On deconvolution problems: numerical aspects

Abstract

An optimal algorithm is described for solving the deconvolution problem of the form ${\bf k}u:=\int_0^tk(t-s)u(s)ds=f(t)$ given the noisy data $f_δ$, $||f-f_δ||\leq δ.$ The idea of the method consists of the representation ${\bf k}=A(I+S)$, 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.