Variance-covariance matrix estimation with LSQR in a parallel programming environment

Knowledge about the gravity eld allows an insight into the structure and dynamics of the earth. It provides the geoid as the most important physical reference surface in geodesy and oceanography. Since 2000, the CHAMP (CHallenging Mini-satellite Payload) mission detects the structure of the global gravity eld, followed by the launch of GRACE (Gravity Recovery And Climate Experiment) in 2002. In 2008, nally, the GOCE (Gravity eld and steady-state Ocean Circulation Explorer) satellite is supposed to be set in orbit. These missions demonstrate satellite-based gravity eld recovery to be at the center of geo-scienti c interest. Interpretation and evaluation of satellite observations are di cult, especially the determination of the unknown gravity eld parameters from a huge amount of measurements. Because of the immense demand for memory and computing time, the occurring systems of equations pose a real numerical challenge. Therefore, High-Performance Computing (HPC) is commonly adopted to overcome computational problems. Basically, parallel programming with MPI and OpenMP routines allows to speed up the solution process considerably. In this thesis, rstly global gravity eld modelling by means of satellite observations is reviewed. Secondly, the LSQR method (Least-Squares using QR factorization) is introduced in detail in order to solve the resulting least-squares problems. Because the LSQR method is an iterative solver, it basically can not provide the variance-covariance information of the parameter estimate. To investigate the approximate computation of the variance-covariance matrix, two methods are introduced. The rst one is based on the generalized inverse of the design matrix. The second approach applies Monte-Carlo integration techniques. Because parallel programming is very helpful to implement such iterative methods, it is necessary to introduce some basic principles and concepts about HPC.