# Construction and application of provable positive and exact cubature formulas

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.

## Authors

• 13 publications
09/24/2020

### Constructing Positive Interpolatory Cubature Formulas

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10/11/2018

### A Constraint Propagation Algorithm for Sums-of-Squares Formulas over the Integers

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07/31/2018

### A simple analysis of flying capacitor converter

The paper aims to emphasise how switched systems can be analysed with el...
05/19/2020

### Proving P!=NP in first-order PA

We show that it is provable in PA that there is an arithmetically defina...
01/03/2020

### Monte-Carlo cubature construction

In numerical integration, cubature methods are effective, in particular ...
07/17/2020

### Stable High Order Quadrature Rules for Scattered Data and General Weight Functions

Numerical integration is encountered in all fields of numerical analysis...
09/07/2020

### Stable High-Order Cubature Formulas for Experimental Data

In many applications, it is impractical—if not even impossible—to obtain...
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