# A conjugate-gradient-type rational Krylov subspace method for ill-posed problems

## TL;DR

This paper introduces a new regularisation scheme for ill-posed problems by extending conjugate-gradient methods to rational Krylov subspaces, achieving order-optimal regularisation.

## Contribution

It presents a novel rational Krylov subspace method based on conjugate gradients for improved regularisation of ill-posed problems.

## Key findings

- Order-optimal regularisation scheme established
- Effective for linear inverse problems
- Extends conjugate-gradient methods to rational Krylov subspaces

## Abstract

Conjugated gradients on the normal equation (CGNE) is a popular method to regularise linear inverse problems. The idea of the method can be summarised as minimising the residuum over a suitable Krylov subspace. It is shown that using the same idea for the shift-and-invert rational Krylov subspace yields an order-optimal regularisation scheme.