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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

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Volume 18, Issue 4


Minimax Rates for Statistical Inverse Problems Under General Source Conditions

Litao Ding / Peter Mathé
Published Online: 2017-12-05 | DOI: https://doi.org/10.1515/cmam-2017-0055


We describe the minimax reconstruction rates in linear ill-posed equations in Hilbert space when smoothness is given in terms of general source sets. The underlying fundamental result, the minimax rate on ellipsoids, is proved similarly to the seminal study by D. L. Donoho, R. C. Liu, and B. MacGibbon [4]. These authors highlighted the special role of the truncated series estimator, and for such estimators the risk can explicitly be given. We provide several examples, indicating results for statistical estimation in ill-posed problems in Hilbert space.

Keywords: Statistical Inverse Problem; General Source Condition; Minimax Rate

MSC 2010: 65J22; 62G20


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About the article

Received: 2017-08-09

Revised: 2017-10-30

Accepted: 2017-11-17

Published Online: 2017-12-05

Published in Print: 2018-10-01

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11331004

Award identifier / Grant number: 11421110002

The first author is supported by the National Natural Science Foundation of China (Grants No. 11331004 and No. 11421110002) and the Program of Introducing Talents of Discipline to Universities (Grant No. B08018).

Citation Information: Computational Methods in Applied Mathematics, Volume 18, Issue 4, Pages 603–608, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2017-0055.

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