Conditional gradients for total variation regularization with PDE constraints: a graph cuts approach

TL;DR

This work addresses PDE-constrained optimization with total variation regularization by reframing TV as a convex-geometric object via the TV ball and exploiting its extremal points to drive an efficient FC-GCG method; the insertion steps are solved with a Dinkelbach-Newton scheme implemented through graph cuts, enabling scalable PDE solves. It proves discrete-to-continuum convergence to an anisotropic TV $\mathrm{TV}_{\varphi}$, with an explicit octagonal anisotropy for planar double-diagonal meshes, linking mesh geometry to regularization. Numerical experiments in 2D and 3D validate fast convergence, demonstrate mesh-induced anisotropy, and illustrate practical performance on challenging PDE-constrained TV problems. Overall, the paper contributes a practical, geometry-aware framework for TV-regularized PDE control with rigorous convergence guarantees and efficient, graph-cut-based subproblem solutions.

Abstract

Total variation regularization has proven to be a valuable tool in the context of optimal control of differential equations. This is particularly attributed to the observation that TV-penalties often favor piecewise constant minimizers with well-behaved jumpsets. On the downside, their intricate properties significantly complicate every aspect of their analysis, from the derivation of first-order optimality conditions to their discrete approximation and the choice of a suitable solution algorithm. In this paper, we investigate a general class of minimization problems with TV-regularization, comprising both continuous and discretized control spaces, from a convex geometry perspective. This leads to a variety of novel theoretical insights on minimization problems with total variation regularization as well as tools for their practical realization. First, by studying the extremal points of the respective total variation unit balls, we enable their efficient solution by geometry exploiting algorithms, e.g. fully-corrective generalized conditional gradient methods. We give a detailed account on the practical realization of such a method for piecewise constant finite element approximations of the control on triangulations of the spatial domain. Second, in the same setting and for suitable sequences of uniformly refined meshes, it is shown that minimizers to discretized PDE-constrained optimal control problems approximate solutions to a continuous limit problem involving an anisotropic total variation reflecting the fine-scale geometry of the mesh.

Figures (12)

• Figure 1: Support and graph of a $P_1(\mathcal{T})$ function with two contiguous nonzero vertex values, on a triangulation of $\mathbb{R}^2$ with equilateral triangles of equal area.
• Figure 2: The set $\partial \mathfrak{B}_{1,2}$ for $\lambda = (4\sqrt{3}+\sqrt{11})^{-1}$ in black, and the $\ell^1$ ball in $\mathbb{R}^2$ in green.
• Figure 3: Our triangulated domain $(\Omega,\mathcal{T})$ in black, and the associated $(s,t)$-graph in red. On the right, an example of $(s,t)$-cut (blue vs green) for the unweighted $(s,t)$-graph, with the edges passing through the cut (i.e. connecting different color nodes) in red.
• Figure 4: Representing the regular 'double diagonal' triangulation of the plane as interactions of periodic lattice systems, with the shaded central square being a complete period. The red segments represent the interactions corresponding to the thicker black edges of the triangulation. The assignment of Proposition \ref{['prop:periodicconv']} is depicted on the left, with the triangles themselves being the Voronoi cells. For the alternative construction on the right using the regular lattice $\frac{1}{2k}\mathbb{Z}^2$ each triangle is assigned counterclockwise to a red point, and the corresponding dashed blue square Voronoi cell has the same area.
• Figure 5: A sketch of the constructions in the proof of Lemma \ref{['lem:discretedegiorgi']}. In this case $\nu=(2,-1)/\sqrt{5}$.

Theorems & Definitions (63)

• Proposition 2.1
• Remark 2.2
• Proposition 2.3
• proof
• Theorem 2.4
• proof
• Definition 2.5
• Theorem 2.6
• proof
• Proposition 2.7