Efficient white noise sampling and coupling for multilevel Monte Carlo with nonnested meshes
When solving stochastic partial differential equations (SPDEs) driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. Here, we present a new sampling technique that can be used to efficiently compute white noise samples in a finite element meth...
| Main Authors: | , , , |
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| Format: | Journal article |
| Language: | English |
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Society for Industrial and Applied Mathematics
2018
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| _version_ | 1826258261931720704 |
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| author | Croci, M Giles, M Rognes, M Farrell, P |
| author_facet | Croci, M Giles, M Rognes, M Farrell, P |
| author_sort | Croci, M |
| collection | OXFORD |
| description | When solving stochastic partial differential equations (SPDEs) driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. Here, we present a new sampling technique that can be used to efficiently compute white noise samples in a finite element method (FEM) and multilevel Monte Carlo (MLMC) setting. The key idea is to exploit the finite element matrix assembly procedure and factorize each local mass matrix independently, hence avoiding the factorization of a large matrix. Moreover, in an MLMC framework, the white noise samples must be coupled between subsequent levels. We show how our technique can be used to enforce this coupling even in the case of nonnested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments. We observe optimal convergence rates for the finite element solution of the elliptic SPDEs of interest in 2D and 3D and we show convergence of the sampled field covariances. In an MLMC setting, a good coupling is enforced and the telescoping sum is respected. |
| first_indexed | 2024-03-06T18:31:10Z |
| format | Journal article |
| id | oxford-uuid:09b3c132-87d5-4dee-8825-a2c0ad21ca73 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T18:31:10Z |
| publishDate | 2018 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:09b3c132-87d5-4dee-8825-a2c0ad21ca732022-03-26T09:19:47ZEfficient white noise sampling and coupling for multilevel Monte Carlo with nonnested meshesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:09b3c132-87d5-4dee-8825-a2c0ad21ca73EnglishSymplectic Elements at OxfordSociety for Industrial and Applied Mathematics2018Croci, MGiles, MRognes, MFarrell, PWhen solving stochastic partial differential equations (SPDEs) driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. Here, we present a new sampling technique that can be used to efficiently compute white noise samples in a finite element method (FEM) and multilevel Monte Carlo (MLMC) setting. The key idea is to exploit the finite element matrix assembly procedure and factorize each local mass matrix independently, hence avoiding the factorization of a large matrix. Moreover, in an MLMC framework, the white noise samples must be coupled between subsequent levels. We show how our technique can be used to enforce this coupling even in the case of nonnested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments. We observe optimal convergence rates for the finite element solution of the elliptic SPDEs of interest in 2D and 3D and we show convergence of the sampled field covariances. In an MLMC setting, a good coupling is enforced and the telescoping sum is respected. |
| spellingShingle | Croci, M Giles, M Rognes, M Farrell, P Efficient white noise sampling and coupling for multilevel Monte Carlo with nonnested meshes |
| title | Efficient white noise sampling and coupling for multilevel Monte Carlo with nonnested meshes |
| title_full | Efficient white noise sampling and coupling for multilevel Monte Carlo with nonnested meshes |
| title_fullStr | Efficient white noise sampling and coupling for multilevel Monte Carlo with nonnested meshes |
| title_full_unstemmed | Efficient white noise sampling and coupling for multilevel Monte Carlo with nonnested meshes |
| title_short | Efficient white noise sampling and coupling for multilevel Monte Carlo with nonnested meshes |
| title_sort | efficient white noise sampling and coupling for multilevel monte carlo with nonnested meshes |
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