Skip to main content
SpringerLink
Log in

Galerkin Methods for Stationary Radiative Transfer Equations with Uncertain Coefficients

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper we study the stationary radiative transfer equation with random coefficients. Galerkin methods are applied, which use orthogonal polynomials associated with the probability distribution of the random variables as basis functions in the random space. Such algorithms have been widely used for kinetic equations with random inputs, however, the corresponding numerical analysis is rare. In this paper we establish regularity theorems describing the smoothness properties of the solution, and investigate the convergence rate of N-term truncated polynomials under the spectral method framework. Numerical tests are conducted to demonstrate our analytical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Albi, G., Pareschi, L., Zanella, M.: Uncertainty quantification in control problems for flocking models. Math. Probl. Eng.(ID 850124), 14 (2015)

  2. Babuska, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45(3), 1005–1034 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Babuska, I., Tempone, R., Zouraris, G.: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42(2), 800–825 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barth, A., Schwab, C., Zollinger, N.: Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients. Numerische Mathematik 119(1), 123–161 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Branicki, M., Majda, A.J.: Fundamental limitations of polynomial chaos for uncertainty quantification in systems with intermittent instabilities. Commun. Math. Sci. 11(1), 55–103 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Case, K.M., Zweifel, P.F.: Existence and uniqueness theorems for the neutron transport equation. J. Math. Phys. 4(11), 1376–1385 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  7. Case, K.M., Zweifel, P.F.: Linear Transport Theory. Addison-Wesley Pub. Co., Boston (1967)

    MATH  Google Scholar 

  8. Cercignani, C.: The Boltzmann Equation and its Applications. Springer, Berlin (2012)

    MATH  Google Scholar 

  9. Chandrasekhar, S.: Radiative Transfer. Dover Books on Intermediate and Advanced Mathematics. Dover Publications, Mineola (1960)

    Google Scholar 

  10. Charrier, J., Scheichl, R., Teckentrup, A.L.: Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods. SIAM J. Numer. Anal. 51(1), 322–352 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chkifa, A., Cohen, A., Schwab, C.: Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs. Journal de Mathématiques Pures et Appliquées (2014)

  12. Choulli, M., Stefanov, P.: An inverse boundary value problem for the stationary transport equation. Osaka J. Math. 36(1), 87–104 (1999)

    MathSciNet  MATH  Google Scholar 

  13. Cohen, A., DeVore, R., Schwab, C.: Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs. Found. Comput. Math. 10(6), 615–646 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cohen, A., Devore, R., Schwab, C.: Analytic regularity and polynomial approximation of parametric and stochastic elliptic pde’s. Anal. Appl. 09(01), 11–47 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology. Springer, Berlin (1993)

    MATH  Google Scholar 

  16. Despres, B., Perthame, B.: Uncertainty propagation; intrusive kinetic formulations of scalar conservation laws. SIAM/ASA J. Uncerta. Quantif. 4(1), 980–1013 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Egger, H., Schlottbom, M.: An Lp theory for stationary radiative transfer. Appl. Anal. 93(6), 1283–1296 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)

    Book  MATH  Google Scholar 

  19. Giles, M.B.: Multilevel Monte Carlo path simulation. Oper. Res. 56(3), 607–617 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gottlieb, D., Xiu, D.: Galerkin method for wave equations with uncertain coefficients. Commun. Comput. Phys 3(2), 505–518 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Gui, W., Babuska, I.: The h, p and h–p versions of the finite element method in 1 dimension. Part 1. The error analysis of the p-version. Numer. Math. 49(6), 577–612 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hou, Y.T., Li, Q., Zhang, P.: Exploring the locally low dimensional structure in solving random elliptic PDEs. SIAM Multisc. Model. Simul. 15(2), 661–695 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hou, Y.T., Li, Q., Zhang, P.: A sparse decomposition of low rank symmetric positive semi-definite matrices. SIAM Multisc. Model. Simul. 15(1), 410–444 (2016)

