On deconvolution problems: numerical aspects

Abstract

An optimal algorithm is described for solving the deconvolution problem of the form $\mathbf{k}u:={\int }_0^{t}k(t-s)u(s)ds=f(t)$ given the noisy data $f_{\delta }$, $||f-f_{\delta }||\le \delta .$ The idea of the method consists of the representation $\mathbf{k}=A(I+S)$, where $S$ is a compact operator, $I+S$ is injective, $I$ is the identity operator, $A$ is not boundedly invertible, and an optimal regularizer is constructed for $A$. The optimal regularizer is constructed using the results of the paper MR 40#5130.

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