On deconvolution methods

Several methods for solving efficiently the one-dimensional deconvolution problem are proposed. The problem is to solve the Volterra equation k u:=∫ 0 tk(t−s)u(s) ds=g(t), 0⩽t⩽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...

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Bibliographic Details
Published in:International journal of engineering science 2003, Vol.41 (1), p.31-43
Main Authors: Ramm, Alexander G., Galstian, Anahit
Format: Article
Language:English
Subjects:
Deconvolution
Exact sciences and technology
Ill-posed problems
Integral equations
Mathematical analysis
Mathematics
Sciences and techniques of general use
Volterra equations
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Summary:Several methods for solving efficiently the one-dimensional deconvolution problem are proposed. The problem is to solve the Volterra equation k u:=∫ 0 tk(t−s)u(s) ds=g(t), 0⩽t⩽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 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.
ISSN:0020-7225
1879-2197
DOI:10.1016/S0020-7225(02)00145-3