Skip to main content
SpringerLink
Log in

Adaptive stochastic Galerkin FEM with hierarchical tensor representations

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

The solution of PDE with stochastic data commonly leads to very high-dimensional algebraic problems, e.g. when multiplicative noise is present. The Stochastic Galerkin FEM considered in this paper then suffers from the curse of dimensionality. This is directly related to the number of random variables required for an adequate representation of the random fields included in the PDE. With the presented new approach, we circumvent this major complexity obstacle by combining two highly efficient model reduction strategies, namely a modern low-rank tensor representation in the tensor train format of the problem and a refinement algorithm on the basis of a posteriori error estimates to adaptively adjust the different employed discretizations. The adaptive adjustment includes the refinement of the FE mesh based on a residual estimator, the problem-adapted stochastic discretization in anisotropic Legendre Wiener chaos and the successive increase of the tensor rank. Computable a posteriori error estimators are derived for all error terms emanating from the discretizations and the iterative solution with a preconditioned ALS scheme of the problem. Strikingly, it is possible to exploit the tensor structure of the problem to evaluate all error terms very efficiently. A set of benchmark problems illustrates the performance of the adaptive algorithm with higher-order FE. Moreover, the influence of the tensor rank on the approximation quality is investigated.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [Wiley], New York (2000)

  2. Babuška, I., Chatzipantelidis, P.: On solving elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 191(37–38), 4093–4122 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Babuška, 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Babuška, I., Tempone, R., Zouraris, G.E.: Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation. Comput. Methods Appl. Mech. Eng. 194(12–16), 1251–1294 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bachmayr, M., Cohen, A., Migliorati, G.: Sparse polynomial approximation of parametric elliptic pdes. Part i: affine coefficients (2015). arXiv:1509.07045

  7. Ballani, J., Grasedyck, L.: Hierarchical tensor approximation of output quantities of parameter-dependent pdes. SIAM/ASA J. Uncertainty Quantification. 3(1), 852–872 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bespalov, A., Powell, C.E., Silvester, D.: Energy norm a posteriori error estimation for parametric operator equations. SIAM J. Sci. Comput. 36(2), A339–A363 (2014)

  9. Braess, D.: Finite Elements: Theory, fast solvers, and applications in elasticity theory (Translated from the German by Schumaker, L.L) Cambridge University Press, Cambridge (2007)

  10. Carstensen, C., Eigel, M., Hoppe, R.H.W., Löbhard, C.: A review of unified a posteriori finite element error control. Numer. Math. Theor. Methods. Appl. 5(4), 509–558 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, P., Quarteroni, A., Rozza, G.: A weighted reduced basis method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 51(6), 3163–3185 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, P., Quarteroni, A., Rozza, G.: Comparison between reduced basis and stochastic collocation methods for elliptic problems. J. Sci. Comput. 59(1), 187–216 (2014)

    Article  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. (Singap.) 9(1), 11–47 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. de Silva, V., Lim, L.H.: Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 30(3), 1084–1127 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Deb, M.K., Babuška, I.M., Oden, J.T.: Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Eng. 190(48), 6359–6372 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sergey, D., Khoromskij, B.N., Litvinenko, A., Matthies, H.G.: Computation of the response surface in the tensor train data format (2014). arXiv:1406.2816

  18. Dolgov, S., Khoromskij, B.N., Litvinenko, A., Matthies, H.G.: Polynomial chaos expansion of random coefficients and the solution of stochastic partial differential equations in the tensor train format (2015). arXiv preprint. arXiv:1503.03210

