Polyharmonic Hardy Spaces on the Klein-Dirac Quadric with Application to Polyharmonic Interpolation and Cubature Formulas

In the present paper we introduce a new concept of Hardy type space naturally defined on the Klein-Dirac quadric. We study different properties of the functions belonging to these spaces, in particular boundary value problems. We apply these new spaces to polyharmonic interpolation and to interpolatory cubature formulas.Comment: 32 page

Fast boundary element methods for the simulation of wave phenomena

This thesis is concerned with the efficient implementation of boundary element methods (BEM) for their application in wave problems. BEM present a particularly useful tool, since they reduce the dimension of the problems by one, resulting in much fewer unknowns. However, this comes at the cost of dense system matrices, whose entries require the integration of singular kernel functions over pairs of boundary elements. Because calculating these four-dimensional integrals by cubature rules is expensive, a novel approach based on singularity cancellation and analytical integration is proposed. In this way, the dimension of the integrals is reduced and closed formulae are obtained for the most challenging cases. This allows for the accurate calculation of the matrix entries while requiring less computational work compared with conventional numerical integration. Furthermore, a new algorithm based on hierarchical low-rank approximation is presented, which compresses the dense matrices and improves the complexity of the method. The idea is to collect the matrices corresponding to different time steps in a third-order tensor and to approximate individual sub-blocks by a combination of analytic and algebraic low-rank techniques. By exploiting the low-rank structure in several ways, the method scales almost linearly in the number of spatial degrees of freedom and number of time steps. The superior performance of the new method is demonstrated in numerical examples.Diese Arbeit befasst sich mit der effizienten Implementierung von Randelementmethoden (REM) für ihre Anwendung auf Wellenprobleme. REM stellen ein besonders nützliches Werkzeug dar, da sie die Dimension der Probleme um eins reduzieren, was zu weit weniger Unbekannten führt. Allerdings ist dies mit vollbesetzten Matrizen verbunden, deren Einträge die Integration singulärer Kernfunktionen über Paare von Randelementen erfordern. Da die Berechnung dieser vierdimensionalen Integrale durch Kubaturformeln aufwendig ist, wird ein neuer Ansatz basierend auf Regularisierung und analytischer Integration verfolgt. Auf diese Weise reduziert sich die Dimension der Integrale und es ergeben sich geschlossene Formeln für die schwierigsten Fälle. Dies ermöglicht die genaue Berechnung der Matrixeinträge mit geringerem Rechenaufwand als konventionelle numerische Integration. Außerdem wird ein neuer Algorithmus beruhend auf hierarchischer Niedrigrangapproximation präsentiert, der die Matrizen komprimiert und die Komplexität der Methode verbessert. Die Idee ist, die Matrizen der verschiedenen Zeitpunkte in einem Tensor dritter Ordnung zu sammeln und einzelne Teilblöcke durch eine Kombination von analytischen und algebraischen Niedrigrangverfahren zu approximieren. Durch Ausnutzung der Niedrigrangstruktur skaliert die Methode fast linear mit der Anzahl der räumlichen Freiheitsgrade und der Anzahl der Zeitschritte. Die überlegene Leistung der neuen Methode wird anhand numerischer Beispiele aufgezeigt

Matrix-valued Quantum Lattice Boltzmann Method

We devise a lattice Boltzmann method (LBM) for a matrix-valued quantum Boltzmann equation, with the classical Maxwell distribution replaced by Fermi-Dirac functions. To accommodate the spin density matrix, the distribution functions become 2 x 2 matrix-valued. From an analytic perspective, the efficient, commonly used BGK approximation of the collision operator is valid in the present setting. The numerical scheme could leverage the principles of LBM for simulating complex spin systems, with applications to spintronics.Comment: 18 page

Control Theoretic Approach To Sampling And Approximation Problems

Thesis (Ph.D.) University of Alaska Fairbanks, 2009We present applications of some methods of control theory to problems of signal processing and optimal quadrature problems. The following problems are considered: construction of sampling and interpolating sequences for multi-band signals; spectral estimation of signals modeled by a finite sum of exponentials modulated by polynomials; construction of optimal quadrature formulae for integrands determined by solutions of initial boundary value problems. A multi-band signal is a function whose Fourier transform is supported on a finite union of intervals. The approach used in Chapter I is based on connections between the sampling and interpolation problem and the problem of the controllability of a dynamical system. We prove that there exist infinitely many sampling and interpolating sequences for signals whose spectra are supported on a union of two disjoint intervals, and provide an algorithm for construction of such sequences. There exist numerous methods for solving the spectral estimation problem. In Chapter II we introduce a new approach to this problem based on the Boundary Control method, which uses the connection between inverse problems of mathematical physics and control theory for partial differential equations. Using samples of the signal at integer moments of time we construct a convolution operator regarded as an input-output map of a linear discrete dynamical system. This system can be identified, and the exponents and amplitudes of the signal can be found from the parameters of the system. We show that the coefficients of the signal can be recovered by solving a generalized eigenvalue problem as in the Matrix Pencil method. Our method allows to consider signals with polynomial amplitudes, and we obtain an exact formula for these amplitudes. In the third chapter we consider an optimal quadrature problem for solutions of initial boundary value problems. The problem of optimization of an error functional over the set of solutions and quadrature weights is a problem of optimal control of partial differential equations. We obtain estimates for the error in quadrature formulae and an optimality condition for quadrature weights

On the computation of Gaussian quadrature rules for Chebyshev sets of linearly independent functions

We consider the computation of quadrature rules that are exact for a Chebyshev set of linearly independent functions on an interval $[a,b]$. A general theory of Chebyshev sets guarantees the existence of rules with a Gaussian property, in the sense that $2l$ basis functions can be integrated exactly with just $l$ points and weights. Moreover, all weights are positive and the points lie inside the interval $[a,b]$. However, the points are not the roots of an orthogonal polynomial or any other known special function as in the case of regular Gaussian quadrature. The rules are characterized by a nonlinear system of equations, and earlier numerical methods have mostly focused on finding suitable starting values for a Newton iteration to solve this system. In this paper we describe an alternative scheme that is robust and generally applicable for so-called complete Chebyshev sets. These are ordered Chebyshev sets where the first $k$ elements also form a Chebyshev set for each $k$. The points of the quadrature rule are computed one by one, increasing exactness of the rule in each step. Each step reduces to finding the unique root of a univariate and monotonic function. As such, the scheme of this paper is guaranteed to succeed. The quadrature rules are of interest for integrals with non-smooth integrands that are not well approximated by polynomials