A note on the degree of ill-posedness for mixed differentiation on the d-dimensional unit cube

被引：3

作者：

Hofmann B. ^{[1
]}

Fischer H.-J. ^{[2
]}

机构：

[1] Faculty of Mathematics, Chemnitz University of Technology, Chemnitz

[2] Dresden

来源：

Journal of Inverse and Ill-Posed Problems | 2023年 / 31卷 / 06期

关键词：

copula density; degree of ill-posedness; mixed differentiation; singular value asymptotics;

D O I：

10.1515/jiip-2023-0025

中图分类号：

O172 [微积分];

学科分类号：

摘要：

Numerical differentiation of a function over the unit interval of the real axis, which is contaminated with noise, by inverting the simple integration operator J mapping in L2 is discussed extensively in the literature. The complete singular system of the compact operator J is explicitly given with singular values σn(J) asymptotically proportional to 1/n. This indicates a degree one of ill-posedness for the associated inverse problem of differentiation. We recall the concept of the degree of ill-posedness for linear operator equations with compact forward operators in Hilbert spaces. In contrast to the one-dimensional case, there is little specific material available about the inverse problem of mixed differentiation, where the d-dimensional analog J d to J, defined over unit d-cube, is to be inverted. In this note, we show for that problem that the degree of ill-posedness stays at one for all dimensions d ϵ ℕ. Some more discussion refers to the two-dimensional case in order to characterize the range of the operator J2. © 2023 Walter de Gruyter GmbH, Berlin/Boston.

引用

页码：949 / 957

页数：8

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