Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Linear convergence of a one-cut conditional gradient method for total variation regularization

Giacomo Cristinelli, José A. Iglesias, Daniel Walter

TL;DR

The authors address TV-regularized convex minimization with a PDE-based forward operator by introducing a one-cut generalized conditional gradient method. Each iteration solves a single prescribed mean-curvature problem and updates the active set via decomposition into indecomposable components, enabling a graph-cut discretization that speeds up computation. They establish global sublinear convergence and, in 2D under structural assumptions, asymptotic linear convergence, supported by a deformation-based stability analysis. Numerical experiments on parabolic and elliptic PDE-constrained problems demonstrate efficient performance and favorable convergence behavior, with substantial practical speedups over previous approaches.

Abstract

We introduce a fully-corrective generalized conditional gradient method for convex minimization problems involving total variation regularization on multidimensional domains. It relies on alternatively updating an active set of subsets of the spatial domain and an iterate given by a conic combination of the associated characteristic functions. Different to previous approaches in the same spirit, the computation of a new candidate set only requires the solution of one prescribed mean curvature problem, instead of the resolution of a fractional minimization task analogous to finding a generalized Cheeger set. After discretization, the former can be realized by a single run of a graph cut algorithm, leading to a significant speedup in practice. We prove the global sublinear convergence of the resulting method, under mild assumptions, and its asymptotic linear convergence in a more restrictive two-dimensional setting which uses results of stability of surfaces of prescribed mean curvature under perturbations of the curvature. Finally, we numerically demonstrate this convergence behavior in some model PDE-constrained minimization problems.

Linear convergence of a one-cut conditional gradient method for total variation regularization

TL;DR

The authors address TV-regularized convex minimization with a PDE-based forward operator by introducing a one-cut generalized conditional gradient method. Each iteration solves a single prescribed mean-curvature problem and updates the active set via decomposition into indecomposable components, enabling a graph-cut discretization that speeds up computation. They establish global sublinear convergence and, in 2D under structural assumptions, asymptotic linear convergence, supported by a deformation-based stability analysis. Numerical experiments on parabolic and elliptic PDE-constrained problems demonstrate efficient performance and favorable convergence behavior, with substantial practical speedups over previous approaches.

Abstract

We introduce a fully-corrective generalized conditional gradient method for convex minimization problems involving total variation regularization on multidimensional domains. It relies on alternatively updating an active set of subsets of the spatial domain and an iterate given by a conic combination of the associated characteristic functions. Different to previous approaches in the same spirit, the computation of a new candidate set only requires the solution of one prescribed mean curvature problem, instead of the resolution of a fractional minimization task analogous to finding a generalized Cheeger set. After discretization, the former can be realized by a single run of a graph cut algorithm, leading to a significant speedup in practice. We prove the global sublinear convergence of the resulting method, under mild assumptions, and its asymptotic linear convergence in a more restrictive two-dimensional setting which uses results of stability of surfaces of prescribed mean curvature under perturbations of the curvature. Finally, we numerically demonstrate this convergence behavior in some model PDE-constrained minimization problems.

Paper Structure

This paper contains 14 sections, 20 theorems, 115 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation and standing assumptions
  3. Existence of minimizers and optimality conditions
  4. A one-cut generalized conditional gradient method
  5. A motivation of Algorithm \ref{['alg:abstractonecut']}
  6. Sublinear convergence
  7. Linear convergence under structural assumptions
  8. Preparatory results
  9. Proof of the main result
  10. Numerical results on triangular meshes with PDE constraints
  11. A parabolic example
  12. An elliptic example
  13. Practical performance and discussion
  14. Extending the prescribed curvature problem from Omega to Rd

Key Result

Theorem 2.1

Problem def:BVprob admits at least one minimizer and we have $K \bar{u}_1=K \bar{u}_2$ for all solutions $\bar{u}_1, \bar{u}_2$ to def:BVprob. Moreover, there is $M_q >0$ such that $\|\bar{u} \|_{L^q(\Omega)} \leq M_q$ for any solution $\bar{u}$ of def:BVprob.

Figures (3)

  • Figure 1: Toy control $u_d$ and output $\bar{u}$ of Algorithm \ref{['alg:abstractonecut']} for the parabolic problem
  • Figure 2: Desired state $y_d$ and output $\bar{u}$ of Algorithm \ref{['alg:abstractonecut']} for the elliptic problem
  • Figure 3: Convergence rate and residual of Algorithm \ref{['alg:abstractonecut']} in the two examples.

Theorems & Definitions (41)

  • Theorem 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Remark 3.1
  • Lemma 3.2
  • ...and 31 more