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In this paper, we propose and study a fast multilevel dimension iteration(MDI) algorithm for computing arbitrary $d$-dimensional integrals based ontensor product approximations. It reduces the computational complexity (interms of the CPU time) of a tensor product method from the exponential order$O(N^d)$ to the polynomial order {\color{black} $O(d^3N^2)$ or better}, where$N$ stands for the number of quadrature points in each coordinate direction. Asa result, the proposed MDI algorithm effectively circumvents the curse of thedimensionality of tensor product methods for high dimensional numericalintegration. The main idea of the proposed MDI algorithm is to compute thefunction evaluations at all integration points in the cluster and iterativelyalong each coordinate direction, so lots of computations for functionevaluations can be reused in each iteration. This idea is also applicable toany quadrature rule whose integration points have a lattice-like structure.

A fast multilevel dimension iteration algorithm for high dimensional numerical integration