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Concept
Monte Carlo integration
Summary
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher-dimensional integrals.
There are different methods to perform a Monte Carlo integration, such as uniform sampling, stratified sampling, importance sampling, sequential Monte Carlo (also known as a particle filter), and mean-field particle methods.
In numerical integration, methods such as the trapezoidal rule use a deterministic approach. Monte Carlo integration, on the other hand, employs a non-deterministic approach: each realization provides a different outcome. In Monte Carlo, the final outcome is an approximation of the correct value with respective error bars, and the correct value is likely to be within those error bars.
The problem Monte Carlo integration addresses is the computation of a multidimensional definite integral
where Ω, a subset of Rm, has volume
The naive Monte Carlo approach is to sample points uniformly on Ω: given N uniform samples,
I can be approximated by
This is because the law of large numbers ensures that
Given the estimation of I from QN, the error bars of QN can be estimated by the sample variance using the unbiased estimate of the variance.
which leads to
As long as the sequence
is bounded, this variance decreases asymptotically to zero as 1/N. The estimation of the error of QN is thus
which decreases as . This is standard error of the mean multiplied with .
This result does not depend on the number of dimensions of the integral, which is the promised advantage of Monte Carlo integration against most deterministic methods that depend exponentially on the dimension.
It is important to notice that, unlike in deterministic methods, the estimate of the error is not a strict error bound; random sampling may not uncover all the important features of the integrand that can result in an underestimate of the error.
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