Adaptive discretization for signal detection in statistical inverse problems

• Published in: Applicable analysis Vol. 94; no. 3; pp. 494 - 505
• Main Author: Mathe, Peter
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 04-03-2015; Taylor & Francis Ltd
• Subjects: Constants; Discretization; Inverse problems; inverse test; Linear equations; linear regularization; Linear systems; Mathematical analysis; projection scheme; Regularization; Regularization methods; Signal detection; Signaling; Statistical tests
• ISSN: 0003-6811; 1563-504X
• DOI: 10.1080/00036811.2014.900662

We discuss statistical tests in inverse problems when the original equation is replaced by a discretized one, i.e. a linear system of equations. Previous studies revealed that using the discretization level as regularizing procedure is possible, but its application is limited unless discretization is restricted to the singular value decomposition, see C. Marteau and P. Mathé, General regularization schemes for signal detection in inverse problems, 2013. General linear regularization may circumvent this, and we propose a regularization of the discretized equations. The discretization level may be chosen adaptively, which may save computational budget. This results in tests which are known to yield the optimal separation rate up to some constant in many cases.