equations. In: MASCOT13 13th Meeting on Applied Scientific Computing and tools, San Lorenzo de El Escorial, Spain (2013)Google Scholar
  7. 7.
    Abgrall, R., Congedo, P.M., Geraci, G.: A high-order non-linear multiresolution scheme for stochastic PDEs. In: European Workshop on High Order Nonlinear Numerical Methods for Evolutionary PDEs: Theory and Applications (HONOM 2013) (2013)Google Scholar
  8. 8.
    Abgrall, R., Congedo, P.M., Geraci, G.: A one-time Truncate and Encode multiresolution stochastic framework. J. Comput. Phys. 257, 19–56 (2014). doi: 10.1016/j.jcp.2013.08.006 MathSciNetCrossRefGoogle Scholar
  9. 9.
    Abgrall, R., Congedo, P.M., Geraci, G., Iaccarino, G.: Adaptive strategy in multiresolution framework for uncertainty quantification. In: Prooceedings of the Summer Program, Center For Turbulence Research, pp. 209–218 (2012). http://ctr.stanford.edu/Summer/SP12/index.html
  10. 10.
    Abgrall, R., Congedo, P.M., Geraci, G., Iaccarino, G.: An adaptive multiresolution semi-intrusive scheme for UQ in compressible fluid problems. Int. J. Numer. Methods Fluids (2015). doi: 10.1002/fld.4030
  11. 11.
    Abgrall, R., Harten, A.: Multiresolution representation in unstructured meshes. SIAM J. Numer. Anal. 35(6), 2128–2146 (1998). doi: 10.1137/S0036142997315056 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Abgrall, R., Sonar, T.: On the use of Mühlbach expansions in the recovery step of ENO methods. Numerische Mathematik. 1–25 (1997). http://www.springerlink.com/index/LTXHR8P0MC3QBQA7.pdf
  13. 13.
    Amat, S., Aràndiga, F., Cohen, A., Donat, R.: Tensor product multiresolution analysis with error control for compact image representation. Signal Process. 82, 587–608 (2002). http://www.sciencedirect.com/science/article/pii/S0165168401002067
  14. 14.
    Amat, S., Busquier, S., Trillo, J.: Nonlinear Harten’s multiresolution on the quincunx pyramid. J. Comput. Appl. Math. 189(1–2), 555–567 (2006). doi: 10.1016/j.cam.2005.03.034. http://linkinghub.elsevier.com/retrieve/pii/S0377042705001433
  15. 15.
    Aràndiga, F., Belda, A.M., Mulet, P.: Point-value WENO multiresolution applications to stable image compression. J. Sci. Comput. 43(2), 158–182 (2010). doi: 10.1007/s10915-010-9351-8 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Arandiga, F., Chiavassa, G., Donat, R.: Harten framework for multiresolution with applications: from conservation laws to image compression. Boletín SEMA 31(31), 73–108 (2009). http://www.sema.org.es/ojs/index.php?journal=sema&page=article&op=view&path[]=174Google Scholar
  17. 17.
    Arandiga, F., Donat, R.: Nonlinear multiscale decompositions: the approach of A. Harten. Numer. Algorithms 23, 175–216 (2000). http://www.springerlink.com/index/N363R0747675J70L.pdf
  18. 18.
    Babuška, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Rev. 52(2), 317 (2010). doi: 10.1137/100786356. http://link.aip.org/link/SIREAD/v52/i2/p317/s1&Agg=doi
  19. 19.
    Bellman, R.E., Richard, B.: Adaptive Control Processes: A Guided Tour. Princeton University Press (1961). http://books.google.com/books?id=POAmAAAAMAAJ&pgis=1
  20. 20.
    Bihari, B.L., Harten, A.: Multiresolution schemes for the numerical solution of 2-D conservation laws I. SIAM J. Sci. Comput. 18(2), 315 (1997). doi: 10.1137/S1064827594278848. http://link.aip.org/link/SJOCE3/v18/i2/p315/s1&Agg=doi
  21. 21.
    Chiavassa, G., Donat, R.: Point value multiscale algorithms for 2D compressible flows. SIAM J. Sci. Comput. 23(3), 805–823 (2001). http://epubs.siam.org/doi/abs/10.1137/S1064827599363988
  22. 22.
    Cohen, A., Dyn, N., Matei, B.: Quasilinear subdivision schemes with applications to ENO interpolation. Appl. Comput. Harmon. Anal. 15, 89–116 (2003). http://www.sciencedirect.com/science/article/pii/S1063520303000617
  23. 23.
    Congedo, P.M., Geraci, G., Abgrall, R.: Semi-intrusive multi-resolution methods for stochastic compressible flows. In: VKI Lecture Series on ‘Uncertainty Quantification in Computational Fluid Dynamics’ (STO-AVT-235), Von Karman Institute For Fluid Dynamics (2014)Google Scholar
  24. 24.
