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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Adaptive regularization of convolution type equati
ons in anisotropic spaces with fractional order of
smoothness - Burenkov\, V (Cardiff University)
DTSTART;TZID=Europe/London:20140214T094500
DTEND;TZID=Europe/London:20140214T103000
UID:TALK50883AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/50883
DESCRIPTION:Co-authors: Tamara Tararykova (Cardiff University
(UK))\, Theophile Logon (Cocody University (Cote d
'Ivoir)) \n\nUnder consideration are multidimensio
nal convolution type equations with kernels whose
Fourier transforms satisfy certain anisotropic con
ditions characterizing their behaviour at infinity
. Regularized approximate solutions are constructe
d by using a priori information about the exact so
lution and the error\, characterized by membership
in some anisotropic Nikol'skii-Besov spaces with
fractional order of smoothness: F\, G respectively
. The regularized solutions are defined in a way w
hich is related to minimizing a Tikhonov smoothing
functional involving the norms of the spaces F an
d G. Moreover\, the choice of the spaces F and G i
s adapted to the properties of the kernel. It is i
mportant that the anisotropic smoothness parameter
of the space F may be arbitrarily small and hence
the a priori regularity assumption on the exact s
olution may be very weak. However\, the regularize
d solutions still converge to the exact one in the
appropriate sense (though\, of course\, the weake
r are the a priori assumptions on the exact soluti
on\, the slower is the convergence). In particular
\, for sufficiently small smoothness parameter of
the space F\, the exact solution is allowed to be
an unbounded function with a power singularity whi
ch is the case in some problems arising in geophys
ics. Estimates are obtained characterizing the smo
otheness of the regularized solutions and the rate
of convergence of the regularized solutions to th
e exact one. Similar results are obtained for the
case of periodic convolution type equations.\n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:Mustapha Amrani
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