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Efficient Implementation of a Smolyak Sparse-grid Scheme with Non-nested Grids Lauvergnat, David


The use of Smolyak sparse grids, $G_{L_s}^{SG}$, in quantum dynamics is an approach that pushes the limits of the current calculations based on a multidimensional grid. The main idea is to substitute a single large direct-product grid by a sum of small direct-product grids (See Eq.1). [1] The number of terms in the sum may be large and is controlled through the Smolyak parameter, $L_s$. In this context, it can be used to perform multidimensional integrals or directly as a multidimensional interpolation scheme. \begin{equation} G_{L_s}^{SG}=\sum_{\mathbf{L}= \left[ L^1,L^2 \cdots L^n \right]}^{L_s-n+1\leqslant |\mathbf{L} |\leqslant L_s} (-1)^{L_s-|\mathbf{L}|}.C_{n-1}^{L_s-|\mathbf{L}|}.G_{L^1}^1 \otimes G_{L^2}^2 \cdots \otimes G_{L^n}^n \end{equation} In quantum dynamics, Avila and Carrington were the pioneers in developing efficient implementations [2-5] and they used a Smolyak grid scheme with 1D-nested grids ($G_{L^i}^i$), which not only reduces the total number of grid points but also avoids the need to perform the calculations on each direct-product grid ($G_{L^1}^1 \otimes G_{L^2}^2 \cdots \otimes G_{L^n}^n$) in the sum (Eq. 1). Our previous implementation[6] avoided the need of 1D-nested grids and was extended to nD-Grids (the $G_{L^i}^i$ could be multidimensional grids). However, it required the explicit summation over direct-product grids (see Eq. 1) so that some identical calculations were performed several times. This had an unfavorable impact on the numerical efficiency, without, however, preventing large calculations from being performed. Recently, we have introduced a new implementation, in which we keep the sums of the direct product grids but perform the calculations sequentially over the various dimensions. Consequently, the repetition of identical calculations is avoided, which considerably improves the numerical efficiency. [1] S. A. Smolyak, Soviet Mathematics Doklady {\bf 4} 240 (1963). [2] G. Avila and T. Carrington, J. Chem. Phys. {\bf 131} 174103 (2009). [3] G. Avila and T. Carrington, J. Chem. Phys. {\bf 134} 054126 (2011). [4] G. Avila and T. Carrington, J. Chem. Phys. {\bf 135} 064101 (2011). [5] G. Avila and T. Carrington, J. Chem. Phys. {\bf 139} 134114 (2013). [6] D.~Lauvergnat and A.~Nauts, Spectrochim. Acta Part A {\bf 119} 18 (2014).

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