I am looking for approaches to transform a more complicated stochastic volatility model such as the one shown in Section 2.2 of Smile Dynamics II to a single-factor model such as the one shown in Section 2.1 of the same paper or some kind of version of Heston. I am interested in this question because I want to do a Monte Carlo based calibration and my model (which has multiple factors) is too slow for this. A reasonable approximation with a simpler (one-factor) model would be much faster and would be acceptable in this context.
In more detail, I am mostly interested in a one-factor approximation of two-factor stochastic volatility models, such that most of the volatility dynamics behaviours modelled by the two-factor model are accurately reprices by the one-factor one. A one factor would have a single Brownian motion driving the stochastic volatility, whereas and $n$-factor model would have $n$ Brownian motions driving the stochastic volatiity.
Have such questions been studied in the literature? Have such questions been studied in some generality or only for some types of models?
I could not find any reference until now, but I am sure there must be such approaches out there.