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- Title
A note on the degree of ill-posedness for mixed differentiation on the d-dimensional unit cube.
- Authors
Hofmann, Bernd; Fischer, Hans-Jürgen
- Abstract
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 L 2 is discussed extensively in the literature. The complete singular system of the compact operator J is explicitly given with singular values σ n (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 ∈ ℕ . Some more discussion refers to the two-dimensional case in order to characterize the range of the operator J 2 .
- Subjects
NUMERICAL functions; INVERSE problems; OPERATOR equations; CUBES; NUMERICAL differentiation
- Publication
Journal of Inverse & Ill-Posed Problems, 2023, Vol 31, Issue 6, p949
- ISSN
0928-0219
- Publication type
Article
- DOI
10.1515/jiip-2023-0025