  Motivated by the recent multilevel sparse kernel-based interpolation (MuSIK)
algorithm proposed in [Georgoulis, Levesley and Subhan, SIAM J. Sci. Comput.,
35(2), pp. A815-A831, 2013], we introduce the new quasi-multilevel sparse
interpolation with kernels (Q-MuSIK) via the combination technique. The Q-MuSIK
scheme achieves better convergence and run time in comparison with classical
quasi-interpolation; namely, the Q-MuSIK algorithm is generally superior to the
MuSIK methods in terms of run time in particular in high-dimensional
interpolation problems, since there is no need to solve large algebraic
systems.
  We subsequently propose a fast, low complexity, high-dimensional quadrature
formula based on Q-MuSIK interpolation of the integrand. We present the results
of numerical experimentation for both interpolation and quadrature in high
dimension.