  19. Dolgov, S.V., Savostyanov, D.V.: Alternating minimal energy methods for linear systems in higher dimensions. SIAM J. Sci. Comput. 36(5), A2248–A2271 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Eigel, M., Gittelson, C.J., Schwab, C., Zander, E.: Adaptive stochastic Galerkin FEM. Comput. Methods Appl. Mech. Eng. 270, 247–269 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Eigel, M., Gittelson, C.J., Schwab, C., Zander, E.: A convergent adaptive stochastic galerkin finite element method with quasi-optimal spatial meshes. ESAIM Math. Model. Numer. Anal. 49(5), 1367–1398 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Eigel, M., Merdon, C.: Local equilibration error estimators for guaranteed error control in adaptive stochastic higher-order galerkin fem. WIAS Preprint (1997) (2014)

  23. Eigel, M., Zander, E.: \(\mathtt{alea}\)—A Python Framework for Spectral Methods and Low-Rank Approximations in Uncertainty Quantification. https://bitbucket.org/aleadev/alea

  24. Ernst, O.G., Mugler, A., Starkloff, H., Ullmann, E.: On the convergence of generalized polynomial chaos expansions. Technical Report 60, DFG Schwerpunktprogramm 1324, (2010)

  25. Espig, M., Hackbusch, W., Khachatryan, A.: On the convergence of alternating least squares optimisation in tensor format Representations. 423, (2015). arXiv:1506.00062

  26. Espig, M., Hackbusch, W., Litvinenko, A., Matthies, H.G., Wähnert, P.: Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats. Preprint, Max Planck Institute for Mathematics in the Sciences (2012)

  27. Espig, M., Hackbusch, W., Litvinenko, A., Matthies, H.G., Zander, E.: Efficient analysis of high dimensional data in tensor formats. In Sparse Grids and Applications, pp. 31–56. Springer (2013)

  28. FEniCS Project - Automated solution of Differential Equations by the Finite Element Method. http://fenicsproject.org

  29. Fischer, J., Otto, F.: A higher-order large-scale regularity theory for random elliptic operators. MPI-MIS preprint Leipzig (2015)

  30. Frauenfelder, P., Christoph, S., Todor, R.A.: Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Eng. 194(2–5), 205–228 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Garcia, L.D., Stillman, M., Sturmfels, B.: Algebraic geometry of bayesian networks. J. Symbolic Comput. 39(3–4), 331–355 (2005). (Special issue on the occasion of MEGA 2003)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ghanem, R.G., Kruger, R.M.: Numerical solution of spectral stochastic finite element systems. Comput. Methods Appl. Mech. Eng. 129(3), 289–303 (1996)

    Article  MATH  Google Scholar 

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

  34. Gittelson, C.J.: Stochastic Galerkin approximation of operator equations with infinite dimensional noise. Technical Report 2011-10, Seminar for Applied Mathematics, ETH Zürich (2011)

  35. Grasedyck, L.: Hierarchical singular value decomposition of tensors. SIAM J. Matrix Anal. Appl. 31(4), 2029–2054 (2010)

  36. Gunzburger, M.D., Webster, C.G., Zhang, G.: Stochastic finite element methods for partial differential equations with random input data. Acta Numer. 23, 521–650 (2014)

  37. Hackbusch, W.: Tensor spaces and numerical tensor calculus. Springer Series in Computational Mathematics, vol. 42. Springer, Heidelberg (2012)

  38. Hackbusch, W.: Numerical tensor calculus. Acta Numer. 23, 651–742 (2014)

    Article  MathSciNet  Google Scholar 

  39. Hackbusch, W., Kühn, S.: A new scheme for the tensor representation. J. Fourier Anal. Appl. 15(5), 706–722 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. Hackbusch, W., Schneider,R.: Tensor spaces and hierarchical tensor representations. In: Extr. Quant. Inf. Complex Syst., pp. 237–261. Springer (2014)

  41. Holtz, S., Rohwedder, T., Schneider, R.: The alternating linear scheme for tensor optimization in the tensor train format. SIAM J. Sci. Comput. 34(2), A683–A713 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Holtz, S., Rohwedder, T., Schneider, R.: On manifolds of tensors of fixed TT-rank. Numerische Mathematik 120(4), 701–731 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  43. Khoromskij, B.N., Oseledets, I.V.: Quantics-TT collocation approximation of parameter-dependent and stochastic elliptic PDEs. Comput. Methods Appl. Math. 10(4), 376–394 (2010)

