Computing weak distances between the 2-sphere and its nonsmooth approximations

A novel algorithm is proposed for quantitative comparisons between compact surfaces embedded in the three-dimensional Euclidian space. The key idea is to identify those objects with the associated surface measures and compute distances between them using the Fourier transform on the ambient space. In particular, the inhomogeneous Sobolev norms of negative order are approximated from data in the frequency space, which amounts to comparing measures after appropriate smoothing. Such Fourier-based distances allow several advantages including high accuracy due to fast-converging numerical quadrature rules, acceleration by the nonuniform fast Fourier transform, parallelization on massively parallel architectures. In numerical experiments, the 2-sphere, which is an example whose Fourier transform is explicitly known, is compared with its icosahedral discretization, and it is observed that the piecewise linear approximations converge to the smooth object at the quadratic rate up to small truncations.