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 the authors in 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.

中文翻译：

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多凸变分正则化的范围条件

摘要 在凸变分正则化的背景下，已知的结果是，在适当的可微性假设下，变分不等式形式的源条件隐含着范围条件，而相反的蕴涵仅在算子受到额外限制的情况下成立。在本文中，我们证明了多凸正则化的类似结果。更准确地说，我们证明作者在 2017 年推导出的变分不等式意味着正则化函数的导数必须位于算子导数的对偶伴随范围内。此外，我们还展示了如何调整对运算符的限制以获得相反的含义。