On deconvolution methods
Alexander G. Ramm, A. Galstian
TL;DR
The paper addresses stable deconvolution for Volterra-type integral equations with noisy data, introducing a Laplace-domain regularization that uses a transform filter $f_N(\lambda)=(1+\lambda/N)^{-m}$ and a general decomposition ${\mathbf k}=A(I+S)$. It proves convergence and rate results under spectral-growth assumptions on $K(\lambda)$, and presents a causality-preserving recursive estimator for discrete measurements with provable convergence and rates under Hölder smoothness. Additional contributions include a structured proof framework for the proposed regularizers and a generalization to operator-valued kernels, including matrix-valued kernels and multi-dimensional systems. Overall, the methods provide practical, theoretically grounded tools for stable deconvolution in both continuous and discrete-data settings, with broad applicability to ill-posed Volterra-type problems.
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.