Studies in numerical quadrature

Various types of quadrature formulae for oscillatory integrals are studied with a view to improving the accuracy of existing techniques. Concentration is directed towards the production of practical algorithms which facilitate the efficient evaluation of integrals of this type arising in applications. [Continues.

Numerical investigation of fermion mass generation in QED

We investigate the dynamical generation of fermion mass in quantum electrodynamics (QED). This non-perturbative study is performed using a truncated set of Schwinger-Dyson equations for the fermion and the photon propagator. First, we study dynamical fermion mass generation in quenched QED with the Curtis-Pennington vertex, which satisfies the Ward-Takahashi identity and moreover ensures the multiplicative renormalizability of the fermion propagator. We apply bifurcation analysis to determine the critical point for a general covariant gauge. In the second part of this work we investigate the dynamical generation of fermion mass in full, unquenched QED. We develop a numerical method to solve the system of three coupled non-linear equations for the dynamical fermion mass, the fermion wavefunction renormalization and the photon renormalization function. Much care is taken to ensure the high accuracy of the solutions. Moreover, we discuss in detail the proper numerical cancellation of the quadratic divergence in the vacuum polarization integral and the requirement of using smooth approximations to the solutions. To achieve this, we improve the numerical method by introducing the Chebyshev expansion method. We apply this method to the bare vertex approximation to unquenched QED to determine the critical coupling for a variety of approximations. This culminates in the detailed, highly accurate, solution of the Schwinger-Dyson equations for dynamical fermion mass generation in QED including both, the photon renormalization function and the fermion wavefunction renormalization in a consistent way, in the bare vertex approximation and, for the first time, using improved vertices. We introduce new improvements to the numerical method, to achieve the accuracy necessary to avoid unphysical quadratic divergences in the vacuum polarization with the Ball-Chiu vertex

Nonlinear Quantizer Design Based on Clenshaw-Curtis Quadrature

Trabalho de Conclusão de Curso (graduação)—Universidade de Brasília, Faculdade de Tecnologia, Departamento de Engenharia Elétrica, 2019.Esta tese visa propor um novo método para projeto de quantizadores não lineares conservadores de momentos estatísticos, baseado na quadratura de Clenshaw-Curtis. Os conceitos básicos de Conversores Analógico Digital são definidos para contextualização do problema discutido e para servir de base para o entendimento dos parâmetros de quantizadores. Então, uma definição formal da Transformada da Incerteza - Unscented Transform (UT) - é proposta para o contexto deste trabalho, e os conceitos básicos de quadratura são aplicados como uma ferramenta matemática para cálculo da UT. Finalmente, a metodologia de projeto do quantizador é detalhada, apresentando a relação entre os nós e pesos de uma quadratura com os parâmetros de quantizadores. O projeto é então aplicado a uma simulação de estudo de caso para verificação dos cálculos teóricos.This thesis aims to provide a novel method for designing nonlinear moment preserving quantizers based on the Clenshaw-Curtis quadrature. The basic concepts of Analog-to-Digital Converters (ADCs) are defined for contextualization of the discussed problem and to serve as a basis for understanding quantizers parameters. Then, a formal definition of the Unscented Transform (UT) is proposed for this work’s context, and the key concepts of quadrature are applied to it as a mathematical tool for UT calculation. Finally, the design method is detailed, presenting the relationship between quadrature’s nodes and weights and the quantizers parameters. This design is applied to a case study simulation, for validation of theoretical calculations

Numerical Methods for Uncertainty Quantification in Gas Network Simulation

When modeling the gas flow through a network, some elements such as pressure control valves can cause kinks in the solution. In this thesis we modify the method of simplex stochastic collocation such that it is applicable to functions with kinks. First, we derive a system of partial differential and algebraic equations describing the gas flow through different elements of a network. Restricting the gas flow to an isothermal and stationary one, the solution can be determined analytically. After introducing some common methods for the forward propagation of uncertainty, we present the method of simplex stochastic collocation to approximate functions of uncertain parameters. By utilizing the information whether a pressure regulator is active or not in the current simulation, we improve the method such that we can prove algebraic convergence rates for functions with kinks. Moreover, we derive two new error estimators for an adaptive refinement and show that multiple refinements are possible. Conclusively, several numerical results for a real gas network are presented and compared with standard methods to demonstrate the significantly better convergence results

