Many applications require multi-dimensional numerical integration, often in the form of a cubature formula. These cubature formulas are desired to be positive and exact for certain finite-dimensional function spaces (and weight functions). Although there are several efficient procedures to construct positive and exact cubature formulas for many standard cases, it remains a challenge to do so in a more general setting. Here, we show how the method of least squares can be used to derive provable positive and exact formulas in a general multi-dimensional setting. Thereby, the procedure only makes use of basic linear algebra operations, such as solving a least squares problem. In particular, it is proved that the resulting least squares cubature formulas are ensured to be positive and exact if a sufficiently large number of equidistributed data points is used. We also discuss the application of provable positive and exact least squares cubature formulas to construct nested stable high-order rules and positive interpolatory formulas. Finally, our findings shed new light on some existing methods for multivariate numerical integration and under which restrictions these are ensured to be successful.

翻译：许多应用需要多维数字集成, 通常以幼稚公式的形式。 这些幼稚公式对于某些有限维功能空间( 和重量函数) 来说, 应该是正的和精确的。 虽然为许多标准案例构建正的和精确的幼稚公式有几种有效的程序, 但是在更笼统的环境下, 这样做仍然是个挑战。 在这里, 我们展示如何使用最小正方形的方法在一般的多维环境中生成可辨的正方形和精确的公式。 因此, 程序只使用基本的线性代数操作, 如解决最小方形问题。 特别是, 事实证明, 如果使用足够多的电子分布数据点, 所产生的最小正方形公式是正方形和精确的。 我们还讨论可辨别的正方形公式的应用, 以构建稳妥的高度规则以及积极的相互间公式。 最后, 我们的发现为多种变数数字整合的现有方法提供了新的线索, 并在这些限制下确保这些方法的成功。