In recent years, a major trend towards solving scientific problems of ever larger scales can be observed in practically all academic and industrial applications. These include the simulation of vast ocean circulation models, global climate prediction models, seismic oil reservoir models spanning hundreds of kilometers, extremely fine-scale combustion models, etc. The representation of these models on a computer requires the solution of a large-scale system of equations that typically consists of millions of unknowns. Due to the huge size of these model calculations, computations are often spread across parallel computer platforms to reduce computational time. Furthermore, only the numerical methods with optimal compute and communication complexity are able to efficiently solve these large scale problems. Krylov methods have been established as the benchmark iterative solvers for sparse linear algebra problems due to their robustness and good performance in function of the number of unknowns. However, present-day Krylov methods are not adapted to scale to future parallel hardware. Hence, new numerical methods have to be designed and analyzed mathematically, taking into account numerical rounding error propagation, which possibly has a detrimental effect on convergence. The aim of this project is to develop and analyze new scalable iterative methods that are numerically stable and resilient to the errors that typically arise in these large-scale computations.