Foundations of Computational Mathematics

, Volume 16, Issue 4, pp 813–874 | Cite as

Tensor-Sparsity of Solutions to High-Dimensional Elliptic Partial Differential Equations

  • Wolfgang Dahmen
  • Ronald DeVore
  • Lars Grasedyck
  • Endre Süli


A recurring theme in attempts to break the curse of dimensionality in the numerical approximation of solutions to high-dimensional partial differential equations (PDEs) is to employ some form of sparse tensor approximation. Unfortunately, there are only a few results that quantify the possible advantages of such an approach. This paper introduces a class \(\Sigma _n\) of functions, which can be written as a sum of rank-one tensors using a total of at most \(n\) parameters, and then uses this notion of sparsity to prove a regularity theorem for certain high-dimensional elliptic PDEs. It is shown, among other results, that whenever the right-hand side \(f\) of the elliptic PDE can be approximated with a certain rate \(\mathcal {O}(n^{-r})\) in the norm of \({\mathrm H}^{-1}\) by elements of \(\Sigma _n\), then the solution \(u\) can be approximated in \({\mathrm H}^1\) from \(\Sigma _n\) to accuracy \(\mathcal {O}(n^{-r'})\) for any \(r'\in (0,r)\). Since these results require knowledge of the eigenbasis of the elliptic operator considered, we propose a second “basis-free” model of tensor-sparsity and prove a regularity theorem for this second sparsity model as well. We then proceed to address the important question of the extent to which such regularity theorems translate into results on computational complexity. It is shown how this second model can be used to derive computational algorithms with performance that breaks the curse of dimensionality on certain model high-dimensional elliptic PDEs with tensor-sparse data.


High-dimensional elliptic PDEs Tensor-sparsity models Regularity theorems Exponential sums of operators Dunford integral Complexity bounds 

