Oct
14

## Multidimensional numerical integration

Most derivative pricing problems have finally come to solve integration numerically, by Simpson, Monte Carlo simulation, etc., however, multi-dimensional integration is time-consuming and prone to error, here I share a Cuba library which offers a choice of four independent routines for multidimensional numerical integration: Vegas, Suave, Divonne, and Cuhre.

http://www.feynarts.de/cuba/

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Vegas is the simplest of the four. It uses importance sampling for variance reduction, but is only in some cases competitive in terms of the number of samples needed to reach a prescribed accuracy. Nevertheless, it has a few improvements over the original algorithm and comes in handy for cross-checking the results of other methods.

Suave is a new algorithm which combines the advantages of two popular methods: importance sampling as done by Vegas and subregion sampling in a manner similar to Miser. By dividing into subregions, Suave manages to a certain extent to get around Vegas' difficulty to adapt its weight function to structures not aligned with the coordinate axes.

Divonne is a further development of the CERNLIB routine D151. Divonne works by stratified sampling, where the partitioning of the integration region is aided by methods from numerical optimization. A number of improvements have been added to this algorithm, the most significant being the possibility to supply knowledge about the integrand. Narrow peaks in particular are difficult to find without sampling very many points, especially in high dimensions. Often the exact or approximate location of such peaks is known from analytic considerations, however, and with such hints the desired accuracy can be reached with far fewer points.

Cuhre employs a cubature rule for subregion estimation in a globally adaptive subdivision scheme. It is hence a deterministic, not a Monte Carlo method. In each iteration, the subregion with the largest error is halved along the axis where the integrand has the largest fourth difference. Cuhre is quite powerful in moderate dimensions, and is usually the only viable method to obtain high precision, say relative accuracies much below 1e-3.

Suave is a new algorithm which combines the advantages of two popular methods: importance sampling as done by Vegas and subregion sampling in a manner similar to Miser. By dividing into subregions, Suave manages to a certain extent to get around Vegas' difficulty to adapt its weight function to structures not aligned with the coordinate axes.

Divonne is a further development of the CERNLIB routine D151. Divonne works by stratified sampling, where the partitioning of the integration region is aided by methods from numerical optimization. A number of improvements have been added to this algorithm, the most significant being the possibility to supply knowledge about the integrand. Narrow peaks in particular are difficult to find without sampling very many points, especially in high dimensions. Often the exact or approximate location of such peaks is known from analytic considerations, however, and with such hints the desired accuracy can be reached with far fewer points.

Cuhre employs a cubature rule for subregion estimation in a globally adaptive subdivision scheme. It is hence a deterministic, not a Monte Carlo method. In each iteration, the subregion with the largest error is halved along the axis where the integrand has the largest fourth difference. Cuhre is quite powerful in moderate dimensions, and is usually the only viable method to obtain high precision, say relative accuracies much below 1e-3.

http://www.feynarts.de/cuba/

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