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.

翻译：灵活Krylov方法是处理反问题的常用手段。特别地，当显式变分正则化超越二范数框架时（例如涉及$0 < p \leq 1$的$\ell_p$范数），这类方法能有效应对相关挑战。此外，内积自由Krylov方法在不适定问题背景下被重新审视，通过采用低精度算术来加速计算并改善内存需求。然而这些方法本质上是拟最小残差法，可与随机数值线性代数工具结合以提升结果质量。本文提出了新型灵活内积自由Krylov方法，包括适用于迭代相关预处理的新型灵活广义Hessenberg方法。同时基于草图求解框架，提出了该方法的随机化版本。文中给出了理论分析，并针对不同变分正则化项进行了数值实验，以展示新方法的性能表现。