  44. Khoromskij, B.N., Schwab, C.: Tensor-structured Galerkin approximation of parametric and stochastic elliptic PDEs. SIAM J. Sci. Comput. 33(1), 364–385 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  45. Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  46. Landsberg, J.M.: Tensors: Geometry and Applications. In: Graduate studies in mathematics, American Mathematical Society, USA (2012)

  47. De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  48. Matthies, H.G., Keese, A.: Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 194(12–16), 1295–1331 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  49. 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)

  50. 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)

  51. Oseledets, I.: Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295–2317 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  52. Oseledets, I.V.: ttpy - A Python Implementation of the TT-Toolbox. https://github.com/oseledets/ttpy

  53. Oseledets, I.V.: Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295–2317 (2011)

  54. Oseledets, I.V., Tyrtyshnikov, E.: TT-cross approximation for multidimensional arrays. Linear Algebra Appl. 432(1), 70–88 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  55. Pellissetti, M.F., Ghanem, R.G.: Iterative solution of systems of linear equations arising in the context of stochastic finite elements. Adv. Eng. Softw. 31(8), 607–616 (2000)

    Article  MATH  Google Scholar 

  56. Powell, C.E., Elman, H.C.: Block-diagonal preconditioning for spectral stochastic finite-element systems. IMA J. Numer. Anal. 29, 350–375 (2009)

  57. Rohwedder, T., Uschmajew, A.: On local convergence of alternating schemes for optimization of convex problems in the tensor train format. SIAM J. Numer. Anal. 51(2), 1134–1162 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  58. Schmidt, E.: Zur Theorie der linearen und nichtlinearen Integralgleichungen. I. Teil: Entwicklung. Mathematische Annalen 63, 433–476 (1907)

    Article  MathSciNet  MATH  Google Scholar 

  59. Schneider, R., Uschmajew, A.: Convergence results for projected line-search methods on varieties of low-rank matrices via Łojasiewicz inequality. SIAM J. Optim. 25(1), 622–646 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  60. Schwab, C., Gittelson, C.J.: Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. Acta Numer. 20, 291–467 (2011)

  61. Szalay, S., Pfeffer, M., Murg, V., Barcza, G., Verstraete, F., Schneider, R., Legeza,Ö.: Tensor product methods and entanglement optimization for ab initio quantum chemistry. Int. J. Quantum Chem. 115(19), 1342–1391 (2015) doi:10.1002/qua.24898

  62. Ullmann, E.: A kronecker product preconditioner for stochastic galerkin finite element discretizations. SIAM J. Sci. Comput. 32(2), 923–946 (2010)

  63. Uschmajew, A., Vandereycken, B.: The geometry of algorithms using hierarchical tensors. Linear Algebra Appl 439(1), 133–166 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  64. Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Teubner Verlag and J. Wiley, Stuttgart (1996)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  66. 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)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

M.P. and R.S. were supported by the DFG project ERA Chemistry; additionally R.S. was supported through Matheon by the Einstein Foundation Berlin.

Author information

Authors and Affiliations

  1. Weierstrass Institute, Mohrenstraße 39, 10117, Berlin, Germany

    Martin Eigel

  2. TU Berlin, Straße des 17, Juni 136, 10623, Berlin, Germany

    Max Pfeffer & Reinhold Schneider

Authors
  1. Martin Eigel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Max Pfeffer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Reinhold Schneider
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Reinhold Schneider.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Eigel, M., Pfeffer, M. & Schneider, R. Adaptive stochastic Galerkin FEM with hierarchical tensor representations. Numer. Math. 136, 765–803 (2017). https://doi.org/10.1007/s00211-016-0850-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-016-0850-x

Keywords

Mathematics Subject Classification

Search

Navigation