Deautoconvolution in the two-dimensional case

Yu  Deng; Bernd  Hofmann; Frank  Werner

doi:doi:10.1553/etna_vol59s24

Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

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Deautoconvolution in the two-dimensional case

Yu Deng,

Bernd Hofmann,

Frank Werner

ETNA - Electronic Transactions on Numerical Analysis, pp. 24-42, 2023/04/19

doi: 10.1553/etna_vol59s24

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doi:10.1553/etna_vol59s24

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 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.

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

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Published Online: 2023/04/19 13:18:39

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Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

Deautoconvolution in the two-dimensional case

Abstract