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1	Approximation of Analytic Functions by Faber Polynomials, the Grunsky Matrix, and a Univalence Criterion Farag, Mina January 2022 (has links) The aim of this thesis is derive a set of polynomials defined on simply connected domains, the Faberpolynomials, in which all analytic function on the domain can be uniformly approximated. Importantconcepts and theorems such as isomorphisms, automorphisms and the Riemann mapping theorem areintroduced. Examples and applications are also included. Furthermore, the thesis will aim to introducean important consequence of the Faber polynomials, the method of the Grunsky inequalities. The first section introduces important properties of analytic functions and the concept of isomor-phisms, in particular the form of all automorphisms of the unit disc will be derived. The second sectionconsiders the Riemann mapping theorem, a theorem that relates any simply connected region that is notall of ℂ to the unit disc. A proof of the theorem beginning with the Arzelá-Ascoli theorem is provided.An application in constructing harmonic functions on arbitrary simply connected regions will be pre-sented. In the third section, definitions and properties of the Faber polynomials are developed; followedby simple examples. The section concludes with a proof and example of the statement that analyticfunctions can be approximated by Faber polynomials. In the fourth and last section of the thesis, themethod of Grunsky inequalities is presented. Starting off, the Grunsky coefficients are defined using theFaber polynomials. Properties of Grunsky coefficients such as the symmetry property and the Grunskyinequalities are then derived. To conclude it will be shown that the Grunsky inequalities provide aunivalence criterion for analytic functions defined on the unit disc. Faber polynomials Approximation of functions Grunsky matrix univalence criterion Mathematics Matematik
2	Crouzeix's Conjecture and the GMRES Algorithm Luo, Sarah McBride 13 July 2011 (has links) (PDF) This thesis explores the connection between Crouzeix's conjecture and the convergence of the GMRES algorithm. GMRES is a popular iterative method for solving linear systems and is one of the many Krylov methods. Despite its popularity, the convergence of GMRES is not completely understood. While the spectrum can in some cases be a good indicator of convergence, it has been shown that in general, the spectrum does not provide sufficient information to fully explain the behavior of GMRES iterations. Other sets associated with a matrix that can also help predict convergence are the pseudospectrum and the numerical range. This work focuses on convergence bounds obtained by considering the latter. In particular, it focuses on the application of Crouzeix's conjecture, which relates the norm of a matrix polynomial to the size of that polynomial over the numerical range, to describing GMRES convergence. GMRES Michel Crouzeix Faber Polynomials Complex Approximation Krylov Subspace Convergence Iterative Methods Linear Systems Mathematics
3	CUDA-based Scientific Computing / Tools and Selected Applications Kramer, Stephan Christoph 22 November 2012 (has links) No description available. 510 CUDA C++ Expression Templates Preconditioning Sparse Matrix Indoor Airflow Quantum Hall Effect Magnetic Focussing Hardy Space Infinite Elements Phase Retrieval Optogenetics Object-oriented Design FFT Dielectric Relaxation Spectroscopy Poisson-Nernst-Planck Equations BEM FEM-BEM Coupling Curvilinear Boundaries Boundary Integral Equation Transparent Boundary Conditions Schrödinger Equation Krylov Methods Parallel Computing GPU Computing Sparse Approximate Inverse Faber Polynomials Polynomial Preconditioner Conformational Sampling of Proteins Protein Folding Higher Order Finite Elements Mathematics (PPN61756535X)

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