Performance Improvements of Common Sparse Numerical Linear Algebra Computations

Manufacturers of computer hardware are able to continuously sustain an unprecedented pace of progress in computing speed of their products, partially due to increased clock rates but also because of ever more complicated chip designs. With new processor families appearing every few years, it is increasingly harder to achieve high performance rates in sparse matrix computations. This research proposes new methods for sparse matrix factorizations and applies in an iterative code generalizations of known concepts from related disciplines. The proposed solutions and extensions are implemented in ways that tend to deliver efficiency while retaining ease of use of existing solutions. The implementations are thoroughly timed and analyzed using a commonly accepted set of test matrices. The tests were conducted on modern processors that seem to have gained an appreciable level of popularity and are fairly representative for a wider range of processor types that are available on the market now or in the near future. The new factorization technique formally introduced in the early chapters is later on proven to be quite competitive with state of the art software currently available. Although not totally superior in all cases (as probably no single approach could possibly be), the new factorization algorithm exhibits a few promising features. In addition, an all-embracing optimization effort is applied to an iterative algorithm that stands out for its robustness. This also gives satisfactory results on the tested computing platforms in terms of performance improvement. The same set of test matrices is used to enable an easy comparison between both investigated techniques, even though they are customarily treated separately in the literature. Possible extensions of the presented work are discussed. They range from easily conceivable merging with existing solutions to rather more evolved schemes dependent on hard to predict progress in theoretical and algorithmic research

Memory Optimization to Build a Schur Complement in an Hybrid Solver

Solving linear system $Ax=b$ in parallel where $A$ is a large sparse matrix is a very recurrent problem in numerical simulations. One of the state-of-the-art most promising algorithm is the hybrid method based on domain decomposition and Schur complement. In this method, a direct solver is used as a subroutine on each subdomain matrix. This approach is subject to serious memory overhead. In this paper, we investigate new techniques to reduce memory consumption during the build of the Schur complement by a direct solver. Our method allows memory peak reduction from 10% to 30% on each processus for typical test cases

Regional Gravity Field Modeling with Adjusted Spherical Cap Harmonics in an Integrated Approach

The main objective of this thesis is to develop an integrated approach for the computation of Height Reference Surfaces (HRS) in the context of GNSS positioning. For this purpose, the method of Digital Finite Element Height Reference Surface software (DFHRS) is extended, allowing the use of physical observations in addition to geometrical observation types. Particular emphasis is put on (i) using Adjusted Spherical Cap Harmonics to locally model the potential, (ii) developing a parameterization of coefficients for a least squares estimation, and (iii) optimizing the combination of data needed to calculate the coefficients. In particular, the selection of the terrestrial gravity measurements, height fitting points with known ellipsoidal and normal heights, and the use of the available global gravity models as additional observations are investigated. One of the main motivations is the need to compute a high precise local potential model with the ability to derive all components related to the potential W. These observation components are gravity , quasigeoid height , the geoid height , deflections of the vertical in the east and north direction ( ), the fitting points and the apriori information in terms of coefficients of a local potential model derived from the developed methods of a mapping of a global one. This thesis provides a method for local and global gravity and geoid modelling. The Spherical Cap Harmonics (SCH) for modeling the Earth potential are introduced in detail, including their relationship to the normal Spherical Harmonics (SH). The different types of Spherical Cap Harmonics, such as Adjusted Spherical Cap Harmonics (ASCH), Translated-Origin Spherical Cap Harmonics (TOSCH) and the Revised Spherical Cap Harmonics (RSCH) are discussed. The ASCH method was chosen in further for modeling the local gravitational potential due to its simple principle, that the integer degree and order Legendre functions are preserved and lead to faster implementation algorithms. The ASCH are used in this thesis to transform the global gravity models like EGM2008 or EIGEN05c to local gravity models, guaranteeing a much smaller number of coefficients and making the calculations faster and easier. Tests are applied to validate the use of ASCH for local gravity and potential modelling, with ASCH coefficients calculated in test areas. These coefficients were used to calculate the values of potential or the gravity for new points and then compared with the real measured values and reference values from global models. The tests include the transformation of global gravity models like EGM2008 and EIGEN05c to ASCH models and the integrated solution of heterogeneous groups of data including terrestrial gravity data, height fitting points and the locally mapped global gravity models. The region of the federal state of Baden-Württemberg in Germany was used as a test area for this thesis to prove the concept. Nearly 15000 terrestrially measured gravity observations were used to implement an ASCH model in degree and order of 300 in order to achieve a resolution of 0.01 mGal that corresponds to the measurement accuracy