Mathematics Subject Classification

35J25 41A25 41A63 41A46 65D99 


  1. 1.
    M. Bachmayr, Adaptive Low-Rank Wavelet Methods and Applications to Two-Electron Schrödinger Equations, PhD Thesis, RWTH Aachen, 2012.Google Scholar
  2. 2.
    M. Bachmayr and W. Dahmen, Adaptive near-optimal rank tensor approximation for high-dimensional operator equations, Foundations of Computational Mathematics. doi: 10.1007/s10208-013-9187-3,
  3. 3.
    D. Bini, M. Capovani, G. Lotti, and F. Romani, \(O(n^{2.7799})\) complexity for \(n\times n\) approximate matrix multiplication. Inform. Process. Lett. 8 (1979), 234–235.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    D. Braess, Nonlinear Approximation Theory, Springer-Verlag, Berlin, 1986.CrossRefzbMATHGoogle Scholar
  5. 5.
    D. Braess and W. Hackbusch, Approximation of \(1/x\) by exponential sums in \( [1,\infty )\), IMA Journal of Numerical Analysis, 25 (2005), 685–697.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    D. Braess and W. Hackbusch, On the efficient computation of high-dimensional integrals and the approximation by exponential sums. In: Multiscale, Nonlinear and Adaptive Approximation, R. DeVore and A. Kunoth, Eds. Springer, Berlin Heidelberg, 2009.Google Scholar
  7. 7.
    H. Brezis, Analyse fonctionnelle: Théorie et applications. Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.Google Scholar
  8. 8.
    A. Cohen, R. DeVore, C. Schwab, Analytic Regularity and Polynomial Approximation of Parametric Stochastic Elliptic PDEs, Analysis and Applications, 9(2011), 11–47.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    A. Cohen, R. DeVore, C. Schwab, Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs, Foundations of Computational Mathematics, 10 (2010), 615–646.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    W. Dahmen and M. Jürgens, Error controlled regularization by projection. ETNA, 25(2006), 67–100.MathSciNetzbMATHGoogle Scholar
  11. 11.
    R. DeVore, Nonlinear approximation. Acta Numerica, 7(1998), 51–150.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    T.J. Dijkema, Ch. Schwab, R. Stevenson, An adaptive wavelet method for solving high-dimensional elliptic PDEs, Constructive Approximation, 30, (3) (2009), 423–455.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    M. Espig, Effiziente Bestapproximation mittels Summen von Elementartensoren in hohen Dimensionen. Doctoral thesis, Univ. Leipzig (2007).Google Scholar
  14. 14.
    L. Figueroa and E. Süli, Greedy approximation of high-dimensional Ornstein-Uhlenbeck operators. Foundations of Computational Mathematics, 12(2012), 573–623.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    L. Figueroa and E. Süli, Greedy approximation of high-dimensional Ornstein–Uhlenbeck operators. arxiv:1103.0726v1 [math.NA]. Available from:
  16. 16.
    I.P. Gavrilyuk, W. Hackbusch and B.N. Khoromskij, Hierarchical Tensor-Product Approximation to the Inverse and Related Operators for High-Dimensional Elliptic Problems. Computing 74 (2005), 131–157.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    I.P. Gavrilyuk, W. Hackbusch and B.N. Khoromskij, Data-sparse approximation of a class of operator-valued functions. Math. Comp. 74 (2005), 681–708.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    L. Grasedyck, Existence and Computation of a Low Kronecker-Rank Approximant to the Solution of a Tensor System with Tensor Right-Hand Side. Computing 72 (2004), 247–265.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    L. Grasedyck, Hierarchical Singular Value Decomposition of Tensors. SIAM J. Matrix Anal. Appl. 31 (2010), 2029–2054.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    P. Grisvard, Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics, 24. Pitman (Advanced Publishing Program), Boston, MA, 1985.Google Scholar
  21. 21.
    W. Hackbusch and B.N. Khoromskij, Low-rank Kronecker-product approximation to multi-dimensional nonlocal operators. Part I. Separable approximation of multi-variate functions, Computing, 76 (2006), 177–202.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    W. Hackbusch and S. Kühn, A New Scheme for the Tensor Representation. J. Fourier Anal. Appl. 15 (2009), 706–722.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    J.-B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of convex analysis. Grundlehren Text Editions. Springer-Verlag, Berlin, 2001.Google Scholar
  24. 24.
    M. Jürgens, A Semigroup Approach to the Numerical Solution of Parabolic Differential Equations. Ph.D. thesis, RWTH Aachen, 2005.Google Scholar
  25. 25.
    M. Jürgens, Adaptive application of the operator exponential, submitted to J. Numer. Math., special issue on Breaking Complexity: Multiscale Methods for Efficient PDE Solvers.Google Scholar
  26. 26.
    B.N. Khoromskij, Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in \({\mathbb{R}}^d\). Constr. Approx. 30 (2009), 599–620.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    W.P. Krijnen, T.K. Dijkstra, and A. Stegeman, On the non-existence of optimal solutions and the occurrence of degeneracy in the Candecomp/Parafac model. Psychometrika 73 (2008), 431–439.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    R. Kress, Linear Integral Equations. Springer Verlag, Berlin, 1999.CrossRefzbMATHGoogle Scholar
  29. 29.
    V. De Silva and L.-H. Lim, Tensor rank and the ill-posedness of the best low-rank approximation problem, SIAM J. Matrix Anal. Appl. 30 (2008), 1084–1127.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    E. Novak and H. Woźniakowski, Tractability of Multivariate Problems, Volume I: Linear Information, EMS Tracts in Mathematics 6, EMS Publ. House, Zürich, 2008.Google Scholar
  31. 31.
    E. Novak and H. Woźniakowski, Approximation of infinitely differentiable multivariate functions is intractable, J. Complexity 25 (2009), 398–404.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Second Edition. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980.Google Scholar
  33. 33.
    C. Schwab and E. Süli, Adaptive Galerkin approximation algorithms for Kolmogorov equations in infinite dimensions, Stochastic Partial Differential Equations: Analysis and Computations 1(1) (2013), 204–239.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    W. Sickel and T. Ullrich, Tensor products of Sobolev-Besov spaces and applications to approximation from the hyperbolic cross. J. Approx. Theory. 161 (2009) 748–786.MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    F. Stenger, Numerical Methods based on Sinc and Analytical Functions. Springer-Verlag, New York, 1993.CrossRefzbMATHGoogle Scholar
  36. 36.
    A.G. Werschulz and H. Woźniakowski, Tight tractability results for a model second-order Neumann problem, Foundations of Computational Mathematics (2014). doi: 10.1007/s10208-014-9195-y.
  37. 37.
    E. Zeidler, Applied Functional Analysis. Applications to Mathematical Physics. Applied Mathematical Sciences, 108, Springer-Verlag, New York, 1995.Google Scholar

Copyright information

© SFoCM 2015

Authors and Affiliations

  • Wolfgang Dahmen
    • 1
  • Ronald DeVore
    • 2
  • Lars Grasedyck
    • 1
  • Endre Süli
    • 3
  1. 1.Institut für Geometrie und Praktische MathematikRWTH AachenAachenGermany
  2. 2.Department of MathematicsTexas A&M UniversityTXUSA
  3. 3.Mathematical InstituteUniversity of OxfordOxfordUK

Personalised recommendations