Evolution and Regularisation of Vacuum Brill Gravitational Waves in Spherical Polar Coordinates

In this thesis the universal collapse of vacuum Brill waves is demonstrated numerically and analytically. This thesis presents the mathematical and numerical methods necessary to regularise and evolve Brill Gravitational Waves in spherical polar coordinates. A Cauchy ADM formulation is used for the time evolution. We find strong evidence that all IVP formulations of pure vacuum Brill gravitational waves collapse to form singularities/black holes, and we do not observe critical black hole mass scaling phenomena in the IVP parameter phase space that has been characterised in non-vacuum systems. A theoretical framework to prove this result analytically is presented. We discuss the meaning of Brill metric variables, the topology of trapped surfaces for various scenarios, and verify other results in the field related to critical values of initial value parameters and black hole formation approaching spatial infinity. The instability of Minkowski (flat) space under Brill wave and more general perturbations is demonstrated. The main numerical tools employed to achieve a stable evolution code are (1) derivation of appropriate regularity conditions on the lapse function and metric function q, (2) the move to a 4th order correct discretisation scheme with appropriate boundary conditions, (3) the use of exponential metric terms, (4) an understanding of the right mix of free versus constrained evolution and (5) the development of appropriate numerical techniques for discretisation and differencing to reduce numerical error, along with a characterisation of condition numbers.Comment: PhD Thesis, 2014, University of Calgary, 368 page

Time-transformations for the event location in discontinuous ODEs

In this paper, we consider numerical methods for the location of events of ordinary differential equations. These methods are based on particular changes of the independent variable, called time-transformations. Such a time-transformation reduces the integration of an equation up to the unknown point, where an event occurs, to the integration of another equation up to a known point. This known point corresponds to the unknown point by means of the time-transformation. This approach extends the one proposed by Dieci and Lopez [BIT 55 (2015), no. 4, 987-1003], but our generalization permits, amongst other things, to deal with situations where the solution approaches the event in a tangential way. Moreover, we also propose to use this approach in a different manner with respect to that of Dieci and Lopez

Numerical Solution of Optimal Control Problems with Explicit and Implicit Switches

This dissertation deals with the efficient numerical solution of switched optimal control problems whose dynamics may coincidentally be affected by both explicit and implicit switches. A framework is being developed for this purpose, in which both problem classes are uniformly converted into a mixed–integer optimal control problem with combinatorial constraints. Recent research results relate this problem class to a continuous optimal control problem with vanishing constraints, which in turn represents a considerable subclass of an optimal control problem with equilibrium constraints. In this thesis, this connection forms the foundation for a numerical treatment. We employ numerical algorithms that are based on a direct collocation approach and require, in particular, a highly accurate determination of the switching structure of the original problem. Due to the fact that the switching structure is a priori unknown in general, our approach aims to identify it successively. During this process, a sequence of nonlinear programs, which are derived by applying discretization schemes to optimal control problems, is solved approximatively. After each iteration, the discretization grid is updated according to the currently estimated switching structure. Besides a precise determination of the switching structure, it is of central importance to estimate the global error that occurs when optimal control problems are solved numerically. Again, we focus on certain direct collocation discretization schemes and analyze error contributions of individual discretization intervals. For this purpose, we exploit a relationship between discrete adjoints and the Lagrange multipliers associated with those nonlinear programs that arise from the collocation transcription process. This relationship can be derived with the help of a functional analytic framework and by interrelating collocation methods and Petrov–Galerkin finite element methods. In analogy to the dual-weighted residual methodology for Galerkin methods, which is well–known in the partial differential equation community, we then derive goal–oriented global error estimators. Based on those error estimators, we present mesh refinement strategies that allow for an equilibration and an efficient reduction of the global error. In doing so we note that the grid adaption processes with respect to both switching structure detection and global error reduction get along with each other. This allows us to distill an iterative solution framework. Usually, individual state and control components have the same polynomial degree if they originate from a collocation discretization scheme. Due to the special role which some control components have in the proposed solution framework it is desirable to allow varying polynomial degrees. This results in implementation problems, which can be solved by means of clever structure exploitation techniques and a suitable permutation of variables and equations. The resulting algorithm was developed in parallel to this work and implemented in a software package. The presented methods are implemented and evaluated on the basis of several benchmark problems. Furthermore, their applicability and efficiency is demonstrated. With regard to a future embedding of the described methods in an online optimal control context and the associated real-time requirements, an extension of the well–known multi–level iteration schemes is proposed. This approach is based on the trapezoidal rule and, compared to a full evaluation of the involved Jacobians, it significantly reduces the computational costs in case of sparse data matrices

Dynamic optimization with path constraints

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1998.Includes bibliographical references (p. 381-391).by William Francis Feehery.Ph.D

Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review

A review of diagonally implicit Runge-Kutta (DIRK) methods applied to rst-order ordinary di erential equations (ODEs) is undertaken. The goal of this review is to summarize the characteristics, assess the potential, and then design several nearly optimal, general purpose, DIRK-type methods. Over 20 important aspects of DIRKtype methods are reviewed. A design study is then conducted on DIRK-type methods having from two to seven implicit stages. From this, 15 schemes are selected for general purpose application. Testing of the 15 chosen methods is done on three singular perturbation problems. Based on the review of method characteristics, these methods focus on having a stage order of two, sti accuracy, L-stability, high quality embedded and dense-output methods, small magnitudes of the algebraic stability matrix eigenvalues, small values of aii, and small or vanishing values of the internal stability function for large eigenvalues of the Jacobian. Among the 15 new methods, ESDIRK4(3)6L[2]SA is recommended as a good default method for solving sti problems at moderate error tolerances

