I am interested in the Hyperasymptotics of multidimensional integrals of the form
$$\mathcal{I}(\lambda) = \int_{\mathbb{R}^n} dz_1 \wedge dz_2 \wedge \dotsi \wedge dz_n \, g(z_1,\dotsi,z_n) \, e^{\frac{i}{\lambda}f(z_1,\dotsi,z_n)}.$$
Picard-Lefschetz theory suggests that we should analytically continue functions f and g and to deform the original contour of integration into a sum of Lefschetz thimbles emerging from "relevant" saddle points (essentially a sum over the steepest descent surfaces emerging form saddle points (both real and imaginary) which their steepest ascent contours intersect the original contour of integration). Then, each of these integrals will have exponentially small non-perturbative contributions from other saddle points and the non-perturbative correction to the original integral comes from the sum of these non-perturbative corrections to each integral over a Lefschetz thimble. But, as the dimension, $n$, grows the number of saddle points also grows and it quickly becomes impractical to find all the relevant saddles, their Lefschetz thimble, and the non-perturbative correction to them.
My question is would it be possible to single out a few saddles that give the largest contribution to the non-perturbative correction of the integral over the original contour of integration ($\mathbb{R}^n$)?
Howls mentions in his paper "Hyperasymptotics for multidimensional integrals, exact remainder terms and the global connection problem" that there have been some advances in this direction by Kaminski and Paris (Kaminski, D. & Paris, R. B. 1997 Preprints from University of Abertay Dundee, UK.) but this reference cannot be found online.