Approximations for Performance Analysis in Wireless Communications and Applications to Reconfigurable Intelligent Surfaces

In the last few decades, the field of wireless communications has witnessed significant technological advancements to meet the needs of today’s modern world. The rapidly emerging technologies, however, are becoming increasingly sophisticated, and the process of investigating their performance and assessing their applicability in the real world is becoming more challenging. That has aroused a relatively wide range of solutions in the literature to study the performance of the different communication systems or even draw new results that were difficult to obtain. These solutions include field measurements, computer simulations, and theoretical solutions such as alternative representations, approximations, or bounds of classic functions that commonly appear in performance analyses. Field measurements and computer simulations have significantly improved performance evaluation in communication theory. However, more advanced theoretical solutions can be further developed in order to avoid using the ex- pensive and time-consuming wireless communications measurements, replace the numerical simulations, which can sometimes be unreliable and suffer from failures in numerical evaluation, and achieve analytically simpler results with much higher accuracy levels than the existing theoretical ones. To this end, this thesis firstly focuses on developing new approximations and bounds using unified approaches and algorithms that can efficiently and accurately guide researchers through the design of their adopted wireless systems and facilitate the conducted performance analyses in the various communication systems. Two performance measures are of primary interest in this study, namely the average error probability and the ergodic capacity, due to their valuable role in conducting a better understanding of the systems’ behavior and thus enabling systems engineers to quickly detect and resolve design issues that might arise. In particular, several parametric expressions of different analytical forms are developed to approximate or bound the Gaussian Q-function, which occurs in the error probability analysis. Additionally, any generic function of the Q-function is approximated or bounded using a tractable exponential expression. Moreover, a unified logarithmic expression is proposed to approximate or bound the capacity integrals that occur in the capacity analysis. A novel systematic methodology and a modified version of the classical Remez algorithm are developed to acquire optimal coefficients for the accompanying parametric approximation or bound in the minimax sense. Furthermore, the quasi-Newton algorithm is implemented to acquire optimal coefficients in terms of the total error. The average symbol error probability and ergodic capacity are evaluated for various applications using the developed tools. Secondly, this thesis analyzes a couple of communication systems assisted with reconfigurable intelligent surfaces (RISs). RIS has been gaining significant attention lately due to its ability to control propagation environments. In particular, two communication systems are considered; one with a single RIS and correlated Rayleigh fading channels, and the other with multiple RISs and non-identical generic fading channels. Both systems are analyzed in terms of outage probability, average symbol error probability, and ergodic capacity, which are derived using the proposed tools. These performance measures reveal that better performance is achieved when assisting the communication system with RISs, increasing the number of reflecting elements equipped on the RISs, or locating the RISs nearer to either communication node. In conclusion, the developed approximations and bounds, together with the optimized coefficients, provide more efficient tools than those available in the literature, with richer capabilities reflected by the more robust closed-form performance analysis, significant increase in accuracy levels, and considerable reduction in analytical complexity which in turns can offer more understanding into the systems’ behavior and the effect of the different parameters on their performance. Therefore, they are expected to lay the groundwork for the investigation of the latest communication technologies, such as RIS technology, whose performance has been studied for some system models in this thesis using the developed tools

