## Multidimensional integrals in ion-atom collisions

19/03/2014

Simulation in physics is full of multidimensional integrals. In the paper M. S. Litsarev and I. V. Oseledets. Low rank approximations for the DEPOSIT computer code. arXiv preprint 1403.4068, 2014 we consider the DEPOSIT code that is suited for the computation of ion-atomic collisons. The main computational bottleneck is the computation of a three-dimensional integral. We exploit its structure by computing low-rank separated representation using cross approximation for a two-dimensional function, and exponential sums approach for the Slater density function. Implementation of this technique decreases the total computational time by a factor of 1000. What is more important, the general concept can be applied to more complicated models (like ion-molecule collision).

$\ \begin{array}{cccccc} \\hline \\hline System & \\gamma\mbox{-Shell} & N_w & T_s ( \\times10^{-3} sec) & T_{D} (sec) & T_{D}/T_s \\\\ \\hline Au^{26+}+ O & 4df^{17} & 74 & 7.94 & 3.89 & 490 \\\\ & 4sp^{8} & 69 & 4.92 & 3.83 & 778 \\\\ & 3d^{10} & 73 & 3.59 & 3.88 & 1080 \\\\ & 3sp^{8} & 72 & 3.81 & 3.82 & 1003 \\\\ & 2sp^{8} & 107 & 2.42 & 3.86 & 1592 \\\\ & 1sp^{2} & 209 & 1.24 & 3.88 & 3120 \\\\ \\hline U^{28+}+ Xe & 5sp^{4} & 62 & 10.1 & 3.94 & 390 \\\\ & 4df^{24} & 70 & 6.05 & 3.90 & 644 \\\\ & 4sp^{8} & 67 & 5.00 & 3.94 & 788 \\\\ & 3d^{10} & 71 & 3.88 & 3.92 & 1011 \\\\ & 3sp^{8} & 70 & 3.52 & 3.90 & 1106 \\\\ & 2sp^{8} & 105 & 1.99 & 3.87 & 1945 \\\\ & 1sp^{2} & 207 & 1.04 & 3.88 & 3723 \\\\ \\hline \\hline \\end{array}$

Time for the old code ($$T_D)$$ versus the time for the new code ($$T_s$$).

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