# Monte Carlo integration of Gaussian integrals

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?

• low discrepancy methods have better rate of convergence; look for Halton Sequence and Integration. – Manfred Weis Jan 27 '15 at 15:03
• The article "High-dimensional integration: The quasi-Monte Carlo way" gives a variety of low discrepancy approaches. But without knowing where the maximum of the integrand lies, it is difficult to be confident. – user25199 Jan 27 '15 at 15:46
• If you know the integrand is Gaussian, why not do it exactly? Can you provide a more specific formulation of the problem you're asking? – josh Jan 27 '15 at 16:54
• I have updated the problem. Thank josh. – Jiyao Chen Jan 28 '15 at 2:05
• What did you do to find the maximum? If the function is sharply-peaked, then it's hard for Monte Carlo methods to find the maximum. – arsmath Jan 28 '15 at 12:18

## 2 Answers

If you are computing an integral, and it has a single sharp peak, then the simplest solution is to use the Laplace approximation, 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.

I am investigating the Monte Carlo integration method to calculate an ill-behaved integral such as Gaussian integral. A general Monte Carlo integration fails to calculate integrals that have the feature like the Gaussian integral.

I think you can use the Wang-Landau sampling for numerical integration. The algorithm is efficient to calculate the ill-behaved integral and higher dimension integral. Now, it's improve version is 1/t algorithm.