The Sparse Grid Combination Technique for Functionals with Applications

The sparse grid method is a special discretisation technique used to solve high dimensional problems. There are a wide range of applications of the sparse grid method in calculating high dimensional integrals and the solution of high dimensional PDEs. The sparse grid combination technique is a kind of method used to approximate the numerical result of the sparse grid method. The general idea of the sparse grid combination technique is to compute a linear combination of approximations of the solution of the problem. The approximations are computed on some anisotropic regular grids. The combination technique is based on the inclusion-exclusion principle. Compared with the sparse grid method, there are two advantages of the combination technique. First, only nodal basis functions are required in combination technique rather than the hierarchical basis functions in sparse grid method. Second, the combination technique is easier for parallelisation. Generalised combination techniques, e.g. the truncated combination technique, the dimension-adaptive combination technique etc, are developed to further reduce the cost when solving a high dimensional problem. For many real world problems, people are interested in some functionals related to the solution of the problem rather than the solution itself. These functionals which capture the important features of the problem are usually key for people to further understand it. When a high dimensional problem is considered, the computational cost of the functionals can be large since the numerical solution of a high dimensional partial differential equation is usually expensive to compute. We apply the generalised combination techniques to reducing the cost of computation of important functionals. Our method is based on the error models of the functionals. We build the error models for some special types of functionals when numerical schemes used to compute the PDEs and the functionals are known. We show the connection between the decay of the surpluses and the error models. By using the connection, we can also apply generalised combination techniques to functionals when we only know their computed surpluses. Numerical experiments are provided to illustrate error models for the functionals and the performance of our generalised combination techniques. Stochastic optimisation problems minimise expectations of random cost functions. Thus they require accurate quadrature methods in order to evaluate the objective, gradient and Hessian which appear in the computation. Two categories of methods are studied here. One is the discretise then optimise method, the other is the optimise then discretise method. For the methods in the first category, the application of the sparse grid methods leads to high quadrature accuracy in approximating the objective. However, the sparse grid surrogates have negative quadrature weights which potentially destroy the convexity of the objective and thus may lead to totally wrong results. We prove that the sparse grid surrogates maintain the convexity of the objective for sufficiently fine grids. For the methods in the second category, it is more flexible for us to choose the numerical schemes which used to approximate the objective, gradient and Hessian. Therefore, the application of the dimension adaptive method is possible and reasonable for optimise then discretise approaches. It further reduces the computational costs and has even better performance compared with the classical sparse grid method for many stochastic optimisation problems. Applications are provided to demonstrate the superiority of our approaches over the classical Monte Carlo and product rule based approaches

Adaptive discrete-ordinates algorithms and strategies

The approaches for discretizing the direction variable in particle transport calculations are the discrete-ordinates method and function-expansion methods. Both approaches are limited if the transport solution is not smooth. Angular discretization errors in the discrete-ordinates method arise from the inability of a given quadrature set to accurately perform the needed integrals over the direction ("angular") domain. We propose that an adaptive discrete-ordinate algorithm will be useful in many problems of practical interest. We start with a "base quadrature set" and add quadrature points as needed in order to resolve the angular flux function. We compare an interpolated angular-flux value against a calculated value. If the values are within a user specified tolerance, the point is not added; otherwise it is. Upon the addition of a point we must recalculate weights. Our interpolatory functions map angular-flux values at the quadrature directions to a continuous function that can be evaluated at any direction. We force our quadrature weights to be consistent with these functions in the sense that the quadrature integral of the angular flux is the exact integral of the interpolatory function (a finite-element methodology that determines coefficients by collocation instead of the usual weightedresidual procedure). We demonstrate our approach in two-dimensional Cartesian geometry, focusing on the azimuthal direction The interpolative methods we test are simple linear, linear in sine and cosine, an Abu-Shumays “base” quadrature with a simple linear adaptive and an Abu-Shumays “base” quadrature with a linear in sine and cosine adaptive. In the latter two methods the local refinement does not reduce the ability of the base set to integrate high-order spherical harmonics (important in problems with highly anisotropic scattering). We utilize a variety of one-group test problems to demonstrate that in all cases, angular discretization errors (including "ray effects") can be eliminated to whatever tolerance the user requests. We further demonstrate through detailed quantitative analysis that local refinement does indeed produce a more efficient placement of unknowns. We conclude that this work introduces a very promising approach to a long-standing problem in deterministic transport, and we believe it will lead to fruitful avenues of further investigation

Numerical analysis of some integral equations with singularities

In this thesis we consider new approaches to the numerical solution of a class of Volterra integral equations, which contain a kernel with singularity of non-standard type. The kernel is singular in both arguments at the origin, resulting in multiple solutions, one of which is differentiable at the origin. We consider numerical methods to approximate any of the (infinitely many) solutions of the equation. We go on to show that the use of product integration over a short primary interval, combined with the careful use of extrapolation to improve the order, may be linked to any suitable standard method away from the origin. The resulting split-interval algorithm is shown to be reliable and flexible, capable of achieving good accuracy, with convergence to the one particular smooth solution.Supported by a college bursary from the University of Chester