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Last edited on
Oct 18, 2021 by JJ
.
Thesis:
Master in Mathematics

Author:
Daniel Rasskasow

Title:
A mollifying approach to the deconvolution problem

Supervisor:
Jan JOHANNES

Abstract:
Deconvolution is a problem that occurs in many statistic influenced fields wherever data is gathered in order to draw conclusions about the underlying population. Such data is naturally erroneous due to inaccurate methods of data collection. The estimation of the density of the data leads to two obstacles. First: For linear, ill- posed, statistical inverse problems like this, regularization methods are needed. Second: The estimation of densities usually takes place in the spectral - in this case their Fourier transform. The numerical realization of an estimated Fourier transform combined with regularization techniques is prone to over- and under- vibrations of the real function. This is better known as the Gibbs phenomenon. In this work, we introduce regularization by mollification which tackles both problems and we compare its asymptotical behaviour to classic approaches like the Tikhonov- or the spectral cut-off regularization both theoretical and practical.

Reference:
L. Simar, P. Maréchal und A. Vanhems. A mollifier approach to the deconvolution of probability densities, TSE Working Paper, n. 18–965, 2018.