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arXiv · 2504.12033

Localization Coefficients of Functions with Applications in Partial Differential Equations

Mirza Karamehmedović·Faouzi Triki
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Abstract

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.

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Mirza Karamehmedović, Faouzi Triki. 2025-04-16. Localization Coefficients of Functions with Applications in Partial Differential Equations. https://arxiv.org/abs/2504.12033

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