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SUMMARY:Variational discretizations of gauge field theories using group-eq
 uivariant interpolation spaces - Melvin Leok (University of California\, S
 an Diego)
DTSTART:20191001T100000Z
DTEND:20191001T110000Z
UID:TALK130561@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:Variational integrators are geometric structure-preserving num
 erical methods that preserve the symplectic structure\, satisfy a discrete
  Noether&#39\;s theorem\, and exhibit exhibit excellent long-time energy s
 tability properties. An exact discrete Lagrangian arises from Jacobi&#39\;
 s solution of the Hamilton-Jacobi equation\, and it generates the exact fl
 ow of a Lagrangian system. By approximating the exact discrete Lagrangian 
 using an appropriate choice of interpolation space and quadrature rule\, w
 e obtain a systematic approach for constructing variational integrators. T
 he convergence rates of such variational integrators are related to the be
 st approximation properties of the interpolation space.<br> <br> Many gaug
 e field theories can be formulated variationally using a multisymplectic L
 agrangian formulation\, and we will present a characterization of the exac
 t generating functionals that generate the multisymplectic relation. By di
 scretizing these using group-equivariant spacetime finite element spaces\,
  we obtain methods that exhibit a discrete multimomentum conservation law.
  We will then briefly describe an approach for constructing group-equivari
 ant interpolation spaces that take values in the space of Lorentzian metri
 cs that can be efficiently computed using a generalized polar decompositio
 n. The goal is to eventually apply this to the construction of variational
  discretizations of general relativity\, which is a second-order gauge fie
 ld theory whose configuration manifold is the space of Lorentzian metrics.
LOCATION:Seminar Room 1\, Newton Institute
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