Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacian

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  • #38133
    Avatar photowv
    Participant

    http://sports-ak.espn.go.com/blog/baltimore-ravens/post/_/id/25428/what-offseason-ravens-john-urschel-is-starting-his-phd-at-mit
    What offseason? Ravens’ John Urschel is starting his PhD at MIT
    2:22 PM ET

    Jamison HensleyESPN Staff Writer

    There’s obviously no “off” in the offseason for Baltimore Ravens offensive lineman John Urschel.

    After ending the season as the Ravens’ starting center, Urschel announced on Twitter he is starting his Doctor of Philosophy degree at the Massachusetts Institute of Technology. He is listed as a graduate student for Spectral Graph Theory, Numerical Linear Algebra and Machine Learning.

    After reading that, it feels like Urschel is more likely to be a character on “The Big Bang Theory” than a lineman pushing a 340-pound nose tackle off the line of scrimmage.

    Urschel is one of the most unique players in the game and is proving that you can be a student-athlete beyond college. The winner of the “academic Heisman” coming out of Penn State, Urschel had his research published in peer-reviewed academic journals in his spare time. His latest one was titled “A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians.”

    In June, Urschel visited the National Security Agency to discuss his passion at “Mathfest.” After he sustained a concussion in August, Urschel was working on high-level math problems just to see where he stood.

    The amazing part is that Urschel is doing all of this while becoming a rising offensive lineman on the Ravens. In his first two seasons, he stepped up when injured starters went down, starting five games at guard as a rookie (including two in the playoffs) and seven games at center in 2015. There is a chance that Urschel will become a full-time starter in 2016 if Kelechi Osemele isn’t re-signed in free agency.

    It would be interesting to see Urschel’s two worlds converge one day if he earns the right to be announced in pregame introductions as “Dr. John Urschel.”

    #38137
    Avatar photozn
    Moderator

    , Urschel had his research published in peer-reviewed academic journals in his spare time.

    I don’t like what’s happening to the game of football.

    Used to be, players were too proud to publish in peer-reviewed academic journals.

    Why don’t they just dress quarterbacks in graduation gowns and have done with it.

    #38144
    Avatar photowv
    Participant

    I have never even heard of an “eigenvalue”.

    w
    v
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    A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians
    John C. Urschel, Xiaozhe Hu, Jinchao Xu, Ludmil T. Zikatanov
    (Submitted on 1 Dec 2014)

    In this paper, we develop a cascadic multigrid algorithm for fast computation of the Fiedler vector of a graph Laplacian, namely, the eigenvector corresponding to the second smallest eigenvalue. This vector has been found to have applications in fields such as graph partitioning and graph drawing. The algorithm is a purely algebraic approach based on a heavy edge coarsening scheme and pointwise smoothing for refinement. To gain theoretical insight, we also consider the related cascadic multigrid method in the geometric setting for elliptic eigenvalue problems and show its uniform convergence under certain assumptions. Numerical tests are presented for computing the Fiedler vector of several practical graphs, and numerical results show the efficiency and optimality of our proposed cascadic multigrid algorithm.

    Comments: 16 pages
    Subjects: Numerical Analysis (math.NA)
    MSC classes: 65F15, 68R10, 65N55
    Cite as: arXiv:1412.0565 [math.NA]
    (or arXiv:1412.0565v1 [math.NA] for this version)
    Submission history
    From: Xiaozhe Hu [view email]
    [v1] Mon, 1 Dec 2014 17:55:12 GMT (29kb)
    http://arxiv.org/abs/1412.0565

    #38147
    Avatar photoAgamemnon
    Participant

    They use it in quantum theory.

    in quantum mechanics, you can only measure the stable state of a system, which is defined to be the eigen state of some operator. For example, p=−ih2π∇. Hence eigen vectors and eigen values play a fundamental role in quantum mechanics

    The Rams should make all their linemen learn algebra. 😉

    Agamemnon

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