Fractional differential problems with numerical anti-reflective boundary conditions: a computational/precision analysis and numerical results
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913369680773120 |
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| author | Sousa, Ercília Tablino-Possio, Cristina Krause, Rolf Serra-Capizzano, Stefano |
| author_facet | Sousa, Ercília Tablino-Possio, Cristina Krause, Rolf Serra-Capizzano, Stefano |
| contents | Twenty years ago the anti-reflective numerical boundary conditions (BCs) were introduced in a context of signal processing and imaging, for increasing the quality of the reconstruction of a blurred signal/image contaminated by noise and for reducing the overall complexity to that of few fast sine transforms i.e. to $O(N\log N)$ real arithmetic operations, where $N$ is the number of pixels. Here for quality of reconstruction we mean a better accuracy and the elimination of boundary artifacts, called ringing effects. Now we propose numerical anti-reflective BCs in the context of nonlocal problems of fractional differential type: the goals are the same i.e. a smaller approximation error and the reduction of boundary artifacts. In the latter setting, we compare various types of numerical BCs, including the anti-symmetric ones considered in the case of fractional differential problems for modeling reasons. More in detail, given important similarities between anti-symmetric and anti-reflective BCs, we compare them from the perspective of computational efficiency, by considering nontruncated and truncated versions and also other standard numerical BCs such as reflective/Neumann. Several numerical tests, tables, and visualizations are provided and critically discussed. The conclusion is that the truncated numerical anti-reflective BCs perform better, both in terms of accuracy and low computational cost. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_16485 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fractional differential problems with numerical anti-reflective boundary conditions: a computational/precision analysis and numerical results Sousa, Ercília Tablino-Possio, Cristina Krause, Rolf Serra-Capizzano, Stefano Numerical Analysis Twenty years ago the anti-reflective numerical boundary conditions (BCs) were introduced in a context of signal processing and imaging, for increasing the quality of the reconstruction of a blurred signal/image contaminated by noise and for reducing the overall complexity to that of few fast sine transforms i.e. to $O(N\log N)$ real arithmetic operations, where $N$ is the number of pixels. Here for quality of reconstruction we mean a better accuracy and the elimination of boundary artifacts, called ringing effects. Now we propose numerical anti-reflective BCs in the context of nonlocal problems of fractional differential type: the goals are the same i.e. a smaller approximation error and the reduction of boundary artifacts. In the latter setting, we compare various types of numerical BCs, including the anti-symmetric ones considered in the case of fractional differential problems for modeling reasons. More in detail, given important similarities between anti-symmetric and anti-reflective BCs, we compare them from the perspective of computational efficiency, by considering nontruncated and truncated versions and also other standard numerical BCs such as reflective/Neumann. Several numerical tests, tables, and visualizations are provided and critically discussed. The conclusion is that the truncated numerical anti-reflective BCs perform better, both in terms of accuracy and low computational cost. |
| title | Fractional differential problems with numerical anti-reflective boundary conditions: a computational/precision analysis and numerical results |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2312.16485 |