Efficient algorithms for multivariate and ∞-Variate integration with exponential weight

Using the Multivariate Decomposition Method (MDM), we develop an efficient algorithm for approximating the8-variate integralfor a class of functions f that are once differentiable with respect to each variable.MDM requires efficient algorithms for d-variate versions of the problem. Such algorithms are provided by Smolyaks construction which is based on efficient algorithms for the univariate integrationDetailed analysis and development of (nearly) optimal quadratures for I (f) is the main contribution of the current paper.