Adaptive regularization of convolution type equations in anisotropic spaces with fractional order of smoothness

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Description:	Burenkov, V (Cardiff University) Friday 14 February 2014, 09:45-10:30


Created:	2014-02-28 14:08
Collection:	Inverse Problems
Publisher:	Isaac Newton Institute
Copyright:	Burenkov, V
Language:	eng (English)


Abstract:	Co-authors: Tamara Tararykova (Cardiff University (UK)), Theophile Logon (Cocody University (Cote d'Ivoir)) Under consideration are multidimensional convolution type equations with kernels whose Fourier transforms satisfy certain anisotropic conditions characterizing their behaviour at infinity. Regularized approximate solutions are constructed by using a priori information about the exact solution 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 which is related to minimizing a Tikhonov smoothing functional involving the norms of the spaces F and G. Moreover, the choice of the spaces F and G is adapted to the properties of the kernel. It is important that the anisotropic smoothness parameter of the space F may be arbitrarily small and hence the a priori regularity assumption on the exact solution may be very weak. However, the regularized solutions still converge to the exact one in the appropriate sense (though, of course, the weaker 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 function with a power singularity which is the case in some problems arising in geophysics. Estimates are obtained characterizing the smootheness of the regularized solutions and the rate of convergence of the regularized solutions to the exact one. Similar results are obtained for the case of periodic convolution type equations.

Abstract:

Co-authors: Tamara Tararykova (Cardiff University (UK)), Theophile Logon (Cocody University (Cote d'Ivoir))
Under consideration are multidimensional convolution type equations with kernels whose Fourier transforms satisfy certain anisotropic conditions characterizing their behaviour at infinity. Regularized approximate solutions are constructed by using a priori information about the exact solution 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 which is related to minimizing a Tikhonov smoothing functional involving the norms of the spaces F and G. Moreover, the choice of the spaces F and G is adapted to the properties of the kernel. It is important that the anisotropic smoothness parameter of the space F may be arbitrarily small and hence the a priori regularity assumption on the exact solution may be very weak. However, the regularized solutions still converge to the exact one in the appropriate sense (though, of course, the weaker 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 function with a power singularity which is the case in some problems arising in geophysics. Estimates are obtained characterizing the smootheness of the regularized solutions and the rate of convergence of the regularized solutions to the exact one. Similar results are obtained for the case of periodic convolution type equations.

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