A unified error analysis for the numerical solution of nonlinear wave-type equations with application to kinetic boundary conditions

In this thesis, a unified error analysis for discretizations of nonlinear first- and second-order wave-type equations is provided. For this, the wave equations as well as their space discretizations are considered as nonlinear evolution equations in Hilbert spaces. The space discretizations are supplemented with Runge-Kutta time discretizations. By employing stability properties of monotone operators, abstract error bounds for the space, time, and full discretizations are derived. Further, for semilinear second-order wave-type equations, an implicit-explicit time integration scheme is presented. This scheme only requires the solution of a linear system of equations in each time step and it is stable under a step size restriction only depending on the nonlinearity. It is proven that the scheme converges with second order in time and in combination with the abstract space discretization of the unified error analysis, corresponding full discretization error bounds are derived. The abstract results are used to derive convergence rates for an isoparametric finite element space discretization of a wave equation with kinetic boundary conditions and nonlinear forcing and damping terms. For the combination of the finite element discretization with Runge-Kutta methods or the implicit-explicit scheme, respectively, error bounds of the resulting fully discrete schemes are proven. The theoretical results are illustrated by numerical experiments

A unified error analysis for spatial discretizations of wave-type equations with applications to dynamic boundary conditions

This thesis provides a unified framework for the error analysis of non-conforming space discretizations of linear wave equations in time-domain, which can be cast as symmetric hyperbolic systems or second-order wave equations. Such problems can be written as first-order evolution equations in Hilbert spaces with linear monotone operators. We employ semigroup theory for the well-posedness analysis and to obtain stability estimates for the space discretizations. To compare the finite dimensional approximations with the original solution, we use the concept of a lift from the discrete to the continuous space. Time integration with the Crank–Nicolson method is analyzed. In this framework, we derive a priori error bounds for the abstract space semi-discretization in terms of interpolation and discretization errors. These error bounds yield previously unkown convergence rates for isoparametric finite element discretizations of wave equations with dynamic boundary conditions in smooth domains. Moreover, our results allow to consider already investigated space discretizations in a unified way. Here it successfully reproduces known error bounds. Among the examples which we dicuss in this thesis are discontinuous Galerkin discretizations of Maxwell’s equations and finite elements with mass lumping for the scalar wave equation

hp-DGFEM on Shape-Irregular Meshes: Reaction-Diffusion Problems

We consider the hp-version of the discontinuous Galerkin finite element method (DGFEM) for second-order elliptic reaction-diffusion equations with mixed Dirichlet and Neumann boundary conditions. For simplicity of the presentation, we only consider boundary-value problems defined on an axiparallel polygonal domain whose solutions are approximated on subdivisions consisting of axiparallel elements. Our main concern is the generalisation of the error analysis of the hp-DGFEM for the case when shape-irregular (anisotropic) meshes and anisotropic polynomial degrees for the element basis functions are used. We shall present a general framework for deriving error bounds for the approximation error and we shall consider two important special cases. In the first of these we derive an error bound that is simultaneously optimal in h and p, for shape-regular elements and isotropic polynomial degrees, provided that the solution belongs to a certain anisotropic Sobolev space. The second result deals with the case where we have a uniform polynomial degree in every space direction and a shape-irregular mesh. Again we derive an error bound that is optimal both in h and in p. For element-wise analytic solutions the method exhibits exponential rates of convergence under p-refinement, in both cases considered. Finally, numerical experiments using shape-regular and shape-irregular elements are presented

Higher moments of the pair correlation function for Sato-Tate sequences

In \cite{BS}, Balasubramanyam and the second named author derived the first moment of the pair correlation function for Hecke angles lying in small subintervals of $[0,1]$ upon averaging over large families of Hecke newforms of weight $k$ with respect to $\Gamma_0(N)$. The goal of this article is to study higher moments of this pair correlation function. For an integer $r \geq 2$, we present bounds for its $r$-th power moments. We apply these bounds to record lower order error terms in the computation of the second and third moments. As a result, one can obtain the convergence of the second and third moments of this pair correlation function for suitably small intervals, and under appropriate growth conditions for the size of the families of Hecke newforms.Comment: This is a significantly revised version incorporating a study of the $r$-th power moments of the above-mentioned pair correlation function for all integers $r \geq 2$. The previous version only discussed the second moment. 47 page

Local properties and augmented Lagrangians in fully nonconvex composite optimization

A broad class of optimization problems can be cast in composite form, that is, considering the minimization of the composition of a lower semicontinuous function with a differentiable mapping. This paper discusses the versatile template of composite optimization without any convexity assumptions. First- and second-order optimality conditions are discussed, advancing the variational analysis of compositions. We highlight the difficulties that stem from the lack of convexity when dealing with necessary conditions in a Lagrangian framework and when considering error bounds. Building upon these characterizations, a local convergence analysis is delineated for a recently developed augmented Lagrangian method, deriving rates of convergence in the fully nonconvex setting.Comment: 42 page

Uniformly-convergent numerical methods for a system of coupled singularly perturbed convection–diffusion equations with mixed type boundary conditions

In this paper, two hybrid difference schemes on the Shishkin mesh are constructed for solving a weakly coupled system of two singularly perturbed convection - diffusion second order ordinary differential equations subject to the mixed type boundary conditions. We prove that the method has almost second order convergence in the supremum norm independent of the diffusion parameter. Error bounds for the numerical solution and also the numerical derivative are established. Numerical results are provided to illustrate the theoretical results. First published online: 24 Oct 201

Finite state machine representation of digital signal processing systems

A new method for implementing digital filters is discussed. The met11od maximises the output signal to noise ratio of a filter by assigning at each of the filter variables an optimal quantization law. A filter optimised for a gaussian process is considered in detail. An error model is developed and applied to first and second order canonic form filter sections. Comparisons are drawn between the gaussian optimised filter and the equivalent fixed point arithmetic filter. The performance of gaussian optimised filters under sinusoidal input signal conditions is considered ; it is found that the gaussian optimised filter exhibits a lower approximation error than the equivalent fixed point arithmetic filter. It is shown that when high order filters are implemented as a cascade of second order sections - with if necessary one first order section - the section ordering has a very small effect on the overall signal to noise r atio performance. A similar result for the pairing of poles and zeroes is found. Bounds on the maximum limit cycle amplitude for first and second order all-pole sections are presented. It is shown that for a first order all-pole the maximum limit cycle amplitude is lower than would be expected in the equivalent fixed point arithmetic filter, whereas , for the second order all- pole the bound is twice as large. Examples of a low-pass , band-pass and wideband differentiating filter,designed using free quantization law techniques,are presented. This new design method leads to a filter whose arithmetic operations can not be performed using fixed point arithmetic hardware. Instead, the filter must be represented as a finite state machine and then implemented using sequential logic circuit synthesis techniques. The logic complexity is found to depend - amongst other considerations - on the so called state (code) assignment. Some preliminary results on this problem are presented for the case of a next state function computed using the AND/EXCLUSIVE- OR (ring-sum) logic expansion. A review of the state assignment techniques in the literature is included. A part of the state assignment problem - for the case of AND/EX'·/OR logic - requires the numerous and consequently rapid computation of the Reed-Muller Transformation. A hardware processor - designed as an add-on to a minicomputer - is described; speed comparisons are drawn with the equivalent software algorithm.Digitisation of this thesis was sponsored by Arcadia Fund, a charitable fund of Lisbet Rausing and Peter Baldwin