    Geraci, G.: Schemes and Strategies to Propagate and Analyze Uncertainties in Computational Fluid Dynamics Applications. Ph.D. thesis, INRIA and University of Bordeaux 1 (2013)Google Scholar
  25. 25.
    Getreuer, P., Meyer, F.G.: ENO multiresolutions schemes with general discretizations. SIAM J. Numer. Anal. 46(6), 2953–2977 (2008). http://epubs.siam.org/doi/pdf/10.1137/060663763
  26. 26.
    Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements. A Spectral Approach. Springer, Berlin (1991)CrossRefzbMATHGoogle Scholar
  27. 27.
    Graham, I., Kuo, F., Nuyens, D., Scheichl, R., Sloan, I.: Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications. J. Comput. Phys. 230(10), 3668–3694 (2011). doi: 10.1016/j.jcp.2011.01.023. http://linkinghub.elsevier.com/retrieve/pii/S0021999111000489
  28. 28.
    Harten, A.: Discrete multi-resolution analysis and generalized wavelets. Appl. Numer. Math. 12(13), 153–192 (1993). doi: 10.1016/0168-9274(93)90117-A MathSciNetzbMATHGoogle Scholar
  29. 29.
    Harten, A.: Adaptive multiresolution schemes for shock computations. J. Comput. Phys. 135(2), 260–278 (1994). doi: 10.1006/jcph.1997.5713. http://linkinghub.elsevier.com/retrieve/pii/S0021999197957132 http://www.sciencedirect.com/science/article/pii/S0021999184711995
  30. 30.
    Harten, A.: Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Commun. Pure Appl. Math. 48(12), 1305–1342 (1995). doi: 10.1002/cpa.3160481201/abstract MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Harten, A.: Multiresolution representation of data: a general framework. SIAM J. Numer. Anal. 33(3), 1205–1256 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Harten, A., Engquist, B., Osher, S.: Uniformly high order accurate essentially non-oscillatory schemes. III. J. Comput. Phys. 71(2), 231–303 (1987). doi: 10.1016/0021-9991(87)90031-3 MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Le Maître, O., Knio, O.: Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics. Springer, Berlin (2010)CrossRefGoogle Scholar
  34. 34.
    Leveque, R.J.: Finite Volume Methods for Conservation Laws and Hyperbolic Systems. Cambridge University Press, Cambridge (2002)CrossRefGoogle Scholar
  35. 35.
    Orszag, S.: Dynamical properties of truncated Wiener-Hermite expansions. Phys. Fluids 10(12), 2603–2613 (1967)CrossRefzbMATHGoogle Scholar
  36. 36.
    Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics. Springer, Berlin (2000)Google Scholar
  37. 37.
    Shu, C.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Tech. Rep. 97, ICASE Report No. 97–65 (1997). http://link.springer.com/chapter/10.1007/BFb0096355
  38. 38.
    Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Mechanics. Springer, Berlin (1997)CrossRefGoogle Scholar
  39. 39.
    Tryoen, J.: Methodes de Galerkin stochastiques adaptatives pour la propagation d’incertitudes parametriques dans les systemes hyperboliques. Ph.D. thesis, Université Paris-Est (2011)Google Scholar
  40. 40.
    Tryoen, J., Le Maître, O., Ern, A.: Adaptive anisotropic spectral stochastic methods for uncertain scalar conservation laws. SIAM J. Sci. Comput. 34, A2459–A2481 (2012)CrossRefzbMATHGoogle Scholar
  41. 41.
    Tryoen, J., Le Maître, O., Ndjinga, M., Ern, A.: Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems. J. Comput. Phys. 229, 6485–6511 (2010). doi: 10.1016/j.jcp.2010.05.007 MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Wan, X., Karniadakis, G.E.: Beyond Wiener Askey expansions: handling arbitrary PDFs. J. Sci. Comput. 27(1–3), 455–464 (2005). doi: 10.1007/s10915-005-9038-8 MathSciNetGoogle Scholar
  43. 43.
    Xiu, D., Karniadakis, G.E.: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187(1), 137–167 (2003). doi: 10.1016/S0021-9991(03)00092-5 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Gianluca Geraci
    • 1
    Email author
  • Pietro Marco Congedo
    • 2
  • Rémi Abgrall
    • 3
  • Gianluca Iaccarino
    • 1
  1. 1.Flow Physics and Computational Engineering, Mechanical Engineering DepartmentStanford UniversityStanfordUSA
  2. 2.INRIA Bordeaux–Sud-OuestTalence CedexFrance
  3. 3.Institut für MathematikUniversität ZürichZurichSwitzerland

Personalised recommendations