Small Superposition Dimension and Active Set Construction for Multivariate Integration Under Modest Error Demand
For researchers in multivariate integration, this provides a practical method to reduce computational cost under modest error demands.
The paper proposes an algorithm for constructing optimal active sets for multivariate integration, achieving significantly smaller sets than existing methods. Experiments show the superposition dimension can be as low as 3 when error demand is ≥10⁻³ and weights decay fast.
Constructing active sets is a key part of the Multivariate Decomposition Method. An algorithm for constructing optimal or quasi-optimal active sets is proposed in the paper. By numerical experiments, it is shown that the new method can provide sets that are significantly smaller than the sets constructed by the already existing method. The experiments also show that the superposition dimension could surprisingly be very small, at most 3, when the error demand is not smaller than $10^{-3}$ and the weights decay sufficiently fast.