Nonlinear estimation for linear inverse problems with error in the operator

TL;DR

This paper introduces two nonlinear methods combining Galerkin regularization and wavelet thresholding for linear inverse problems with unknown operators, achieving rate-optimality and adaptivity over Besov classes.

Contribution

The paper proposes novel nonlinear estimation techniques for inverse problems with unknown operators, demonstrating their optimality and adaptivity in a statistical framework.

Findings

Methods are rate-optimal for Besov classes.

Performance depends on operator sparsity.

Achieves adaptivity over a range of function spaces.

Abstract

We study two nonlinear methods for statistical linear inverse problems when the operator is not known. The two constructions combine Galerkin regularization and wavelet thresholding. Their performances depend on the underlying structure of the operator, quantified by an index of sparsity. We prove their rate-optimality and adaptivity properties over Besov classes.