This thesis is a study on the implementation of the Gaussian Markov Random Field (GMRF) for random sample generation and also the Multilevel Monte Carlo (MLMC) method to reduce the computational costs involved with doing uncertainty quantification studies. The GMRF method is implemented in different programming environments in order to evaluate the potential performance enhancements given varying levels of language abstraction. It is seen that the GMRF method can be used to generate Gaussian Fields with a Mat{\'e}rn type covariance function and reduces the computational requirements for large scale problems. Speedups of as much as 1000 can be observed when compared to the standard Cholesky Decomposition sample generation method, even for a relatively small problem size. The MLMC method was shown to be at least 6 times faster than the standard Monte Carlo method and the speedup increases with grid size. It is also seen that in any Monte Carlo type methods, a Krylov subspace type solver is almost always recommended together with a suitable preconditioner for robust sampling. This thesis also studies the ease of implementation of these methods in varying levels of programming abstraction. The methods are implemented in different languages ranging from the most common language used by mathematicians (MATLAB), to the more performance oriented language (C++-PETSc/MPI), and ends with one of the newest programming concept (ExaStencils). The GMRF method featured in this thesis also is one of the earliest application to be implemented in ExaStencils.