Volume 59, pp. 24-42, 2023.

Deautoconvolution in the two-dimensional case

Yu Deng, Bernd Hofmann, and Frank Werner

Abstract

There is extensive mathematical literature on the inverse problem of deautoconvolution for a function with support in the unit interval $[0,1]\subset \mathbb{R}$, but little is known about the multidimensional situation. This article tries to fill this gap with analytical and numerical studies on the reconstruction of a real function of two real variables over the unit square from observations of its autoconvolution on $[0,2{]}^2\subset {\mathbb{R}}^2$ (full data case) or on $[0,1{]}^2$ (limited data case). In an $L^2$-setting, twofoldness and uniqueness assertions are proven for the deautoconvolution problem in 2D. Moreover, its ill-posedness is characterized and illustrated. Extensive numerical case studies give an overview of the behaviour of stable approximate solutions to the two-dimensional deautoconvolution problem obtained by Tikhonov-type regularization with different penalties and the iteratively regularized Gauss–Newton method.

Full Text (PDF) [2.1 MB], BibTeX , DOI: 10.1553/etna_vol59s24

Key words

deautoconvolution, inverse problem, ill–posedness, case studies in 2D, Tikhonov-type regularization, iteratively regularized Gauss–Newton method

AMS subject classifications

47J06, 65R32, 45Q05, 47A52, 65J20