# Evaluating integrals over a domain in $R^n$ using space filling curves

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 $f : R^n \rightarrow R$ over a domain $X \subset R^n$. The function $f$ is a joint probability density function (pdf) and its integral over the domain $X$ gives the probability of the region:

$$Prob(x \in X) = \int_{X\subset R^n} f(x) dx$$

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:

$$\int_{X\subset R^n} f(x) dx = \int_a^b f(g(t)) dt$$

where $g : R \rightarrow R^n$ 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?

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This looks pretty relevant. –  Guess who it is. Dec 20 '11 at 8:16