Convergence rates for regularization functionals with polyconvex integrands

Abstract

Convergence rates results for variational regularization methods typically assume the regularization functional to be convex. While this assumption is natural for scalar-valued functions, it can be unnecessarily strong for vector-valued ones. In this paper we focus on regularization functionals with polyconvex integrands. Even though such functionals are nonconvex in general, it is possible to derive linear convergence rates with respect to a generalized Bregman distance, an idea introduced by Grasmair in 2010. As a case example we consider the image registration problem.

A Range Condition for Polyconvex Variational Regularization

“…Elements of ∂ poly R(u) are called W poly -subgradients of R at u. Concerning existence of W poly -subgradients we have shown the following result in [10].…”

“…This section is a brief summary of the most important prerequisites from [10]. For N, n ∈ N we will frequently identify matrices in R N ×n with vectors in R N n .…”

“…We denote the effective domain of R by dom R = {u ∈ U : R(u) < +∞}. Following [7,10,12] we define the W poly -subdifferential of R at u ∈ dom R as…”

“…In this article we prove the analogous result for polyconvex regularization. More precisely, we show that the variational inequality derived in [10] 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.…”

“…By means of the corresponding W poly -Bregman distance we were then able to translate the convergence rates result of [8] to the polyconvex setting. The source condition derived in [10] reads…”