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Multilevel tensor approximation of PDEs with random data

  • Jonas Ballani
  • Daniel Kressner
  • Michael D. PetersEmail author
Article
  • 128 Downloads

Abstract

In this paper, we introduce and analyze a new low-rank multilevel strategy for the solution of random diffusion problems. Using a standard stochastic collocation scheme, we first approximate the infinite dimensional random problem by a deterministic parameter-dependent problem on a high-dimensional parameter domain. Given a hierarchy of finite element discretizations for the spatial approximation, we make use of a multilevel framework in which we consider the differences of the solution on two consecutive finite element levels at the collocation points. We then address the approximation of these high-dimensional differences by adaptive low-rank tensor techniques. This allows to equilibrate the error on all levels by exploiting regularity and additional low-rank structure of the solution. We arrive at an explicit representation in a low-rank tensor format of the approximate solution on the entire parameter domain, which can be used for, e.g., the direct and cheap computation of statistics. Numerical results are provided in order to illustrate the approach.

References

  1. 1.
    Babuška, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Rev. 52(2), 317–355 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bachmayr, M., Cohen, A., Dahmen, W.: Parametric PDEs: sparse or low-rank approximations? (2016). arXiv:1607.04444
  3. 3.
    Ballani, J., Grasedyck, L.: Hierarchical tensor approximation of output quantities of parameter-dependent PDEs. SIAM/ASA J. Uncertain. Quantif. 3(1), 393–416 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ballani, J., Grasedyck, L., Kluge, M.: Black box approximation of tensors in hierarchical Tucker format. Linear Algebra Appl. 438(2), 639–657 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ballani, J., Kressner, D.: Reduced basis methods: from low-rank matrices to low-rank tensors. SIAM J. Sci. Comput. 38(4), A2045–A2067 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bangerth, W., Hartmann, R., Kanschat, G.: deal.II – a general purpose object oriented finite element library. ACM Trans. Math. Softw., 33(4):24/1–24/27 (2007)Google Scholar
  7. 7.
    Barth, A., Schwab, C., Zollinger, N.: Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients. Numer. Math. 119(1), 123–161 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bungartz, H.-J., Griebel, M.: Sparse grids. Acta Numer. 13, 147–269 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Charrier, J.: Strong and weak error estimates for elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal. 50(1), 216–246 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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
  11. 11.
    Chen, P., Quarteroni, A., Rozza, G.: Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by stokes equations. Numer. Math. 133(1), 67–102 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cohen, A., DeVore, R., Schwab, C.: Convergence rates of best \(N\)-term Galerkin approximations for a class of elliptic sPDEs. Found. Comput. Math. 10, 615–646 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    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. SIAM/ASA J. Uncertain. Quantif. 3(1), 1109–1135 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Doostan, A., Iaccarino, G.: A least-squares approximation of partial differential equations with high-dimensional random inputs. J. Comput. Phys. 228(12), 4332–4345 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Eigel, M., Pfeffer, M., Schneider, R.: Adaptive stochastic Galerkin FEM with hierarchical tensor representations. Technical report 2015/29, TU Berlin (2015)Google Scholar
  16. 16.
    Espig, M., Hackbusch, W., Litvinenko, A., Matthies, H.G., Wähnert, P.: Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats. Comput. Math. Appl. 67(4), 818–829 (2014)Google Scholar
  17. 17.
    Espig, M., Hackbusch, W., Litvinenko, A., Matthies, H.G., Zander, E.: Efficient analysis of high dimensional data in tensor formats. Sparse Grids and Applications. volume 88 of Lecture Notes in Computational Science and Engineering, pp. 31–56. Springer, Berlin-Heidelberg (2013)Google Scholar
  18. 18.
    Fejér, L.: Mechanische Quadraturen mit positiven Cotesschen Zahlen. Math. Z. 37(1), 287–309 (1933)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Giles, M.: Multilevel Monte Carlo path simulation. Oper. Res. 56(3), 607–617 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Giles, M., Waterhouse, B.: Multilevel quasi-Monte Carlo path simulation. Radon series. Comp. Appl. Math. 8, 1–18 (2009)zbMATHGoogle Scholar
  21. 21.
    Grasedyck, L.: Hierarchical singular value decomposition of tensors. SIAM J. Matrix Anal. Appl. 31, 2029–2054 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Grasedyck, L., Kressner, D., Tobler, C.: A literature survey of low-rank tensor approximation techniques. GAMM-Mitt. 36(1), 53–78 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Griebel, M., Harbrecht, H., Peters, M.: Multilevel quadrature for elliptic parametric partial differential equations on non-nested meshes. arXiv:1509.09058 (2015)
