High dimensional polynomial interpolation on sparse grids

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 652065 interpolation points.