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Numerische Mathematik

, Volume 136, Issue 3, pp 765–803 | Cite as

Adaptive stochastic Galerkin FEM with hierarchical tensor representations

  • Martin Eigel
  • Max Pfeffer
  • Reinhold SchneiderEmail author
Article

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.

Keywords

Partial differential equations with random coefficients Tensor representation Tensor train Uncertainty quantification Stochastic finite element methods Operator equations Adaptive methods ALS Low-rank Reduced basis methods 

Mathematics Subject Classification

35R60 47B80 60H35 65C20 65N12 65N22 65J10 

Notes

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.

References

  1. 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)Google Scholar
  2. 2.
    Babuška, I., Chatzipantelidis, P.: On solving elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 191(37–38), 4093–4122 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bachmayr, M., Cohen, A., Migliorati, G.: Sparse polynomial approximation of parametric elliptic pdes. Part i: affine coefficients (2015). arXiv:1509.07045
  7. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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)Google Scholar
  9. 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)Google Scholar
  10. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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)MathSciNetCrossRefzbMATHGoogle 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. (Singap.) 9(1), 11–47 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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. 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. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Eigel, M., Gittelson, C.J., Schwab, C., Zander, E.: Adaptive stochastic Galerkin FEM. Comput. Methods Appl. Mech. Eng. 270, 247–269 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Eigel, M., Merdon, C.: Local equilibration error estimators for guaranteed error control in adaptive stochastic higher-order galerkin fem. WIAS Preprint (1997) (2014)Google Scholar
  23. 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. 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)Google Scholar
  25. 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. 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)Google Scholar
  27. 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)Google Scholar
  28. 28.
    FEniCS Project - Automated solution of Differential Equations by the Finite Element Method. http://fenicsproject.org
  29. 29.
    Fischer, J., Otto, F.: A higher-order large-scale regularity theory for random elliptic operators. MPI-MIS preprint Leipzig (2015)Google Scholar
  30. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 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)CrossRefzbMATHGoogle Scholar
  33. 33.
    Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)Google Scholar
  34. 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)Google Scholar
  35. 35.
    Grasedyck, L.: Hierarchical singular value decomposition of tensors. SIAM J. Matrix Anal. Appl. 31(4), 2029–2054 (2010)Google Scholar
  36. 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)Google Scholar
  37. 37.
    Hackbusch, W.: Tensor spaces and numerical tensor calculus. Springer Series in Computational Mathematics, vol. 42. Springer, Heidelberg (2012)Google Scholar
  38. 38.
    Hackbusch, W.: Numerical tensor calculus. Acta Numer. 23, 651–742 (2014)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Hackbusch, W., Kühn, S.: A new scheme for the tensor representation. J. Fourier Anal. Appl. 15(5), 706–722 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Hackbusch, W., Schneider,R.: Tensor spaces and hierarchical tensor representations. In: Extr. Quant. Inf. Complex Syst., pp. 237–261. Springer (2014)Google Scholar
  41. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Holtz, S., Rohwedder, T., Schneider, R.: On manifolds of tensors of fixed TT-rank. Numerische Mathematik 120(4), 701–731 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 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)Google Scholar
  44. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Landsberg, J.M.: Tensors: Geometry and Applications. In: Graduate studies in mathematics, American Mathematical Society, USA (2012)Google Scholar
  47. 47.
    De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 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)Google Scholar
  50. 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)Google Scholar
  51. 51.
    Oseledets, I.: Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295–2317 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Oseledets, I.V.: ttpy - A Python Implementation of the TT-Toolbox. https://github.com/oseledets/ttpy
  53. 53.
    Oseledets, I.V.: Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295–2317 (2011)Google Scholar
  54. 54.
    Oseledets, I.V., Tyrtyshnikov, E.: TT-cross approximation for multidimensional arrays. Linear Algebra Appl. 432(1), 70–88 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 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)CrossRefzbMATHGoogle Scholar
  56. 56.
    Powell, C.E., Elman, H.C.: Block-diagonal preconditioning for spectral stochastic finite-element systems. IMA J. Numer. Anal. 29, 350–375 (2009)Google Scholar
  57. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Schmidt, E.: Zur Theorie der linearen und nichtlinearen Integralgleichungen. I. Teil: Entwicklung. Mathematische Annalen 63, 433–476 (1907)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Schwab, C., Gittelson, C.J.: Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. Acta Numer. 20, 291–467 (2011)Google Scholar
  61. 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. 62.
    Ullmann, E.: A kronecker product preconditioner for stochastic galerkin finite element discretizations. SIAM J. Sci. Comput. 32(2), 923–946 (2010)Google Scholar
  63. 63.
    Uschmajew, A., Vandereycken, B.: The geometry of algorithms using hierarchical tensors. Linear Algebra Appl 439(1), 133–166 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Teubner Verlag and J. Wiley, Stuttgart (1996)zbMATHGoogle Scholar
  65. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 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)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Weierstrass InstituteBerlinGermany
  2. 2.TU BerlinBerlinGermany

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