Adaptive discretization for signal detection in statistical inverse problems

Abstract

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.

Keywords:

• inverse problems
• inverse test
• linear regularization
• projection scheme

AMS Subject Classifications:

• 62G05
• 62K20

Notes

Dedicated to Bernd Hofmann on the occasion of his 60th birthday.

1 The paradigm here is that ‘Absence of evidence is not evidence of absence’, and we refer to [4].