Approximation of Random Evolution Equations of Parabolic type
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910751006916608 |
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| author | Klioba, Katharina Seifert, Christian |
| author_facet | Klioba, Katharina Seifert, Christian |
| contents | In this paper, we present an abstract framework to obtain convergence rates for the approximation of random evolution equations corresponding to a random family of forms determined by finite-dimensional noise. The full discretization error in space, time, and randomness is considered, where polynomial chaos expansion (PCE) is used for the semi-discretization in randomness. The main result are regularity conditions on the random forms under which convergence of polynomial order in randomness is obtained depending on the smoothness of the coefficients and the Sobolev regularity of the initial value. In space and time, the same convergence rates as in the deterministic setting are achieved. To this end, we derive error estimates for vector-valued PCE as well as a quantified version of the Trotter--Kato theorem for form-induced semigroups. We apply the abstract framework to an anisotropic diffusion model with random diffusion coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_07660 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximation of Random Evolution Equations of Parabolic type Klioba, Katharina Seifert, Christian Functional Analysis Numerical Analysis Analysis of PDEs Probability 47D06, 47N40, 65J08, 35K90, 41A25 In this paper, we present an abstract framework to obtain convergence rates for the approximation of random evolution equations corresponding to a random family of forms determined by finite-dimensional noise. The full discretization error in space, time, and randomness is considered, where polynomial chaos expansion (PCE) is used for the semi-discretization in randomness. The main result are regularity conditions on the random forms under which convergence of polynomial order in randomness is obtained depending on the smoothness of the coefficients and the Sobolev regularity of the initial value. In space and time, the same convergence rates as in the deterministic setting are achieved. To this end, we derive error estimates for vector-valued PCE as well as a quantified version of the Trotter--Kato theorem for form-induced semigroups. We apply the abstract framework to an anisotropic diffusion model with random diffusion coefficients. |
| title | Approximation of Random Evolution Equations of Parabolic type |
| topic | Functional Analysis Numerical Analysis Analysis of PDEs Probability 47D06, 47N40, 65J08, 35K90, 41A25 |
| url | https://arxiv.org/abs/2404.07660 |