Applications of Special Functions in High Order Finite Element Methods

In this thesis, we optimize different parts of high order finite element methods by application of special functions and symbolic computation. In high order finite element methods, orthogonal polynomials like the Jacobi polynomials are deeply rooted. A broad classical theory of these polynomials is known. Moreover, with modern computer algebra software we can extend this knowledge even further. Here, we apply this knowledge and software for different special functions to derive new recursive relations of local matrix entries. This massively optimizes the assembly time of local high order finite element matrices. Furthermore, the introduced algorithm is in optimal complexity. Moreover, we derive new high order dual functions, which result in fast interpolation operators. Lastly, efficient recursive algorithms for hanging node constraint matrices provided by this new dual functions are given

Holonomic Bessel modules and generating functions

We have solved a number of holonomic PDEs derived from the Bessel modules which are related to the generating functions of classical Bessel functions and the difference Bessel functions recently discovered by Bohner and Cuchta. This $D$-module approach both unifies and extends generating functions of the classical and the difference Bessel functions. It shows that the algebraic structures of the Bessel modules and related modules determine the possible formats of Bessel's generating functions studied in this article. As a consequence of these $D$-modules structures, a number of new recursion formulae, integral representations and new difference Bessel polynomials have been discovered. The key ingredients of our argument involve new transmutation formulae related to the Bessel modules and the construction of $D$-linear maps between different appropriately constructed submodules. This work can be viewed as $D$-module approach to Truesdell's $F$-equation theory specialised to Bessel functions. The framework presented in this article can be applied to other special functions.Comment: 97 pages including one blank pag

Quantum Electrodynamic Bound-State Calculations and Large-Order Perturbation Theory. - (This manuscript is also available - in the form of a book - from Shaker Verlag GmbH, Postfach 101818, 52018 Aachen, Germany world-wide web address: http://www.shaker.de, electronic-mail address: [email protected]. It has been posted on the web sites of Dresden University of Technology with the permission of the publisher.)

The accurate calculation of atomic spectra, including radiative corrections, is one of the rather challenging tasks in theoretical physics. The entire formalism of quantum (gauge) field theory, augmented by the difficulties of the bound-state formalism, is needed for an accurate understanding of the relevant physics at the level of current high-precision spectroscopy. In this thesis, several calculations in this area are described in detail. Investigations on large-order perturbation-theory effects (and predictive limits of perturbation theory) supplement these investigations. In the context of applications, numerical algorithms for the acceleration of the convergence of series are discussed

Sparse Signal Recovery and Detection Utilizing Side Information

In this dissertation, we investigate the signal recovery and detection task for compressive sensing and wireless spectrum sensing.First, we investigate the compressive sensing task for the difference frames of videos.Exploiting the clustered property, we design an effective structural aware reconstruction technique that is capable of eliminating isolated nonzero noisy pixels, and promoting undiscovered signal coefficients.Further, we develop a novel optimization based method for the compressive sensing of binary sparse signals. We formulate the reconstruction task as a least square minimization procedure, and propose a novel regularization term based on the weighted sum of ell_1 norm and ell_infty norm.Moreover, we study the compressive sensing for asymmetrical signals.We devise an efficient algorithm that greatly improves the reconstruction quality of asymmetrical sparse signals.Further, we investigate sparse reconstruction of clustered sparse signals with asymmetrical features.We develop a powerful technique that is capable of taking inference of the signal, estimating the mixture density, and exploiting the clustered features.Finally, we investigate the spectrum sensing task for cognitive radio.We develop an eigenvalue based technique that notably improve the primary user detection performance under finite number of sensors and samples.Electrical Engineerin