Abstract
The sparse grid approach uses basis functions which are tensor products of univariate basis functions. This gives a connection between sparse grids and the tensor space setting. On the other hand, in the tensor calculus one is interested in suitable representations of tensors (cf. Hackbusch, Tensor spaces and numerical tensor calculus, Monograph in preparation). Then the storage size is in particular described by certain rank parameters. Limited rank requires an approximation of the tensors (functions). In particular, function approximations like polynomial interpolation may lead to such representations. Here, we exploit the sparse grid approximation. We show that the involved rank is not the sparse grid complexity \(O(N{\log }^{d-1}N),\) but \(O({N{}^{(d-1)/d}\log }^{d-3}N).\)
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Notes
- 1.
For simplicity, we assume r j  = r for all directions j. 
- 2.
For simplicity, we assume that \(\dim ({V }_{(\lambda )}) = {n}_{\lambda } = {2}^{\lambda }{n}_{0}\). Depending on the boundary condition, the true number can be n λ − 1 or n λ + 1. 
- 3.
When we consider (11) as r-term representation, we ignore that b i for different i may still involve identical \({b}_{{i}_{j}}.\) On the other hand, the factor a[i] may be included into the first factor \({b}_{{i}_{1}}.\) Hence, the storage requirement is \(\#{\mathbf{I}}_{\mathrm{sparse}} \cdot d \cdot {n}_{\mathcal{l}}\).
- 4.
Since the sets
may overlap, one must take care that each v i is associated to only one
- 5.
\({\tau }_{d}(N) < d \cdot \#T\) may occur, since
may hold for \(k\neq {k}^{{\prime}}.\) An example is the decomposition from Fig. 1, where τ2(16) = 7. 
- 6.
\({\prod }_{j\neq m}\) abbreviates the product over \(\{1,\ldots ,d - 1\}\setminus \{m\}.\)
References
Braess, D. and Hackbusch, W.: On the efficient computation of high-dimensional integrals and the approximation by exponential sums. In: Multiscale, nonlinear and adaptive approximation (R. DeVore, A. Kunoth, eds.), Springer Berlin 2009, pp. 39–74.
Bungartz, H.J., Griebel, M.: Sparse grids. Acta Numerica 13, 147–269 (2004)
De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21, 1253–1278 (2000)
De Silva, V., Lim, L.H.: Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 30, 1084–1127 (2008)
M. Espig: Effiziente Bestapproximation mittels Summen von Elementartensoren in hohen Dimensionen. Doctoral thesis, University Leipzig, 2008
W. Hackbusch: Tensor spaces and numerical tensor calculus. Springer Berlin, 2012.
Khoromskij, B., Khoromskaia, V.: Multigrid accelerated tensor approximation of function related multidimensional arrays. SIAM J. Sci. Comput. 31, 3002–3026 (2009)
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Hackbusch, W. (2012). The Use of Sparse Grid Approximation for the r-Term Tensor Representation. In: Garcke, J., Griebel, M. (eds) Sparse Grids and Applications. Lecture Notes in Computational Science and Engineering, vol 88. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31703-3_7
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DOI: https://doi.org/10.1007/978-3-642-31703-3_7
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