A Monotone Discretization for the Fractional Obstacle Problem and Its Improved Policy Iteration
In recent years, the fractional Laplacian has attracted the attention of many researchers, the corresponding fractional obstacle problems have been widely applied in mathematical finance, particle systems, and elastic theory. Furthermore, the monotonicity of the numerical scheme is beneficial for nu...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-08-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/7/8/634 |
| _version_ | 1797584672023117824 |
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| author | Rubing Han Shuonan Wu Hao Zhou |
| author_facet | Rubing Han Shuonan Wu Hao Zhou |
| author_sort | Rubing Han |
| collection | DOAJ |
| description | In recent years, the fractional Laplacian has attracted the attention of many researchers, the corresponding fractional obstacle problems have been widely applied in mathematical finance, particle systems, and elastic theory. Furthermore, the monotonicity of the numerical scheme is beneficial for numerical stability. The purpose of this work is to introduce a monotone discretization method for addressing obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over bounded Lipschitz domains. Through successful monotone discretization of the fractional Laplacian, the monotonicity is preserved for the fractional obstacle problem and the uniform boundedness, existence, and uniqueness of the numerical solutions of the fractional obstacle problems are proved. A policy iteration is adopted to solve the discrete nonlinear problems, and the convergence after finite iterations can be proved through the monotonicity of the scheme. Our improved policy iteration, adapted to solution regularity, demonstrates superior performance by modifying discretization across different regions. Numerical examples underscore the efficacy of the proposed method. |
| first_indexed | 2024-03-10T23:55:02Z |
| format | Article |
| id | doaj.art-1d890786816f435b852b6a142723a755 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T23:55:02Z |
| publishDate | 2023-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-1d890786816f435b852b6a142723a7552023-11-19T01:11:57ZengMDPI AGFractal and Fractional2504-31102023-08-017863410.3390/fractalfract7080634A Monotone Discretization for the Fractional Obstacle Problem and Its Improved Policy IterationRubing Han0Shuonan Wu1Hao Zhou2School of Mathematical Sciences, Peking University, Beijing 100871, ChinaSchool of Mathematical Sciences, Peking University, Beijing 100871, ChinaSchool of Mathematical Sciences, Peking University, Beijing 100871, ChinaIn recent years, the fractional Laplacian has attracted the attention of many researchers, the corresponding fractional obstacle problems have been widely applied in mathematical finance, particle systems, and elastic theory. Furthermore, the monotonicity of the numerical scheme is beneficial for numerical stability. The purpose of this work is to introduce a monotone discretization method for addressing obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over bounded Lipschitz domains. Through successful monotone discretization of the fractional Laplacian, the monotonicity is preserved for the fractional obstacle problem and the uniform boundedness, existence, and uniqueness of the numerical solutions of the fractional obstacle problems are proved. A policy iteration is adopted to solve the discrete nonlinear problems, and the convergence after finite iterations can be proved through the monotonicity of the scheme. Our improved policy iteration, adapted to solution regularity, demonstrates superior performance by modifying discretization across different regions. Numerical examples underscore the efficacy of the proposed method.https://www.mdpi.com/2504-3110/7/8/634obstacle problemfractional Laplacianbounded Lipschitz domainmonotone discretizationpolicy iteration |
| spellingShingle | Rubing Han Shuonan Wu Hao Zhou A Monotone Discretization for the Fractional Obstacle Problem and Its Improved Policy Iteration Fractal and Fractional obstacle problem fractional Laplacian bounded Lipschitz domain monotone discretization policy iteration |
| title | A Monotone Discretization for the Fractional Obstacle Problem and Its Improved Policy Iteration |
| title_full | A Monotone Discretization for the Fractional Obstacle Problem and Its Improved Policy Iteration |
| title_fullStr | A Monotone Discretization for the Fractional Obstacle Problem and Its Improved Policy Iteration |
| title_full_unstemmed | A Monotone Discretization for the Fractional Obstacle Problem and Its Improved Policy Iteration |
| title_short | A Monotone Discretization for the Fractional Obstacle Problem and Its Improved Policy Iteration |
| title_sort | monotone discretization for the fractional obstacle problem and its improved policy iteration |
| topic | obstacle problem fractional Laplacian bounded Lipschitz domain monotone discretization policy iteration |
| url | https://www.mdpi.com/2504-3110/7/8/634 |
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