I was doing a physical problem, and then it comes to this Gaussian integral. The dimension of the integral is very large (dimension = 300~600), and it is difficult to find the maximum of the integrand. Is it possible to use Monte Carlo technique to do this integration?


To make the problem more specific, I write it in the following. I need to calculate C = A/B, where $B = \int d\bar{\phi\_i}d\phi\_i e^{-\phi\_i^\*\*\phi\_i}f(\phi\_i^\*,\phi\_i)$, $A = \int d\bar{\phi\_i}d\phi\_i e^{-\phi\_i^\*\*\phi\_i}f(\phi\_i^\*,\phi\_i)\*H(\phi\_i^\*,\phi\_i)$. The number of $\phi\_i$, i.e., the dimension of the integration, can change from $10^2$ to $10^3$. The function $f(\phi\_i^\*,\phi\_i)$ is positive definite, however it is difficult to find the maximum of $e^{-\phi\_i^\*\*\phi\_i}f(\phi\_i^\*,\phi\_i)$ and $f$ itself is very complex.


The problem is much like the original problem considered by Metropolis, who invented importance sampling with otheres. Is it really possible to calculate by Monte Carlo directly?



Monte Carlo integration of Gaussian integrals
If you are computing an integral, and it has a single sharp peak, then the simplest solution is to use the [Laplace approximation](http://en.wikipedia.org/wiki/Laplace%27s_method), which provides a good approximation for this exact type of problem. Monte Carlo methods in general have trouble with sharply-peaked integrands unless they start near the peak.


