Efficient algorithms for multivariate and \(\infty\)-variate integration with exponential weight
The authors present an efficient algorithm for \(\infty\)-variate integration with exponential weight function \[ I_{\infty}(f) = \lim_{d\to \infty} \int_{{\mathbb R}_+^d} f(x_1,\dots,x_d,0,0,\dots)\, \exp(-\sum_{j=1}^d x_j)\,dx\,, \] where the function \(f\) is differentiable with respect to each variable and vanishes if there is at least one \(x_j =0\). The \(d\)-variate problem is solved by the \(d\)-variate integration method of \textit{S. A. Smolyak} [Dokl. Akad. Nauk SSSR 148, 1042--1045 (1963; Zbl 0202.39901)] which requires efficient algorithms for the univariate integral \[ I(f) = \int_0^{\infty} f(x)\, \exp(-x)\, dx. \] Using weighted trapezoidal rules, (nearly) optimal quadratures for \(I(f)\) are presented in detail.
univariate integral with exponential weight
scientific article; zbMATH DE number 6363122
scientific article