# Small Superposition Dimension and Active Set Construction for Multivariate Integration Under Modest Error Demand

## TL;DR

This paper introduces an algorithm for constructing optimal active sets in multivariate integration, revealing that the superposition dimension can be surprisingly small under certain error and weight decay conditions.

## Contribution

It proposes a new method for active set construction that produces significantly smaller sets and demonstrates that the superposition dimension is often very small for modest error demands.

## Key findings

- Active sets can be much smaller with the new method.
- Superposition dimension is at most 3 for error demands ≥ 10^{-3}.
- Fast weight decay leads to smaller active sets.

## Abstract

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.