Flexible inner-product free Krylov methods for inverse problems

Flexible Krylov methods are a common standpoint for inverse problems. In particular, they are used to address the challenges associated with explicit variational regularization when it goes beyond the two-norm, for example involving an $\ell_p$ norm for $0 < p \leq 1$. Moreover, inner-product free Krylov methods have been revisited in the context of ill-posed problems, to speed up computations and improve memory requirements by means of using low precision arithmetics. However, these are effectively quasi-minimal residual methods, and can be used in combination with tools from randomized numerical linear algebra to improve the quality of the results. This work presents new flexible and inner-product free Krylov methods, including a new flexible generalized Hessenberg method for iteration-dependent preconditioning. Moreover, it introduces new randomized versions of the methods, based on the sketch-and-solve framework. Theoretical considerations are given, and numerical experiments are provided for different variational regularization terms to show the performance of the new methods.