On convergence rates of adaptive ensemble Kalman inversion for linear ill-posed problems

TL;DR

This paper analyzes the convergence rates of an adaptive ensemble Kalman inversion method for linear ill-posed problems, introducing a Nyström-based sampling scheme and proving order optimal regularization properties.

Contribution

It introduces a Nyström-based sampling scheme and an adaptive ensemble Kalman inversion method with proven order optimal regularization performance.

Findings

The adaptive scheme achieves order optimal regularization.

Nyström sampling improves practical performance.

Numerical results validate theoretical findings.

Abstract

In this paper we discuss a deterministic form of ensemble Kalman inversion as a regularization method for linear inverse problems. By interpreting ensemble Kalman inversion as a low-rank approximation of Tikhonov regularization, we are able to introduce a new sampling scheme based on the Nystr\"om method that improves practical performance. Furthermore, we formulate an adaptive version of ensemble Kalman inversion where the sample size is coupled with the regularization parameter. We prove that the proposed scheme yields an order optimal regularization method under standard assumptions if the discrepancy principle is used as a stopping criterion. The paper concludes with a numerical comparison of the discussed methods for an inverse problem of the Radon transform.