Gradient Flows on Sparse Representations for PDE‐Regularized Inverse Problems

ABSTRACT

In this short paper, we summarize the theory of Atomic Gradient Flows (AGF) introduced in (Amend, Carioni and Zemas, arxiv:2603.25675, 2026), which provides a framework to construct gradient flows on sparse representations for convex regularized optimization problems. Then, we apply such theory to the case of inverse problems regularized with the total variation of a scalar partial differential operator. Building on the results in (Bredies, Carioni, Calc. Var. Partial Differ. Equ., 2020), we show that sparse representations can be characterized as linear combinations of fundamental solutions applied to Dirac deltas. By analyzing the geometric properties of the associated atoms, we verify that the assumptions of the general AGF framework are satisfied in this setting. As a consequence, we prove that the corresponding AGFs are well‐defined, enjoy key structural properties (e.g., uniqueness and contraction estimates) and admit a lifting to the space of probability measures on atoms. Finally, we prove that in this specific case, AGFs reduce to Euclidean gradient flows on the underlying domain and can be easily implemented to numerically compute solutions to inverse problems regularized with the total variation norm of the Laplacian.