MonteCarloIntegration

MonteCarloIntegration is a numerical technique for estimating definite integrals by random sampling. It is particularly useful for high-dimensional problems or when the integrand is difficult to handle analytically, offering a simple algorithmic approach that can scale with complexity rather than with the smoothness of the function.

In its simplest form, consider an integral I = ∫_D f(x) dx over a domain D with finite volume Vol(D). If one draws N points X1, X2, ..., XN uniformly from D, the integral can be estimated by I_hat = Vol(D) × (1/N) ∑_{i=1}^N f(X_i). This is known as naive Monte Carlo integration. More generally, if sampling from a probability density p(x) supported on D, the integral can be written as I = E_p[f(X)/p(X)], and the estimator becomes I_hat = (1/N) ∑_{i=1}^N f(X_i)/p(X_i). This form allows importance sampling when p is chosen to approximate the shape of f.

Convergence is governed by the central limit theorem: I_hat converges to I with standard error that scales as σ/√N, where σ^2 is the variance of the sampled quantity under p. The convergence rate is dimensionally favorable compared to grid-based methods, but the constant can deteriorate in high dimensions. Variance reduction techniques, such as importance sampling, stratified sampling, control variates, and antithetic variates, are commonly employed to improve efficiency.

Limitations include potential slow convergence for functions with large variance, challenging domains, or poor choice of sampling distribution. MonteCarloIntegration remains a foundational tool in fields ranging from physics and engineering to finance and computer graphics.