A range condition for polyconvex variational regularization

TL;DR

This paper extends the understanding of source and range conditions from convex to polyconvex variational regularization, establishing new implications between variational inequalities and operator range conditions.

Contribution

It proves that variational inequalities imply specific range conditions for polyconvex regularization and adapts operator restrictions for the converse implication.

Findings

Variational inequalities imply range conditions in polyconvex regularization.

The derivative of the regularization functional lies in the range of the dual-adjoint of the operator.

Adapted operator restrictions allow the converse implication to hold.

Abstract

In the context of convex variational regularization it is a known result that, under suitable differentiability assumptions, source conditions in the form of variational inequalities imply range conditions, while the converse implication only holds under an additional restriction on the operator. In this article we prove the analogous result for polyconvex regularization. More precisely, we show that the variational inequality derived by Kirisits, Scherzer (2017) implies that the derivative of the regularization functional must lie in the range of the dual-adjoint of the derivative of the operator. In addition, we show how to adapt the restriction on the operator in order to obtain the converse implication.