Karhunen-Loéve Decomposition of Isotropic Gaussian Random Fields Using a Tensor Approximation of Autocovariance Kernel

  • Michal BérešEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11087)


Applications of random fields typically require a generation of random samples or their decomposition. In this contribution, we focus on the decomposition of the isotropic Gaussian random fields on a two or three-dimensional domain. The preferred tool for the decomposition of the random field is the Karhunen-Loéve expansion. The Karhunen-Loéve expansion can be approximated using the Galerkin method, where we encounter two main problems. First, the calculation of each element of the Galerkin matrix is expensive because it requires an accurate evaluation of multi-dimensional integral. The second problem consists of the memory requirements, originating from the density of the matrix. We propose a method that overcomes both problems. We use a tensor-structured approximation of the autocovariance kernel, which allows its separable representation. This leads to the representation of the matrix as a sum of Kronecker products of matrices related to the one-dimensional problem, which significantly reduces the storage requirements. Moreover, this representation dramatically reduces the computation cost, as we only calculate two-dimensional integrals.


Random fields sampling Karhunen-Loève decomposition Tensor approximation Numerical integration 



This work was supported by The Ministry of Education, Youth and Sports from the National Programme of Sustainability (NPU II) project “IT4Innovations excellence in science - LQ1602”. The work was also partially supported by the project LD15105 “Ultrascale computing in geo-sciences”. The authors were also supported by Grant of SGS No. SP2017/56, VŠB - Technical University of Ostrava, Czech Republic.


  1. 1.
    Lord, G.J., Powell, C.E., Shardlow, T.: An Introduction to Computational Stochastic PDEs. Cambridge University Press, New York (2014)CrossRefGoogle Scholar
  2. 2.
    Aune, E., Eidsvik, J., Pokern, Y.: Iterative numerical methods for sampling from high dimensional Gaussian distributions. Stat. Comput. 23(4), 501–521 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chow, E., Saad, Y.: Preconditioned Krylov subspace methods for sampling multivariate Gaussian distributions. SIAM J. Sci. Comput. 36(2), A588–A608 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Mrovec, M.: Tensor approximation of Slater-type orbital basis functions. Adv. Electr. Electron. Eng. 15(2), 314–321 (2017)zbMATHGoogle Scholar
  5. 5.
    Mrovec, M.: Low-rank tensor representation of Slater-type and hydrogen-like orbitals. Appl. Math. 62(6), 679–698 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Duffy, M.G.: Quadrature over a pyramid or cube of integrands with a singularity at a vertex. SIAM J. Numer. Anal. 19(6), 1260–1262 (1982)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hoeksema, R.J., Kitanidis, P.K.: Analysis of the spatial structure of properties of selected aquifers. Water Resour. Res. 21(4), 563–572 (1985)CrossRefGoogle Scholar
  8. 8.
    Light, W.A., Cheney, E.W.: Approximation Theory in Tensor Product Spaces. Springer, New York (1985). Scholar
  9. 9.
    Da Prato, G., Zabczyk, J.: Second Order Partial Differential Equations in Hilbert Spaces. Cambridge University Press, Cambridge (2002)CrossRefGoogle Scholar
  10. 10.
    Oliveira, S.P., Wisniewski, F., Azevedo, J.S.: A wavelet Galerkin approximation of Fredholm integral eigenvalue problems with bidimensional Haar functions. Proc. Ser. Braz. Soc. Comput. Appl. Math. 2(1), 010060-1–010060-6 (2014)Google Scholar
  11. 11.
    Blaheta, R., Béreš, M., Domesová, S.: A study of stochastic FEM method for porous media flow problem. In: Applied Mathematics in Engineering and Reliability: Proceedings of the 1st International Conference on Applied Mathematics in Engineering and Reliability, p. 281 (2016)CrossRefGoogle Scholar
  12. 12.
    Béreš, M., Domesová, S.: The stochastic Galerkin method for Darcy flow problem with log-normal random field coefficients. Adv. Electr. Electron. Eng. 15(2), 267–279 (2017)Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Electrical Engineering and Computer ScienceVŠB - Technical University of OstravaOstravaCzech Republic
  2. 2.IT4Innovations National Supercomputing CenterVŠB - Technical University of OstravaOstravaCzech Republic
  3. 3.Institute of Geonics of the CASOstravaCzech Republic

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