Participants:

Maya Neytcheva (Scientific Computing, Uppsala University)

Niklas Fors (MSc student, Scientific Computing, Uppsala University)

Problem description In recent years computing has undoubtedly become a third branch of research, commonly referred to as Scientific Computing, joining the traditional practices of theoretical research and laboratory or field experiments. Even more, due to the availability of large high performance computers, the science and engineering community has the opportunity to simulate very complex physical phenomena with scales and accuracy yet unattained, experiencing in this way the advantages of computer simulations over experimental tests. Together with the growth in computing power, the interest and demand in fast and reliable numerical solution methods to perform the simulations has much increased. In the very core of the above mentioned scientific computations lies the task to solve systems of linear equations, stand-alone or as a part of a nonlinear problem, and is required within all numerical simulations, irrelevant which is the scientific field. What makes this task to be of continuous importance and interest for both numerical analysts and practitioners, is the fact that the size of the linear systems (or the number of degrees of freedom of the unknown solution) becomes very large (of order of several hundreds of millions) when we address nowadays problems of utmost importance, for instance in weather pollution, climate changes, geophysics, (bio-)medical applications to name a few. The large scale of the computations adds additional requirements to the classical robustness and accuracy of the solution methods, namely, high computational efficiency and reduced memory demands. Therefore, a lot of research is devoted to the so-called iterative solution methods as a feasible alternative to the traditionally used direct solution methods, which latter have very high demands for computer resources, in particular for three-dimensional applications. The iterative solution methods have already been shown to be an attractive alternative of the direct solvers and to be the methods of choice for many important applications. There are many successful stories to cite, in particular when the so-called optimal methods have been used (such as various multilevel and multigrid techniques). Here optimal is used in the sense that the total computational complexity of the iterative solution method is linearly (up to some constant) proportional to the number of degrees of freedom. However, the complexity of the target problems has now increased even more, which has lead to seeking new ways to further enhance the efficiency of the iterative solution methods. This project will deal with the latter topic, namely, study and apply novel techniques in the construction of iterative solution methods, utilizing some extra available (method- or problem-dependent) information.