# Tikhonov Regularization Within Ensemble Kalman Inversion

## TL;DR

This paper enhances ensemble Kalman inversion by integrating Tikhonov regularization with Sobolev penalties, improving solution stability and incorporating prior information, supported by theoretical analysis and numerical experiments.

## Contribution

It introduces a Tikhonov-regularized ensemble Kalman inversion method and analyzes its continuous-time limit, including ensemble collapse and consensus behavior.

## Key findings

- Tikhonov regularization improves inversion stability.
- The method incorporates prior information effectively.
- Numerical experiments demonstrate enhanced performance.

## Abstract

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.