2003/01/31 by Ramm, Alexander G., Galstian, A.
#45D05 #45L05 #45P05 #65R20 #65R30 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.math/0301382
Several methods for solving efficiently the one-dimensional deconvolution problem are proposed. The problem is to solve the Volterra equation \mathbf k u:=∫0t k(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 \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.