Deautoconvolution in the two-dimensional case

Abstract

There is extensive mathematical literature on the inverse problem of deautoconvolution for a function with support in the unit interval [0, 1] ⊂ 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 ⊂ 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.

On uniqueness and ill-posedness for the deautoconvolution problem in the multi-dimensional case

“…We are going to extend, with respect to the reconstruction of real functions with n real variables, assertions on uniqueness, ambiguity and ill-posedness that previously had been proven in the one-dimensional case. We also complement and generalize findings of our recent paper [10], where such results have been stated for the two-dimensional case. Our focus is on the reconstruction of a square integrable real function x = x(t) with t = (t 1 , t 2 , .…”

“…The discretization is achieved via the composite midpoint rule, and the corresponding discretized nonlinear optimization problem (5.1) is solved by using a damped Newton method. More details and a conceptional algorithm can be found in [10].…”

“…There, we restrict ourselves for simplicity to the classical variant of quadratic Tikhonov regularization for nonlinear operator equations along the lines of the seminal paper [12] and best possible regularization parameters. For a more detailed numerical study in the two-dimensional case we refer to [10].…”