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On deconvolution methods

Alexander G. Ramm·A. Galstian
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Abstract

Several methods for solving efficiently the one-dimensional deconvolution problem are proposed. The problem is to solve the Volterra equation ${\mathbf k} u:=\int_0^t k(t-s)u(s)ds=g(t),\quad 0\leq t\leq T$. The data, $g(t)$, are noisy. Of special practical interest is the case when the data are noisy and known at a discrete set of times. A general approach to the deconvolution problem is proposed: represent ${\mathbf k}=A(I+S)$, where a method for a stable inversion of $A$ is known, $S$ is a compact operator, and $I+S$ is injective. This method is illustrated by examples: smooth kernels $k(t)$, and weakly singular kernels, corresponding to Abel-type of integral equations, are considered. A recursive estimation scheme for solving deconvolution problem with noisy discrete data is justified mathematically, its convergence is proved, and error estimates are obtained for the proposed deconvolution method.

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Alexander G. Ramm, A. Galstian. 2003-01-31. On deconvolution methods. https://arxiv.org/abs/math/0301382

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