  For generic Hamiltonian systems we derive predictions for dynamical tunneling
from regular to chaotic phase-space regions. In contrast to previous
approaches, we account for the resonance-assisted enhancement of
regular-to-chaotic tunneling in a non-perturbative way. This provides the
foundation for future semiclassical complex-path evaluations of
resonance-assisted regular-to-chaotic tunneling. Our approach is based on a new
class of integrable approximations which mimic the regular phase-space region
and its dominant nonlinear resonance chain in a mixed regular--chaotic system.
We illustrate the method for the standard map.