  24. 24.
    Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus. Springer, Berlin (2012)CrossRefzbMATHGoogle Scholar
  25. 25.
    Hackbusch, W., Börm, S.: \({\cal{H}}^2\)-matrix approximation of integral operators by interpolation. Appl. Numer. Math. 43(1–2), 129–143 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hackbusch, W., Kühn, S.: A new scheme for the tensor representation. J. Fourier Anal. Appl. 15(5), 706–722 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Haji Ali, A.L., Nobile, F., Tamellini, L., Tempone, R.: Multi-index stochastic collocation for random pdes. arXiv preprint arXiv:1508.07467 (2015)
  28. 28.
    Harbrecht, H., Peters, M., Siebenmorgen, M.: On multilevel quadrature for elliptic stochastic partial differential equations. In: Garcke, J., Griebel, M. (eds.) Sparse Grids and Applications. Lecture Notes in Computational Science and Engineering, vol. 88, pp. 161–179. Springer, Berlin-Heidelberg (2012)Google Scholar
  29. 29.
    Harbrecht, H., Peters, M., Siebenmorgen, M.: Multilevel accelerated quadrature for PDEs with log-normally distributed diffusion coefficient. SIAM/ASA J. Uncertain. Quantif. 4(1), 520–551 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Heinrich, S.: The multilevel method of dependent tests. In: Advances in stochastic simulation methods (St. Petersburg, 1998), Stat. Ind. Technol., pp. 47–61. Birkhäuser, Boston, (2000)Google Scholar
  31. 31.
    Heinrich, S.: Multilevel Monte Carlo methods. In: Lecture Notes in Large Scale Scientific Computing, pp. 58–67. Springer, London (2001)Google Scholar
  32. 32.
    Hille, E., Phillips, R.S.: Functional Analysis and Semi-Groups, vol. 31. American Mathematical Society, Providence (1957)zbMATHGoogle Scholar
  33. 33.
    Hoang, V.A., Schwab, C.: N-term Wiener chaos approximation rate for elliptic PDEs with lognormal Gaussian random inputs. Math. Models Methods Appl. Sci. 4(24), 797–826 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Khoromskij, B.N., Oseledets, I.: Quantics-TT collocation approximation of parameter-dependent and stochastic elliptic PDEs. Comp. Meth. in Applied Math. 10(4), 376–394 (2010)MathSciNetzbMATHGoogle Scholar
  35. 35.
    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
  36. 36.
    Kressner, D., Tobler, C.: Low-rank tensor Krylov subspace methods for parameterized linear systems. SIAM J. Matrix Anal. Appl. 32(4), 1288–1316 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Kuo, Frances Y., Schwab, Christoph, Sloan, Ian H.: Multi-level quasi-Monte Carlo finite element methods for a class of elliptic PDEs with random coefficients. Found. Comput. Math. 15(2), 411–449 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Lee, K., Elman, H.C.: A preconditioned low-rank projection method with a rank-reduction scheme for stochastic partial differential equations. arXiv:1605.05297 (2016)
  39. 39.
    Matthies, H.G., Zander, E.: Solving stochastic systems with low-rank tensor compression. Linear Algebra Appl. 436(10), 3819–3838 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Nouy, A.: Low-rank methods for high-dimensional approximation and model order reduction. arXiv preprint arXiv:1511.01554 (2015)
  41. 41.
    Nouy, A.: Low-rank tensor methods for model order reduction. arXiv preprint arXiv:1511.01555, 7 (2015)
  42. 42.
    Oseledets, I.V., Tyrtyshnikov, E.E.: TT-cross approximation for multidimensional arrays. Linear Algebra Appl. 432(1), 70–88 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Rivlin, T.J.: The Chebyshev Polynomials. Wiley, Chichester (1974)zbMATHGoogle Scholar
  44. 44.
    Savostyanov, D.V., Oseledets, I.V.: Fast adaptive interpolation of multi-dimensional arrays in tensor train format. In: Proceedings of 7th International Workshop on Multidimensional Systems (nDS). IEEE (2011)Google Scholar
  45. 45.
    Schwab, C., Gittelson, C.J.: Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. Acta Numer. 20, 291–467 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Schwab, C., Todor, R.: Karhunen-Loève approximation of random fields by generalized fast multipole methods. J. Comput. Phys. 217, 100–122 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Teckentrup, A.L., Jantsch, P., Webster, C.G., Gunzburger, M.: A multilevel stochastic collocation method for partial differential equations with random input data. SIAM/ASA J. Uncertain. Quantif. 3(1), 1046–1074 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Tobler, C.: Low-rank Tensor Methods for Linear Systems and Eigenvalue Problems. PhD thesis, ETH Zürich (2012)Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Jonas Ballani
    • 1
  • Daniel Kressner
    • 1
  • Michael D. Peters
    • 2
    Email author
  1. 1.MATHICSE-ANCHP, École Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.Department of Mathematics and Computer ScienceUniversity of BaselBaselSwitzerland

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