Browsing by Author "Steinerberger, S"

Now showing 1 - 3 of 3
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    Carrier frequencies, holomorphy and unwinding
    (SIAM Journal on Mathematical Analysis, 2017-12-17) Wu, H; Steinerberger, S; Coifman, R
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    Detecting localized eigenstates of linear operators
    (2017-11-30) Lu, J; Steinerberger, S
    We describe a way of detecting the location of localized eigenvectors of a linear system $Ax = \lambda x$ for eigenvalues $\lambda$ with $|\lambda|$ comparatively large. We define the family of functions $f_{\alpha}: \left\{1.2. \dots, n\right\} \rightarrow \mathbb{R}_{}$ $$ f_{\alpha}(k) = \log \left( \| A^{\alpha} e_k \|_{\ell^2} \right),$$ where $\alpha \geq 0$ is a parameter and $e_k = (0,0,\dots, 0,1,0, \dots, 0)$ is the $k-$th standard basis vector. We prove that eigenvectors associated to eigenvalues with large absolute value localize around local maxima of $f_{\alpha}$: the metastable states in the power iteration method (slowing down its convergence) can be used to predict localization. We present a fast randomized algorithm and discuss different examples: a random band matrix, discretizations of the local operator $-\Delta + V$ and the nonlocal operator $(-\Delta)^{3/4} + V$.
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    On Zeroes of Random Polynomials and an Application to Unwinding
    (International Mathematics Research Notices) Steinerberger, S; Wu, HT