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

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.

An extended Krylov subspace method for decoding edge-based compressed images by homogeneous diffusion

“…They also turned out to be useful in inverse problems, in general (e.g. [7,9,20,32]). If A is an operator or an arbitrarily large matrix with a field-of-values in the left complex half-plane, the matrix exponential times a vector can be approximated reliably for an arbitrary time t > 0 under reasonable assumptions on the vector (cf.…”