An efficient and fast sparse grid algorithm for high-dimensional numerical integration
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This paper is concerned with developing an efficient numerical algorithm for fast implementation of the sparse grid method for computing the d-dimensional integral of a given function. The new algorithm, called the MDI-SG (multilevel dimension iteration sparse grid) method, implements the sparse grid method based on a dimension iteration/reduction procedure, it does not need to store the integration points, neither does it compute the function values independently at each integration point, instead, it re-uses the computation for function evaluations as much as possible by performing the function evaluations at all integration points in a cluster and iteratively along coordinate directions. It is shown numerically that the computational complexity (in terms of CPU time) of the proposed MDI-SG method is of polynomial order O(Nd^3 ) or better, compared to the exponential order O(N(log N)^d-1) for the standard sparse grid method, where N denotes the maximum number of integration points in each coordinate direction. As a result, the proposed MDI-SG method effectively circumvents the curse of dimensionality suffered by the standard sparse grid method for high-dimensional numerical integration.
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