Time-fractional diffusion equations with randomness, and efficient numerical estimations of expected values
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916377344868352 |
|---|---|
| author | Dick, Josef Gao, Hecong McLean, William Mustapha, Kassem |
| author_facet | Dick, Josef Gao, Hecong McLean, William Mustapha, Kassem |
| contents | In this work, we explore a time-fractional diffusion equation of order $α\in (0,1)$ with a stochastic diffusivity parameter. We focus on efficient estimation of the expected values (considered as an infinite dimensional integral on the parametric space corresponding to the random coefficients) of linear functionals acting on the solution of our model problem. To estimate the expected value computationally, the infinite expansions of the random parameter need to be truncated. Then we approximate the high-dimensional integral over the random field using a high-order quasi-Monte Carlo method. This follows by approximating the deterministic solution over the space-time domain via a second-order accurate time-stepping scheme in combination with a spatial discretization by Galerkin finite elements. Under reasonable regularity assumptions on the given data, we show some regularity properties of the continuous solution and investigate the errors from estimating the expected value. We report on numerical experiments that complement the theoretical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00893 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Time-fractional diffusion equations with randomness, and efficient numerical estimations of expected values Dick, Josef Gao, Hecong McLean, William Mustapha, Kassem Numerical Analysis In this work, we explore a time-fractional diffusion equation of order $α\in (0,1)$ with a stochastic diffusivity parameter. We focus on efficient estimation of the expected values (considered as an infinite dimensional integral on the parametric space corresponding to the random coefficients) of linear functionals acting on the solution of our model problem. To estimate the expected value computationally, the infinite expansions of the random parameter need to be truncated. Then we approximate the high-dimensional integral over the random field using a high-order quasi-Monte Carlo method. This follows by approximating the deterministic solution over the space-time domain via a second-order accurate time-stepping scheme in combination with a spatial discretization by Galerkin finite elements. Under reasonable regularity assumptions on the given data, we show some regularity properties of the continuous solution and investigate the errors from estimating the expected value. We report on numerical experiments that complement the theoretical results. |
| title | Time-fractional diffusion equations with randomness, and efficient numerical estimations of expected values |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2409.00893 |