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UID:20251007T003509EDT-2241MTuo4M@132.216.98.100
DTSTAMP:20251007T043509Z
DESCRIPTION:Seminar CRM CAMP In Nonlinear Analysis\n	En ligne / Web. Pour s'
 inscrire\, veuillez visiter / For registration\, please visit: http://crm.
 math.ca/camp-nonlineaire/\n\nTitle: Beyond Exponential Complexity of Newto
 n-Galerkin Validation Methods: A Polynomial-Time Newton-Picard Validation 
 Algorithm for linear ODEs.\n\nAbstract: A wide range of techniques have be
 en developed to compute validated numerical solutions to various kind of e
 quations (e.g.\, ODE\, PDE\, DDE) arising in computer-assisted proofs. Amo
 ng them are Newton-Galerkin a posteriori validation techniques\, which pro
 vide error bounds for approximate solutions by using the contraction map p
 rinciple in a suitable coefficient space (e.g.\, Fourier or Chebyshev). Mo
 re precisely\, a contracting Newton-like operator is constructed by trunca
 ting and inverting the inverse Jacobian of the equation. While these techn
 iques were extensively used in cutting-edge works in the community\, we sh
 ow that they suffer from an exponential running time w.r.t. the input equa
 tion. We illustrate this shortcomings on simple linear ODEs\, where a 'lar
 ge' parameter in the equation leads to an intractable instance for Newton-
 Galerkin validation algorithms. From this observation\, we build a new val
 idation scheme\, called Newton-Picard\, which breaks this complexity barri
 er. The key idea consists in replacing the inverse Jacobian not by a finit
 e-dimensional truncated matrix as in Newton-Galerkin\, but by an integral 
 operator with a polynomial approximation of the so-called resolvent kernel
 . Moreover\, this method is also less basis-dependent and more suitable to
  be formalized in a computer proof assistant towards a fully certified imp
 lementation in the future.\n\nWeb site : http://crm.umontreal.ca/camp-nonl
 inear/\n
DTSTART:20210223T150000Z
DTEND:20210223T160000Z
SUMMARY:Florent Bréhard\, Uppsala University
URL:/mathstat/channels/event/florent-brehard-uppsala-u
 niversity-328395
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