Localization Coefficients of Functions with Applications in Partial Differential Equations
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910912659587072 |
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| author | Karamehmedović, Mirza Triki, Faouzi |
| author_facet | Karamehmedović, Mirza Triki, Faouzi |
| contents | We identify shortcomings in two popular measures of localization of functions: the $L^p-L^q$ participation ratio and the mass concentration comparison. We then introduce a novel localization measure for functions on bounded subsets of $\mathbf{R}^d$, $d=1,2,3,\dots$, based on a Wasserstein metric. For efficient computation, we prove the equality of this measure with a suitable Sobolev norm in dimension one. We demonstrate our approach by numerical experiments in one and two dimensions. Finally, we discuss and mitigate challenges arising from boundary effects. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12033 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Localization Coefficients of Functions with Applications in Partial Differential Equations Karamehmedović, Mirza Triki, Faouzi Analysis of PDEs 49Q22, 46E35, 35Bxx We identify shortcomings in two popular measures of localization of functions: the $L^p-L^q$ participation ratio and the mass concentration comparison. We then introduce a novel localization measure for functions on bounded subsets of $\mathbf{R}^d$, $d=1,2,3,\dots$, based on a Wasserstein metric. For efficient computation, we prove the equality of this measure with a suitable Sobolev norm in dimension one. We demonstrate our approach by numerical experiments in one and two dimensions. Finally, we discuss and mitigate challenges arising from boundary effects. |
| title | Localization Coefficients of Functions with Applications in Partial Differential Equations |
| topic | Analysis of PDEs 49Q22, 46E35, 35Bxx |
| url | https://arxiv.org/abs/2504.12033 |