Mathematics – Numerical Analysis
Scientific paper
2003-01-31
Mathematics
Numerical Analysis
Scientific paper
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.
Galstian Anahit
Ramm Alexander G.
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