  We are interested in mesh-free formulas based on the Monte-Carlo methodology
for the approximation of multi-dimensional integrals, and we investigate their
accuracy when the functions belong to a reproducing-kernel space. A kernel
typically captures regularity and qualitative properties of functions "beyond"
the standard Sobolev regularity class. We are interested in the issue whether
quantitative error bounds can be a priori guaranteed in applications (e.g.
mathematical finance but also scientific computing and machine learning). Our
main contribution is a numerical study of the error discrepancy function based
on a comparison between several numerical strategies, when one varies the
choice of the kernel, the number of approximation points, and the dimension of
the problem. We consider two strategies in order to localize to a bounded set
the standard kernels defined in the whole Euclidian space (exponential,
multiquadric, Gaussian, truncated), namely, on one hand the class of periodic
kernels defined via a discrete Fourier transform on a lattice and, on the other
hand, a class of transport-based kernels. First of all, relying on a Poisson
formula on a lattice, together with heuristic arguments, we discuss the
derivation of theoretical bounds for the discrepancy function of periodic
kernels. Second, for each kernel of interest, we perform the numerical
experiments that are required in order to generate the optimal distributions of
points and the discrepancy error functions. Our numerical results allow us to
validate our theoretical observations and provide us with quantitative
estimates for the error made with a kernel-based strategy as opposed to a
purely random strategy.
