  We introduce and study a novel tensor approach for fast and accurate
assembled summation of a large number of lattice-allocated potentials
represented on 3D $N\times N \times N$ grid with the computational requirements
only \emph{weakly dependent} on the number of summed potentials. It is based on
the assembled low-rank canonical tensor representations of the collected
potentials using pointwise sums of shifted canonical vectors representing the
single generating function, say the Newton kernel. For a sum of electrostatic
potentials over $L\times L \times L$ lattice embedded in a box the required
storage scales linearly in the 1D grid-size, $O(N )$, while the numerical cost
is estimated by $O(N L)$. For periodic boundary conditions, the storage demand
remains proportional to the 1D grid-size of a unit cell, $n=N/L$, while the
numerical cost reduces to $O(N)$, that outperforms the FFT-based Ewald-type
summation algorithms of complexity $O(N^3 \log N)$. The complexity in the grid
parameter $N$ can be reduced even to the logarithmic scale $O(\log N)$ by using
data-sparse representation of canonical $N$-vectors via the quantics tensor
approximation. For justification, we prove an upper bound on the quantics ranks
for the canonical vectors in the overall lattice sum. The presented approach is
beneficial in applications which require further functional calculus with the
lattice potential, say, scalar product with a function, integration or
differentiation, which can be performed easily in tensor arithmetics on large
3D grids with 1D cost. Numerical tests illustrate the performance of the tensor
summation method and confirm the estimated bounds on the tensor ranks.
