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Localization Coefficients of Functions with Applications in Partial Differential Equations

Mirza Karamehmedović, Faouzi Triki

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.

Localization Coefficients of Functions with Applications in Partial Differential Equations

Abstract

We identify shortcomings in two popular measures of localization of functions: the participation ratio and the mass concentration comparison. We then introduce a novel localization measure for functions on bounded subsets of , , 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.

Paper Structure

This paper contains 11 sections, 3 theorems, 54 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Mass concentration comparisons and Lebesgue norm-based localization coefficients
  3. A new measure of localization
  4. Measuring localization using the Wasserstein-2 distance
  5. Localization in terms of a Sobolev norm
  6. Compensating for the boundary effect
  7. Further numerical examples of $\beta(u,\Omega)$
  8. Localization of eigenfunctions of a Sturm-Liouville operator
  9. Localization of Neumann-Poincaré eigenfunctions on curves
  10. Distance-to-boundary measure of localization in dimension two
  11. Conclusion and outlook

Key Result

Proposition 1

Let $u, \, v \in L^\infty(\Omega)\setminus\{0\}$. Then $u \prec v$ if and only if $\alpha_{1,\infty}(v,\Omega) \leq \alpha_{1,\infty}(u,\Omega)$.

Figures (14)

  • Figure 1: The distribution function $\mu_{u_{a,d}}$ of $u_{a,d}$ from \ref{['eqn:uad']}--\ref{['eqn:j2']} is independent of $d\in[2b,1-a-b]$, and so is any $L^p$ norm of $u_{a,d}$. Thus mass concentration comparisons involving $\mu_{u_{a,d}}$ and the constant-valued function $1$, as well as all participation ratios $\alpha_{p,q}(u_{a,d},(0,1))$, are invariant to changes in $d$, in spite of the fact that $u_{a,d}$ delocalizes as $d$ grows.
  • Figure 2: $\beta_{a,d}=\beta(u_{a,d},(0,1))=W_2(\mu_{u_{a,d}},\lambda)$ vs. $S_{a,d}=\|\mu_{u_{a,d}}-\lambda\|_{\dot H^{-1}(\lambda)}$ for $u_{a,d}$ from \ref{['eqn:uad']}--\ref{['eqn:j2']}, and $\beta_{\sigma}=\beta(u_{\sigma},(0,1))=W_2(\mu_{u_{\sigma}},\lambda)$ vs. $S_{\sigma}=\|\mu_{u_{\sigma}}-\lambda\|_{\dot H^{-1}(\lambda)}$ for $u_{\sigma}$ from \ref{['eqn:usigma']}: first as function of $d\in[2b,1-a-b]$, with $a=0.1$ and $b=0.05$; next as function of $d\in[b,1/2-b]$, for symmetric densities with $b=0.05$; next as function of $a\in[b,1-d-b]$, when translating a fixed density profile with $b=0.05$ and $d=0.3$; and finally as function of $\sigma\in[0,1]$.
  • Figure 3: Wasserstein-2 optimal transport cost vs. periodized optimal transport cost between $u_{a,d}$ of \ref{['eqn:uad']}--\ref{['eqn:j2']} and the constant-valued function $1$, as function of $d\in[2b, 1 - a - b]$, with $a = 0.1$ and $b = 0.05$.
  • Figure 4: Extended-domain Wasserstein-2 optimal transport cost between $u_{a,d}$ of \ref{['eqn:uad']}--\ref{['eqn:j2']} and the constant-valued function $1/5$, as function of $d\in[2b, 1 - a - b]$, with $a = 0.1$ and $b = 0.05$.
  • Figure 5: The metric for the Laplacian of the numerical example in Section \ref{['sec:leslo']}.
  • ...and 9 more figures

Theorems & Definitions (10)

  • Definition 1: Comparison of mass concentrations
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 1
  • Definition 2
  • Lemma 1
  • proof
  • Remark 2