Editing Numerical integration (section)

== Multidimensional integrals ==
The quadrature rules discussed so far are all designed to compute one-dimensional integrals. To compute integrals in multiple dimensions, one approach is to phrase the multiple integral as repeated one-dimensional integrals by applying [[Fubini's theorem]] (the tensor product rule). This approach requires the function evaluations to [[exponential growth|grow exponentially]] as the number of dimensions increases. Three methods are known to overcome this so-called ''[[curse of dimensionality]]''.

A great many additional techniques for forming multidimensional cubature integration rules for a variety of weighting functions are given in the monograph by Stroud.<ref name="StroudBook">{{cite book |last1=Stroud |first1=A. H. |title=Approximate Calculation of Multiple Integrals |url=https://archive.org/details/approximatecalcu0000stro_b8j7 |url-access=registration |date=1971 |publisher=Prentice-Hall Inc. |location=Cliffs, NJ|isbn=9780130438935 }}</ref>
Integration on the [[sphere]] has been reviewed by Hesse et al. (2015).<ref>Kerstin Hesse, Ian H. Sloan, and Robert S. Womersley: Numerical Integration on the Sphere. In W. Freeden et al. (eds.), Handbook of Geomathematics, Springer: Berlin 2015, {{doi|10.1007/978-3-642-54551-1_40}}</ref>

=== Monte Carlo ===
{{main|Monte Carlo integration}}

[[Monte Carlo method]]s and [[quasi-Monte Carlo method]]s are easy to apply to multi-dimensional integrals. They may yield greater accuracy for the same number of function evaluations than repeated integrations using one-dimensional methods.{{Citation needed|date=November 2018}}

A large class of useful Monte Carlo methods are the so-called [[Markov chain Monte Carlo]] algorithms, which include the [[Metropolis–Hastings algorithm]] and [[Gibbs sampling]].

=== Sparse grids ===
[[Sparse grid]]s were originally developed by Smolyak for the quadrature of high-dimensional functions. The method is always based on a one-dimensional quadrature rule, but performs a more sophisticated combination of univariate results. However, whereas the tensor product rule guarantees that the weights of all of the cubature points will be positive if the weights of the quadrature points were positive, Smolyak's rule does not guarantee that the weights will all be positive.

=== Bayesian quadrature ===
[[Bayesian quadrature]] is a statistical approach to the numerical problem of computing integrals and falls under the field of [[probabilistic numerics]]. It can provide a full handling of the uncertainty over the solution of the integral expressed as a [[Gaussian process]] posterior variance.