  Krylov subspace methods are a powerful family of iterative solvers for linear
systems of equations, which are commonly used for inverse problems due to their
intrinsic regularization properties. Moreover, these methods are naturally
suited to solve large-scale problems, as they only require matrix-vector
products with the system matrix (and its adjoint) to compute approximate
solutions, and they display a very fast convergence. Even if this class of
methods has been widely researched and studied in the numerical linear algebra
community, its use in applied medical physics and applied engineering is still
very limited. e.g. in realistic large-scale Computed Tomography (CT) problems,
and more specifically in Cone Beam CT (CBCT). This work attempts to breach this
gap by providing a general framework for the most relevant Krylov subspace
methods applied to 3D CT problems, including the most well-known Krylov solvers
for non-square systems (CGLS, LSQR, LSMR), possibly in combination with
Tikhonov regularization, and methods that incorporate total variation (TV)
regularization. This is provided within an open source framework: the
Tomographic Iterative GPU-based Reconstruction (TIGRE) toolbox, with the idea
of promoting accessibility and reproducibility of the results for the
algorithms presented. Finally, numerical results in synthetic and real-world 3D
CT applications (medical CBCT and {\mu}-CT datasets) are provided to showcase
and compare the different Krylov subspace methods presented in the paper, as
well as their suitability for different kinds of problems.

Numerical Analysis;Numerical Analysis