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- W4225516114 abstract "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." @default.
- W4225516114 created "2022-05-05" @default.
- W4225516114 creator A5005268833 @default.
- W4225516114 date "2022-05-20" @default.
- W4225516114 modified "2023-09-23" @default.
- W4225516114 title "Construction and application of provable positive and exact cubature formulas" @default.
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- W4225516114 doi "https://doi.org/10.1093/imanum/drac017" @default.
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