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SUMMARY:Adaptive regularization of convolution type equations in anisotrop
 ic spaces with fractional order of smoothness - Burenkov\, V (Cardiff Univ
 ersity)
DTSTART:20140214T094500Z
DTEND:20140214T103000Z
UID:TALK50883@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:Co-authors: Tamara Tararykova (Cardiff University (UK))\, Theo
 phile Logon (Cocody University (Cote d'Ivoir)) \n\nUnder consideration are
  multidimensional convolution type equations with kernels whose Fourier tr
 ansforms satisfy certain anisotropic conditions characterizing their behav
 iour at infinity. Regularized approximate solutions are constructed by usi
 ng a priori information about the exact solution and the error\, character
 ized by membership in some anisotropic Nikol'skii-Besov spaces with fracti
 onal order of smoothness: F\, G respectively. The regularized solutions ar
 e defined in a way which is related to minimizing a Tikhonov smoothing fun
 ctional involving the norms of the spaces F and G. Moreover\, the choice o
 f the spaces F and G is adapted to the properties of the kernel. It is imp
 ortant that the anisotropic smoothness parameter of the space F may be arb
 itrarily small and hence the a priori regularity assumption on the exact s
 olution may be very weak. However\, the regularized solutions still conver
 ge to the exact one in the appropriate sense (though\, of course\, the wea
 ker are the a priori assumptions on the exact solution\, 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 functio
 n with a power singularity which is the case in some problems arising in g
 eophysics. Estimates are obtained characterizing the smootheness of the re
 gularized solutions and the rate of convergence of the regularized solutio
 ns to the exact one. Similar results are obtained for the case of periodic
  convolution type equations.\n
LOCATION:Seminar Room 1\, Newton Institute
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