We study polynomial interpolation on a d-dimensional cube, where d
is large. We suggest to use the least solution at sparse grids with
the extrema of the Chebyshev polynomials.
The polynomial exactness of this method is almost optimal.
Our error bounds show that the method is universal, i.e.,
almost optimal for many different function spaces.
We report on numerical experiments for d=10 using up
to 652 065 interpolation points.
Submitted for publication.