Rounding error using low precision approximate random variables
For numerical approximations to stochastic differential equations using the Euler Maruyama scheme, we propose incorporating approximate random variables computed using low precisions, such as single and half precision. We propose and justify a model for the rounding error incurred, and produce an av...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Society for Industrial and Applied Mathematics
2024
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| _version_ | 1826313468804857856 |
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| author | Giles, M Sheridan-Methven, O |
| author_facet | Giles, M Sheridan-Methven, O |
| author_sort | Giles, M |
| collection | OXFORD |
| description | For numerical approximations to stochastic differential equations using the Euler Maruyama scheme, we propose incorporating approximate random variables computed using low precisions, such as single and half precision. We propose and justify a model for the rounding error incurred, and produce an average case error bound for two and four way differences, appropriate for regular and nested multilevel Monte Carlo estimations. Our rounding error model recovers and extends the statistical model by Arciniega and Allen [1], while bounding the size systematic and biased rounding errors are permitted to be. By considering the variance structure of multilevel Monte Carlo correction terms in various precisions with and without a Kahan compensated summation, we compute the potential speed ups offered from the various precisions. We find single precision offers the potential for approximate speed improvements by a factor of 7 across a wide span of discretisation levels. Half precision offers comparable improvements for several levels of coarse simulations, and even offers improvements by a factor of 10–12 for the very coarsest few levels, which are likely to dominate higher order methods such as the Milstein scheme. |
| first_indexed | 2024-04-09T03:56:51Z |
| format | Journal article |
| id | oxford-uuid:6b5c9aa4-4709-49fa-b126-711ae88e637c |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:15:29Z |
| publishDate | 2024 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:6b5c9aa4-4709-49fa-b126-711ae88e637c2024-07-16T17:00:52ZRounding error using low precision approximate random variablesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:6b5c9aa4-4709-49fa-b126-711ae88e637cEnglishSymplectic ElementsSociety for Industrial and Applied Mathematics2024Giles, MSheridan-Methven, OFor numerical approximations to stochastic differential equations using the Euler Maruyama scheme, we propose incorporating approximate random variables computed using low precisions, such as single and half precision. We propose and justify a model for the rounding error incurred, and produce an average case error bound for two and four way differences, appropriate for regular and nested multilevel Monte Carlo estimations. Our rounding error model recovers and extends the statistical model by Arciniega and Allen [1], while bounding the size systematic and biased rounding errors are permitted to be. By considering the variance structure of multilevel Monte Carlo correction terms in various precisions with and without a Kahan compensated summation, we compute the potential speed ups offered from the various precisions. We find single precision offers the potential for approximate speed improvements by a factor of 7 across a wide span of discretisation levels. Half precision offers comparable improvements for several levels of coarse simulations, and even offers improvements by a factor of 10–12 for the very coarsest few levels, which are likely to dominate higher order methods such as the Milstein scheme. |
| spellingShingle | Giles, M Sheridan-Methven, O Rounding error using low precision approximate random variables |
| title | Rounding error using low precision approximate random variables |
| title_full | Rounding error using low precision approximate random variables |
| title_fullStr | Rounding error using low precision approximate random variables |
| title_full_unstemmed | Rounding error using low precision approximate random variables |
| title_short | Rounding error using low precision approximate random variables |
| title_sort | rounding error using low precision approximate random variables |
| work_keys_str_mv | AT gilesm roundingerrorusinglowprecisionapproximaterandomvariables AT sheridanmethveno roundingerrorusinglowprecisionapproximaterandomvariables |