Parallel algorithms for non-classical problems with big computational costs

This research focuses on parallel algorithms, which help to solve limited memory and computational time problems. In the second chapter of the dissertation parallel algorithms for various most recent numerical methods were studied and compared. These methods are based on the general approach: the given non-local differential problem with fractional powers of the Laplacian is transformed to a local differential problem of elliptic or pseudo-parabolic type, but formulated in a higher dimensional space R^(d+1), if Ω ⊂ R^d . The scalability and convergence analysis of parallel algorithms was performed. Recommendations to achieve a given accuracy for the provided fractional power coefficient were specified. In the third chapter the detailed analysis of absorbing boundary conditions for the linear Schrödinger equation was performed. Recommendations for constructing absorbing boundary conditions for the one-dimensional Schrödinger equation using methods based on the approximation of exact transparent boundary conditions by rational functions were presented. The proposed methodology has shown, that it is possible to find the accurate absorbing boundary conditions for four qualitatively different tasks. In the fourth chapter a three-level parallelisation scheme was proposed. The possibilities of this methodology are demonstrated for solving local optimization problems. The proposed three-level scheme increases the amount of computational resources, which can be used efficiently