# Tikhonov Regularization Within Ensemble Kalman Inversion

arXiv: https://arxiv.org/abs/1901.10382 · HTML: https://mlsanity.com/paper/1901.10382/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jan 2019.

## Built on by (most cited first, 13 of 13)

- [Ensemble Kalman Diffusion Guidance: A Derivative-free Method for Inverse Problems](https://mlsanity.com/paper/2409.20175/) (2024)
- [Reduced-Order Autodifferentiable Ensemble Kalman Filters](https://mlsanity.com/paper/2301.11961/) (2023)
- [Neural Dynamical Operator: Continuous Spatial-Temporal Model with Gradient-Based and Derivative-Free Optimization Methods](https://mlsanity.com/paper/2311.11798/) (2023)
- [Scaling Up Bayesian Uncertainty Quantification for Inverse Problems using Deep Neural Networks](https://mlsanity.com/paper/2101.03906/) (2021)
- [Gradient-free training of neural ODEs for system identification and control using ensemble Kalman inversion](https://mlsanity.com/paper/2307.07882/) (2023)
- [Efficient, Multimodal, and Derivative-Free Bayesian Inference With Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2406.17263/) (2024)
- [Efficient Bayesian Physics Informed Neural Networks for Inverse Problems via Ensemble Kalman Inversion](https://mlsanity.com/paper/2303.07392/) (2023)
- [The Rank-Reduced Kalman Filter: Approximate Dynamical-Low-Rank Filtering In High Dimensions](https://mlsanity.com/paper/2306.07774/) (2023)
- [Blade: A Derivative-free Bayesian Inversion Method using Diffusion Priors](https://mlsanity.com/paper/2510.10968/) (2025)
- [Connections between sequential Bayesian inference and evolutionary dynamics](https://mlsanity.com/paper/2411.16366/) (2024)
- [The Ensemble Kalman Inversion Race](https://mlsanity.com/paper/2511.15853/) (2025)
- [Scalable Bayesian Physics-Informed Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2501.08501/) (2025)
- [Gradient-Free Sequential Bayesian Experimental Design via Interacting Particle Systems](https://mlsanity.com/paper/2504.13320/) (2025)

## Abstract (arXiv)

Ensemble Kalman inversion is a parallelizable methodology for solving inverse or parameter estimation problems. Although it is based on ideas from Kalman filtering, it may be viewed as a derivative-free optimization method. In its most basic form it regularizes ill-posed inverse problems through the subspace property: the solution found is in the linear span of the initial ensemble employed. In this work we demonstrate how further regularization can be imposed, incorporating prior information about the underlying unknown. In particular we study how to impose Tikhonov-like Sobolev penalties. As well as introducing this modified ensemble Kalman inversion methodology, we also study its continuous-time limit, proving ensemble collapse; in the language of multi-agent optimization this may be viewed as reaching consensus. We also conduct a suite of numerical experiments to highlight the benefits of Tikhonov regularization in the ensemble inversion context.