    Article  MATH  Google Scholar 

  24. Hu, J., Jin, S., Li, Q.: Asymptotic-preserving schemes for multiscale hyperbolic and kinetic equations. In: Abgrall, R., Shu, C.-W. (eds.) Handbook of Numerical Methods for Hyperbolic Problems. North Holland/Elsevier (2017)

  25. Jin, S.: Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review. Rivista di Matematica della Universita di Parma 3, 177–216 (2012)

    MathSciNet  MATH  Google Scholar 

  26. Jin, S., Xiu, D., Zhu, X.: Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings. J. Comput. Phys. 289, 35–52 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Knutson, A., Tao, T.: Honeycombs and sums of hermitian matrices. arXiv:math/0009048v1 [math.RT] (2000)

  28. Li, Q., Wang, L.: Polynomial interpolation of burgers’ equation with randomness. arXiv:1708.04332 [math] (2017)

  29. Li, Q., Wang, L.: Uniform regularity for linear kinetic equations with random input based on hypocoercivity. SIAM/ASA J. Uncertain. Quantif. (2017)

  30. Minnich, A.J., Chen, G., Mansoor, S., Yilbas, B.S.: Quasiballistic heat transfer studied using the frequency-dependent boltzmann transport equation. Phys. Rev. B 84, 235207 (2011)

    Article  Google Scholar 

  31. Mishra, S., Risebro, N.H., Schwab, C., Tokareva, S.: Numerical solution of scalar conservation laws with random flux functions. SIAM/ASA J. Uncertain. Quantif. 4, 552–591 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mishra, S., Schwab, C.: Sparse tensor multi-level monte carlo finite volume methods for hyperbolic conservation laws with random initial data. Math. Comp. 81, 1979–2018 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Mishra, S., Schwab, C., Šukys, J.: Multi-level Monte Carlo Finite Volume Methods for Uncertainty Quantification in Nonlinear Systems of Balance Laws, p. 225294. Springer, Cham (2013)

    MATH  Google Scholar 

  34. Nobile, F., Tempone, R., Webster, C.G.: An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46(5), 2411–2442 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  35. 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. 46(5), 2309–2345 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Schwab, C., Todor, R.-A.: Sparse finite elements for elliptic problems with stochastic loading. Numerische Mathematik 95(4), 707–734 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wiener, N.: The homogeneous chaos. Am. J. Math. 60(4), 897–936 (1938)

    Article  MathSciNet  MATH  Google Scholar 

  38. Xiu, D.: Generalized (Weiner-Askey) Polynomial Chaos. Ph.D. thesis, Brown University (2004)

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

    MATH  Google Scholar 

  40. Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118–1139 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  41. Xiu, D., Karniadakis, G.: The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  42. Xiu, D., Karniadakis, G.E.: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187(1), 137–167 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhang, G., Gunzburger, M.: Error analysis of a stochastic collocation method for parabolic partial differential equations with random input data. SIAM J. Numer. Anal. 50(4), 1922–1940 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

X. Zhong is supported by the start-up funds from Zhejiang University and funds from Recruitment Program for Young Professionals (No. 588020-X01702/105). X. Zhong is also supported in part by the Funds for Creative Research Groups of NSFC (No. 11621101). Q. Li is supported by the start-up funds from UW-Madison and NSF 1619778.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China

    Xinghui Zhong

  2. Department of Mathematics, University of Wisconsin-Madison, Madison, WI, 53705, USA

    Qin Li

Authors
  1. Xinghui Zhong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xinghui Zhong.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhong, X., Li, Q. Galerkin Methods for Stationary Radiative Transfer Equations with Uncertain Coefficients. J Sci Comput 76, 1105–1126 (2018). https://doi.org/10.1007/s10915-018-0652-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-018-0652-7

Keywords

Search

Navigation