论文标题
在二维情况下的断路卷
Deautoconvolution in the two-dimensional case
论文作者
论文摘要
关于单位间隔$ [0,1] \ subset \ mathbb r $的功能，关于deautoconvolution的反面问题有广泛的数学文献，但对多维情况知之甚少。本文试图通过分析和数值研究来填补这一空白，以从单位正方形上重建两个真实变量的实际功能，这是从$ [0,2]^2 \ subset \ subset \ mathbb r^2 $（完整数据案例）或$ [0,1]^2 $（有限的数据案例）上的$ [0,2]^2 \ subset \ subset \ subset \ subset \ subset \ subset \ subset \ subset \ subset \ subset \ subset \ subset \ subset \ subset \ subset \ subset \ subset \ mathbb r^2 $（有限的数据案例）。在$ l^2 $ setting中，双重性和独特性的断言被证明是2D中的Deautoconvolution问题。此外，它的不良性是特征和说明的。广泛的数值案例研究概述了稳定的近似解决方案对Tikhonov-type正规化所获得的二维脱氧卷积问题的行为，并具有不同的惩罚和迭代的正则高斯 - 纽顿方法。
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.