Eine gitterfreie Raum-Zeit-Kollokationsmethode für gekoppelte Probleme auf Gebieten mit komplizierten Rändern

In der vorliegenden Arbeit wird eine neuartige gitterfreie Raum-Zeit-Kollokationsmethode (engl. STMCM) zur Lösung von Systemen partieller und gewöhnlicher Differentialgleichungen durch eine konsistente Diskretisierung in Raum und Zeit als Alternative zu den etablierten netzbasierten Verfahren vorgeschlagen. Die STMCM gehört zur Klasse der tatsächlich gitterfreien Methoden, die nur mit Punktwolken ohne a priori Netzkonnektivität arbeiten und kein Diskretisierungsnetz benötigen. Das Verfahren basiert auf der Interpolating Moving Least Squares Methode, die eine vereinfachte Erfüllung der Randbedingungen durch die von den Kernfunktionen erfüllte Kronecker-Delta-Eigenschaft ermöglicht, was beim größten Teil anderer netzfreier Verfahren nicht der Fall ist. Ein Regularisierungsverfahren zur Bewältigung des beim Aufbau der Kernfunktionen auftretenden Singularitätsproblems, sowie zur Berechnung aller benötigten Ableitungen der Kernfunktionen wird dargelegt. Ziel ist es dabei, eine Methode zu entwickeln, die die Einfachheit der Verfahren zur Lösung partieller Differentialgleichungen in starker Form mit den Vorteilen der gitterfreien Verfahren, insbesondere mit Blick auf gekoppelte Probleme des Ingenieurwesens mit sich bewegenden Grenzflächen, verknüpft. Die vorgeschlagene Methode wird zunächst zur Lösung linearer und nichtlinearer partieller sowie gewöhnlicher Differentialgleichungen angewendet. Dabei werden deren Konvergenz- und Genauigkeitseigenschaften untersucht. Die implizite Rekonstruktion der Gebiete mit komplizierten Rändern als Abbildungsstrategie zur Punktwolken-Streuung wird durch die Interpolation von Punktwolkendaten in zwei und drei Raumdimensionen demonstriert. Anhand der Modelle zur Simulation von Biofilm- und Tumor-Wachstumsprozessen werden Anwendungsbeispiele aus dem Bereich der Umweltwissenschaften und der Medizintechnik dargestellt.In this thesis an innovative Space-Time Meshfree Collocation Method (STMCM) for solving systems of nonlinear ordinary and partial differential equations by a consistent discretization in both space and time is proposed as an alternative to established mesh-based methods. The STMCM belongs to the class of truly meshfree methods, i.e. the methods which do not have any underlying mesh, but work on a set of nodes only without an a priori node-to-node connectivity. The STMCM is constructed using the Interpolating Moving Least Squares technique, allowing a simplified implementation of boundary conditions due to fulfilment of the Kronecker delta property by the kernel functions, which is not the case for the major part of other meshfree methods. A regularization technique to overcome the singularity-by-construction problem and compute all necessary derivatives of the kernel functions is presented. The goal is to design a method that combines the simplicity and straightforwardness of the strong-form computational techniques with the advantages of meshfree methods over the classical ones, especially for coupled engineering problems involving moving interfaces. The proposed STMCM is applied to linear and nonlinear partial and ordinary differential equations of different types and its accuracy and convergence properties are studied. The power of the technique is demonstrated by implicit reconstruction of domains with complex boundaries via interpolation of point cloud data in two and three space dimensions as a `mapping' strategy for distribution of computational points within such domains. Applications from the fields of environmental and medical engineering are presented by means of a mathematical model for simulating a biofilm growth and a nonlinear model of tumour growth processes

Imaging Seismic Reflections

The goal of reflection seismic imaging is making images of the Earth subsurface using surface measurements of reflected seismic waves. Besides the position and orientation of subsurface reflecting interfaces it is a challenge to recover the size or amplitude of the discontinuities. We investigate two methods or techniques of reflection seismic imaging in order to improve the amplitude. First we study the \textsl{one-way wave equation} with attention to the amplitude of the waves. In our study of \textsl{reverse-time migration} the amplitude refers to the amplitude of the image. Though the approach in the thesis is formal, our results have practical implications for seismic imaging algorithms, which improve or correct the amplitudes. The one-way wave equation is a $1^{\rm st}$-order equation that describes wave propagation in a predetermined direction. We derive the equation and identify the steps that determine the wave amplitude. We introduce a \textsl{symmetric square root operator} and a wave field normalization operator and show that they provide the correct amplitude. The idea to use a symmetric square root is generally applicable. Our amplitude claims are numerically verified. The one-way wave equation is an application of \textsl{pseudo-differential operators}. \textsl{Reverse-time migration} (RTM) is an imaging method that uses simulations of the source and receiver wave fields through a slowly varying estimate of the subsurface medium. The receiver wave is an in reverse time continued field that matches the measurements. An \textsl{imaging condition} transforms the fields into an image of the small scale medium contrast. We investigate the linearized inverse problem and use the RTM procedure to reconstruct the perturbation of the medium. We model the scattering event by the scattering operator, which maps the medium perturbation to the scattered wave. We propose an approximate inverse of the scattering operator, derive a novel imaging condition and show that it yields a reconstruction of the perturbation with correct amplitude. The study extensively uses the theory of \textsl{Fourier integral operators} (FIO). Besides that we globally characterize the scattering operator as a FIO, we also obtain a local expression to calculate the amplitude explicitly. The claims are confirmed and illustrated by numerical simulations