# Solving Large-Scale Optimization Problems with a Convergence Rate Independent of Grid Size

arXiv: https://arxiv.org/abs/1805.09453 · HTML: https://mlsanity.com/paper/1805.09453/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 May 2018.

## Built on by (most cited first, 5 of 5)

- [Bridging Mean-Field Games and Normalizing Flows with Trajectory Regularization](https://mlsanity.com/paper/2206.14990/) (2022)
- [Efficient Natural Gradient Descent Methods for Large-Scale PDE-Based Optimization Problems](https://mlsanity.com/paper/2202.06236/) (2022)
- [Unsupervised Solution Operator Learning for Mean-Field Games via Sampling-Invariant Parametrizations](https://mlsanity.com/paper/2401.15482/) (2024)
- [A Natural Primal-Dual Hybrid Gradient Method for Adversarial Neural Network Training on Solving Partial Differential Equations](https://mlsanity.com/paper/2411.06278/) (2024)
- [Beckman Defense](https://mlsanity.com/paper/2301.01495/) (2023)

## Abstract (arXiv)

We present a primal-dual method to solve L1-type non-smooth optimization problems independently of the grid size. We apply these results to two important problems : the Rudin-Osher-Fatemi image denoising model and the L1 earth mover's distance from optimal transport. Crucially, we provide analysis that determines the choice of optimal step sizes and we prove that our method converges independently of the grid size. Our approach allows us to solve these problems on grids as large as 4096 by 4096 in a few minutes without parallelization.