Construction and application of provable positive and exact cubature formulas

Abstract

Abstract Many applications require multi-dimensional numerical integration, often in the form of a cubature formula (CF). These CFs 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 CFs 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 (LSs) 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 an LSs problem. In particular, it is proved that the resulting LSs CFs 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 LSs CFs to construct nested stable high-order rules and positive interpolatory formulas. Finally, our findings shed new light on some existing methods for multi-variate numerical integration and under which restrictions these are ensured to be successful.