Journal of Scientific Computing

, Volume 76, Issue 2, pp 1105–1126 | Cite as

Galerkin Methods for Stationary Radiative Transfer Equations with Uncertain Coefficients

  • Xinghui ZhongEmail author
  • Qin Li


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.


Uncertainty quantification Radiative transfer equation Generalized polynomial chaos Convergence rate 



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.


  1. 1.
    Albi, G., Pareschi, L., Zanella, M.: Uncertainty quantification in control problems for flocking models. Math. Probl. Eng.(ID 850124), 14 (2015)Google Scholar
  2. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Case, K.M., Zweifel, P.F.: Existence and uniqueness theorems for the neutron transport equation. J. Math. Phys. 4(11), 1376–1385 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Case, K.M., Zweifel, P.F.: Linear Transport Theory. Addison-Wesley Pub. Co., Boston (1967)zbMATHGoogle Scholar
  8. 8.
    Cercignani, C.: The Boltzmann Equation and its Applications. Springer, Berlin (2012)zbMATHGoogle Scholar
  9. 9.
    Chandrasekhar, S.: Radiative Transfer. Dover Books on Intermediate and Advanced Mathematics. Dover Publications, Mineola (1960)Google Scholar
  10. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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)Google Scholar
  12. 12.
    Choulli, M., Stefanov, P.: An inverse boundary value problem for the stationary transport equation. Osaka J. Math. 36(1), 87–104 (1999)MathSciNetzbMATHGoogle Scholar
  13. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology. Springer, Berlin (1993)zbMATHGoogle Scholar
  16. 16.
    Despres, B., Perthame, B.: Uncertainty propagation; intrusive kinetic formulations of scalar conservation laws. SIAM/ASA J. Uncerta. Quantif. 4(1), 980–1013 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Egger, H., Schlottbom, M.: An Lp theory for stationary radiative transfer. Appl. Anal. 93(6), 1283–1296 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)CrossRefzbMATHGoogle Scholar
  19. 19.
    Giles, M.B.: Multilevel Monte Carlo path simulation. Oper. Res. 56(3), 607–617 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gottlieb, D., Xiu, D.: Galerkin method for wave equations with uncertain coefficients. Commun. Comput. Phys 3(2), 505–518 (2008)MathSciNetzbMATHGoogle Scholar
  21. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 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)CrossRefzbMATHGoogle Scholar
  24. 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)Google Scholar
  25. 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)MathSciNetzbMATHGoogle Scholar
  26. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Knutson, A., Tao, T.: Honeycombs and sums of hermitian matrices. arXiv:math/0009048v1 [math.RT] (2000)
  28. 28.
    Li, Q., Wang, L.: Polynomial interpolation of burgers’ equation with randomness. arXiv:1708.04332 [math] (2017)
  29. 29.
    Li, Q., Wang, L.: Uniform regularity for linear kinetic equations with random input based on hypocoercivity. SIAM/ASA J. Uncertain. Quantif. (2017)Google Scholar
  30. 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)CrossRefGoogle Scholar
  31. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 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)zbMATHGoogle Scholar
  34. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Schwab, C., Todor, R.-A.: Sparse finite elements for elliptic problems with stochastic loading. Numerische Mathematik 95(4), 707–734 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wiener, N.: The homogeneous chaos. Am. J. Math. 60(4), 897–936 (1938)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Xiu, D.: Generalized (Weiner-Askey) Polynomial Chaos. Ph.D. thesis, Brown University (2004)Google Scholar
  39. 39.
    Xiu, D.: Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, Princeton (2010)zbMATHGoogle Scholar
  40. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Xiu, D., Karniadakis, G.: The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Xiu, D., Karniadakis, G.E.: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187(1), 137–167 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 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)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.School of Mathematical SciencesZhejiang UniversityHangzhouChina
  2. 2.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA

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