BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Nicola Guglielmi (Gran Sasso Science Institute) DTSTART:20260330T111500Z DTEND:20260330T120000Z DTSTAMP:20260417T004418Z UID:cam/93 DESCRIPTION:Title: The power of rank-1 perturbations in matrix nearness problems.\nby Nicola Guglielmi (Gran Sasso Science Institute) as part of CAM seminar\n\nLectur e held in MV:L14.\n\nAbstract\nThis talk concerns matrix nearness problems related to eigenvalues\, singular values\, and pseudospectra. Such proble ms arise in a wide range of applications\, including dynamical systems\, w here they appear in questions of robust stability and control\, as well as graph-based problems such as clustering and ranking. These problems lead to algorithms based on structured matrix perturbations that move eigenvalu es\, singular values\, or Rayleigh quotients to prescribed locations.\n\nR emarkably\, the optimal perturbations are often of rank one\, or projectio ns of rank-1 matrices onto a given linear structure\, such as a prescribed sparsity pattern. In the approach presented here\, these optimal rank-1 p erturbations are computed by a two-level iteration combining eigenvalue op timization with root-finding.\n\nThe eigenvalue optimization problem\, wit h equality or inequality constraints on the perturbation size\, is solved using gradient-based differential equations for rank-1 matrices. In practi ce\, this amounts to evolving two vectors that represent the rank-1 matrix toward a stationary point. The root-finding component determines the opti mal perturbation size by solving a scalar nonlinear equation. The two algo rithmic components can either be nested or applied in an alternating fashi on. Numerical experiments illustrate the efficiency of the approach for se veral eigenvalue- and singular-value–based nearness problems. \nThe talk is largely based on a recent monograph written jointly with Christian Lub ich.\n LOCATION:https://researchseminars.org/talk/cam/93/ END:VEVENT END:VCALENDAR