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A Note on the Construction of L-Fold Sparse Tensor Product Spaces

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Abstract

In the present paper, we consider the construction of general sparse tensor product spaces in arbitrary space dimensions when the single subdomains are of different dimensionality and the associated ansatz spaces possess different approximation properties. Our theory extends the results from Griebel and Harbrecht (Math. Comput., 2013) for the construction of two-fold sparse tensor product space to arbitrary L-fold sparse tensor product spaces.

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Notes

  1. Here and subsequently, the summation limits are in general no natural numbers and must of course be rounded properly. We leave this to the reader to avoid cumbersome floor/ceil-notations.

  2. Otherwise, we apply the induction to an appropriate permutation of the spatial dimensions.

  3. Otherwise, we apply the induction to an appropriate permutation of the spatial dimensions.

References

  1. Allaire, G., Briane, M.: Multiscale convergence and reiterated homogenisation. Proc. R. Soc. Edinb. A 126(2), 297–342 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balescu, R.: Statistical Dynamics, Matter Out of Equilibrium. Imperial College Press, London (1997)

    Book  MATH  Google Scholar 

  3. Barrett, J.W., Knezevic, D., Süli, E.: Kinetic models of dilute polymers: analysis, approximation and computation. In: 11th School on Mathematical Theory in Fluid Mechanics, Kacov, Czech Republic, 22–29 May 2009. Necas Center for Mathematical Modeling, Prague (2009)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Chegini, N., Stevenson, R.: Adaptive wavelet schemes for parabolic problems. Sparse matrices and numerical results. SIAM J. Numer. Anal. 49, 182–212 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cioranescu, D., Damlamian, A., Griso, G.: The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40(4), 1585–1620 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cioranescu, D., Damlamian, A., Donato, P., Griso, G., Zaki, R.: The periodic unfolding method in domains with holes. SIAM J. Math. Anal. 44, 718–760 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dahmen, W.: Wavelet and multiscale methods for operator equations. Acta Numer. 6, 55–228 (1997)

    Article  MathSciNet  Google Scholar 

  9. Dung, D.: Continuous algorithms in n-term approximation and non-linear width. Constr. Approx. 102, 217–242 (2000)

    MATH  Google Scholar 

  10. Griebel, M., Hamaekers, J.: Tensor product multiscale many-particle spaces with finite-order weights for the electronic Schrödinger equation. Z. Phys. Chem. 224, 527–543 (2010)

    Article  Google Scholar 

  11. Griebel, M., Harbrecht, H.: On the construction of sparse tensor product spaces. Math. Comp. 82(282), 975–994 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Griebel, M., Knapek, S.: Optimized general sparse grid approximation spaces for operator equations. Math. Comput. 78(268), 2223–2257 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Griebel, M., Oeltz, D.: A sparse grid space-time discretization scheme for parabolic problems. Computing 81(1), 1–34 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Griebel, M., Oswald, P., Schiekofer, T.: Sparse grids for boundary integral equations. Numer. Math. 83(2), 279–312 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Griebel, M., Oeltz, D., Vassilevski, P.: Space-time approximation with sparse grids. SIAM J. Sci. Comput. 28, 701–727 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Harbrecht, H., Li, J.: A fast deterministic method for stochastic elliptic interface problems based on low-rank approximation. Research Report No. 2011-24, Seminar für Angewandte Mathematik, ETH, Zürich (2011)

  17. Harbrecht, H., Schneider, R., Schwab, C.: Sparse second moment analysis for elliptic problems in stochastic domains. Numer. Math. 109, 167–188 (2008)

    Article  MathSciNet  Google Scholar 

  18. Harbrecht, H., Peters, M., Siebenmorgen, M.: Combination technique based k-th moment analysis of elliptic problems with random diffusion. Preprint No. 2011-2, Mathematisches Institut, Universität, Basel, Switzerland (2011)

  19. Hoang, V.H., Schwab, C.: High-dimensional finite elements for elliptic problems with multiple scales. Multiscale Model. Simul. 3, 168–194 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Le Bris, C., Lelièvre, T.: Multiscale modelling of complex fluids: a mathematical initiation. In: Multiscale Modeling and Simulation in Science. Lecture Notes in Computational Science and Engineering, vol. 66, pp. 49–138. Springer, Berlin (2009)

    Chapter  Google Scholar 

  21. Lozinski, A., Owens, R.G., Phillips, T.N.: The Langevin and Fokker–Planck equations in polymer rheology. In: Glowinski, R. (ed.) Handbook of Numerical Analysis XVI/XVII. Elsevier/North-Holland, Amsterdam (2010)

    Google Scholar 

  22. Schwab, C., Stevenson, R.: Space-time adaptive wavelet methods for parabolic evolution problems. Math. Comput. 78(267), 1293–1318 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Schwab, C., Todor, R.-A.: Sparse finite elements for elliptic problems with stochastic loading. Numer. Math. 95(4), 707–734 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  24. Schwab, C., Todor, R.-A.: Sparse finite elements for stochastic elliptic problems. Higher order moments. Computing 71, 43–63 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Widmer, G., Hiptmair, R., Schwab, C.: Sparse adaptive finite elements for radiative transfer. J. Comput. Phys. 227(12), 6071–6105 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yserentant, H.: Sparse grid spaces for the numerical solution of the electronic Schrödinger equation. Numer. Math. 101, 381–389 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zeiser, A.: Wavelet approximation in weighted Sobolev spaces of mixed order with applications to the electronic Schrödinger equation. Constr. Approx. 35, 293–322 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zenger, C.: Sparse grids. In: Parallel Algorithms for Partial Differential Equations, Kiel, 1990. Notes Numer. Fluid Mech., vol. 31, pp. 241–251. Vieweg, Braunschweig (1991)

    Google Scholar 

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Authors and Affiliations

  1. Institut für Numerische Simulation, Universität Bonn, Wegelerstr. 6, 53115, Bonn, Germany

    Michael Griebel

  2. Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051, Basel, Switzerland

    Helmut Harbrecht

Authors
  1. Michael Griebel
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  2. Helmut Harbrecht
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Correspondence to Helmut Harbrecht.

Additional information

Communicated by Wolfgang Dahmen.

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Griebel, M., Harbrecht, H. A Note on the Construction of L-Fold Sparse Tensor Product Spaces. Constr Approx 38, 235–251 (2013). https://doi.org/10.1007/s00365-012-9178-7

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  • DOI: https://doi.org/10.1007/s00365-012-9178-7

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