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19 avril 2023
- handle: 10670/1.dqelbt
- issn: 1068-9613
- doi: 10.1553/etna_vol59s24
info:eu-repo/semantics/openAccess
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Bernd Hofmann et al., « Deautoconvolution in the two-dimensional case. ETNA - Electronic Transactions on Numerical Analysis », Elektronisches Publikationsportal der Österreichischen Akademie der Wissenschafte, ID : 10.1553/etna_vol59s24
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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 numericalcase 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.