Localization coefficients of functions with applications in partial differential equations

Abstract

Abstract We identify shortcomings in two popular measures of localization of functions: the $$L^p\text {-}L^q$$ L p - L q participation ratio and the mass concentration comparison. We then introduce a novel localization measure for functions on bounded subsets of $$\varvec{R}^d$$ R d , $$d=1,2,3,\dots $$ d = 1 , 2 , 3 , ⋯ , 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.