Approximation of High-Dimensional Rank One Tensors

Abstract

Many real world problems are high-dimensional in that their solution is a function which depends on many variables or parameters. This presents a computational challenge since traditional numerical techniques are built on model classes for functions based solely on smoothness. It is known that the approximation of smoothness classes of functions suffers from the so-called ‘curse of dimensionality’. Avoiding this curse requires new model classes for real world functions that match applications. This has led to the introduction of notions such as sparsity, variable reduction, and reduced modeling. One theme that is particularly common is to assume a tensor structure for the target function. This paper investigates how well a rank one function f(x 1,…,x d )=f 1(x 1)⋯f d (x d ), defined on Ω=[0,1]d can be captured through point queries. It is shown that such a rank one function with component functions f j in \(W^{r}_{\infty}([0,1])\) can be captured (in L ) to accuracy O(C(d,r)N r) from N well-chosen point evaluations. The constant C(d,r) scales like d dr. The queries in our algorithms have two ingredients, a set of points built on the results from discrepancy theory and a second adaptive set of queries dependent on the information drawn from the first set. Under the assumption that a point zΩ with nonvanishing f(z) is known, the accuracy improves to O(dN r).

This is a preview of subscription content, access via your institution.

Algorithm 1
Algorithm 2

References

  1. 1.

    Bungartz, H.-J., Griebel, M.: Sparse grids. Acta Numer. 13, 147–269 (2004)

    Article  MathSciNet  Google Scholar 

  2. 2.

    Dumitrescu, A., Jiang, M.: On the largest empty axis-parallel box amidst n points. Algorithmica (2012). doi:10.1007/s00453-012-9635-5

    MathSciNet  Google Scholar 

  3. 3.

    Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus. Springer Series in Computational Mathematics, vol. 42. Springer, Berlin (2012)

    Google Scholar 

  4. 4.

    Halton, J.H.: On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer. Math. 2, 84–90 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  5. 5.

    Hammersley, J.M.: Monte Carlo methods for solving multivariable problems. Ann. N.Y. Acad. Sci. 86, 844–874 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems, Volume I: Linear Information. EMS Tracts in Mathematics, vol. 6. Eur. Math. Soc., Zurich (2008)

    Google Scholar 

  7. 7.

    Novak, E., Woźniakowski, H.: Approximation of infinitely differentiable multivariate functions is intractable. J. Complex. 25, 398–404 (2009)

    Article  MATH  Google Scholar 

  8. 8.

    Rote, G., Tichy, R.F.: Quasi-Monte-Carlo methods and the dispersion of point sequences. Math. Comput. Model. 23, 9–23 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    Sickel, W., Ullrich, T.: Tensor products of Sobolev–Besov spaces and applications to approximation from the hyperbolic cross. J. Approx. Theory 161, 748–786 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Smolyak, S.A.: Quadrature and interpolation formulas tensor products of certain classes of functions. Sov. Math. Dokl. 4, 240–243 (1963)

    Google Scholar 

  11. 11.

    Temlyakov, V.: Approximation of Periodic Functions. Nova Science Publishers, New York (1993)

    Google Scholar 

Download references

Acknowledgements

This research was supported by the Office of Naval Research Contracts ONR N00014-08-1-1113, ONR N00014-09-1-0107, and ONR N00014-11-1-0712; the AFOSR Contract FA95500910500; the NSF Grants DMS-0810869, and DMS 0915231; and the DFG Special Priority Program SPP-1324. This research was done when R.D. was a visiting professor at RWTH and the AICES Graduate Program. This publication is based in part on work supported by Award No. KUS-C1-016-04 made by King Abdullah University of Science and Technology (KAUST).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Wolfgang Dahmen.

Additional information

Communicated by Vladimir N. Temlyakov.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bachmayr, M., Dahmen, W., DeVore, R. et al. Approximation of High-Dimensional Rank One Tensors. Constr Approx 39, 385–395 (2014). https://doi.org/10.1007/s00365-013-9219-x

Download citation

Keywords

  • Query algorithms
  • High-dimensional approximation
  • Separable functions
  • Rate of approximation

Mathematics Subject Classification (2000)

  • 41A25
  • 65D15