Why Simple Quadrature is just as good as Monte Carlo
© 2020 Walter de Gruyter GmbH, Berlin/Boston. We motive and calculate Newton-Cotes quadrature integration variance and compare it directly with Monte Carlo (MC) integration variance. We find an equivalence between deterministic quadrature sampling and random MC sampling by noting that MC random samp...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
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Walter de Gruyter GmbH
2021
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| Online Access: | https://hdl.handle.net/1721.1/134373.2 |
| _version_ | 1811070750346444800 |
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| author | Vanslette, Kevin Al Alsheikh, Abdullatif Youcef-Toumi, Kamal |
| author2 | Massachusetts Institute of Technology. Department of Mechanical Engineering |
| author_facet | Massachusetts Institute of Technology. Department of Mechanical Engineering Vanslette, Kevin Al Alsheikh, Abdullatif Youcef-Toumi, Kamal |
| author_sort | Vanslette, Kevin |
| collection | MIT |
| description | © 2020 Walter de Gruyter GmbH, Berlin/Boston. We motive and calculate Newton-Cotes quadrature integration variance and compare it directly with Monte Carlo (MC) integration variance. We find an equivalence between deterministic quadrature sampling and random MC sampling by noting that MC random sampling is statistically indistinguishable from a method that uses deterministic sampling on a randomly shuffled (permuted) function. We use this statistical equivalence to regularize the form of permissible Bayesian quadrature integration priors such that they are guaranteed to be objectively comparable with MC. This leads to the proof that simple quadrature methods have expected variances that are less than or equal to their corresponding theoretical MC integration variances. Separately, using Bayesian probability theory, we find that the theoretical standard deviations of the unbiased errors of simple Newton-Cotes composite quadrature integrations improve over their worst case errors by an extra dimension independent factor α N - 12. This dimension independent factor is validated in our simulations. |
| first_indexed | 2024-09-23T08:40:58Z |
| format | Article |
| id | mit-1721.1/134373.2 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T08:40:58Z |
| publishDate | 2021 |
| publisher | Walter de Gruyter GmbH |
| record_format | dspace |
| spelling | mit-1721.1/134373.22022-09-22T07:14:47Z Why Simple Quadrature is just as good as Monte Carlo Vanslette, Kevin Al Alsheikh, Abdullatif Youcef-Toumi, Kamal Massachusetts Institute of Technology. Department of Mechanical Engineering © 2020 Walter de Gruyter GmbH, Berlin/Boston. We motive and calculate Newton-Cotes quadrature integration variance and compare it directly with Monte Carlo (MC) integration variance. We find an equivalence between deterministic quadrature sampling and random MC sampling by noting that MC random sampling is statistically indistinguishable from a method that uses deterministic sampling on a randomly shuffled (permuted) function. We use this statistical equivalence to regularize the form of permissible Bayesian quadrature integration priors such that they are guaranteed to be objectively comparable with MC. This leads to the proof that simple quadrature methods have expected variances that are less than or equal to their corresponding theoretical MC integration variances. Separately, using Bayesian probability theory, we find that the theoretical standard deviations of the unbiased errors of simple Newton-Cotes composite quadrature integrations improve over their worst case errors by an extra dimension independent factor α N - 12. This dimension independent factor is validated in our simulations. 2021-12-07T16:08:24Z 2021-10-27T20:04:41Z 2021-12-07T16:08:24Z 2020 2020-08-14T14:35:56Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/134373.2 en 10.1515/mcma-2020-2055 Monte Carlo Methods and Applications Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/octet-stream Walter de Gruyter GmbH arXiv |
| spellingShingle | Vanslette, Kevin Al Alsheikh, Abdullatif Youcef-Toumi, Kamal Why Simple Quadrature is just as good as Monte Carlo |
| title | Why Simple Quadrature is just as good as Monte Carlo |
| title_full | Why Simple Quadrature is just as good as Monte Carlo |
| title_fullStr | Why Simple Quadrature is just as good as Monte Carlo |
| title_full_unstemmed | Why Simple Quadrature is just as good as Monte Carlo |
| title_short | Why Simple Quadrature is just as good as Monte Carlo |
| title_sort | why simple quadrature is just as good as monte carlo |
| url | https://hdl.handle.net/1721.1/134373.2 |
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