Is there a methodology to simplify evaluation of multi-dimensional integrals using space-filling curves parameterized by a scalar parameter?
I am interested in evaluating the integral of a function $f :\mathbb{R}^n \rightarrow \mathbb{R}$ over a domain $X \subset \mathbb{R}^n$. The function $f$ is a joint probability density function (pdf) and its integral over the domain $X$ gives the probability of the region:
$$ \text{Prob}(x \in X) = \int_{X\subset \mathbb{R}^n} f(x) dx$$
I would like to use a Monte Carlo approach to evaluate the integral. However, instead of sampling the multi-dimensional space, I want to be able to use a space-filling curve and sample a scalar parameter to obtain:
$$ \int_{X\subset \mathbb{R}^n} f(x) dx = \int_a^b f(g(t)) dt,$$
where $g : \mathbb{R} \rightarrow \mathbb{R}^n$ and $t \in [a, b]$.
Can one construct such a function $g(t)$ and get a set of samples of $t$ over a finite range $[a,b]$ to approximate the probability integral?
