In this thesis, we will examine the properties of subelliptic jets in the Engel group of step 3. Step-2 groups, such as the Heisenberg group, do not provide insight into the general abstract calculations. This thesis then, is the first explicit non-trivial computation of the abstract results.

This thesis presents different control design approaches for stabilizing networks of quasi-linear hyperbolic partial differential equations. These equations are usually conservative which gives them interesting properties to design stabilizing control laws. Two main design approaches are developed: a methodology based on entropies and Lyapunov functions and a methodology based on the Riemann invariants. The stability theorems are illustrated using numerical simulations. Two practical applications of these methodologies are presented. Netword of navigation channels are modelled using Saint-Venant equations (also known as the Shallow Water Equations). The stabilization problem of such system has an industrial importance in order to satisfy the navigation constraints and to optimize the production of electricity in hydroelectric plants, usually located at each hydraulic gates. A second application deals with the regulation of water waves in moving tanks. This problem is also modelled by a modified version of the shallow water equations and appears in a number of industrial fields which deal with liquid moving parts.

Using recent progress on moment problems, and their connections with semidefinite optimization, we present in this paper a new methodology based on semidefinite optimization, to obtain a hierarchy of upper and lower bounds on both linear and certain nonlinear functionals defined on solutions of linear partial differential equations. We apply the proposed methods to examples of PDEs in one and two dimensions with very encouraging results. We also provide computation evidence that the semidefinite constraints are critically important in improving the quality of the bounds, that is without them the bounds are weak. / Singapore-MIT Alliance (SMA)

We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced-basis approximations -- Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation -- relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures -- methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage -- in which, given a new parameter value, we calculate the output of interest and associated error bound -- depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. / Singapore-MIT Alliance (SMA)

This thesis consists of five papers and it mainly addresses the theory and schemes to approximate the quadrature domains, QDs. The first deals with the uniqueness and some qualitative properties of the two QDs. The concept of two phase QDs, is more complicated than its one counterpart and consequently introduces significant and interesting open. We present two numerical schemes to approach the one phase QDs in the paper. The first method is based on the properties of the free boundary the level set techniques. We use shape optimization analysis to construct second method. We illustrate the efficiency of the schemes on a variety of experiments. In the third paper we design two finite difference methods for the approximation of the multi phase QDs. We prove that the second method enjoys monotonicity, consistency and stability and consequently it is a convergent scheme by Barles-Souganidis theorem. We also present various numerical simulations in the case of Dirac measures. We introduce the QDs in a sub domain of and Rn study the existence and uniqueness along with a numerical scheme based on the level set method in the fourth paper. In the last paper we study the tangential touch for a semi-linear problem. We prove that there is just one phase free boundary points on the flat part of the fixed boundary and it is also shown that the free boundary is a uniform C1-graph up to that part. / Denna avhandling består av fem artiklar och behandlar främst teori och numeriska metoder för att approximera "quadrature domians", QDs. Den första artikeln behandlar entydighet och allmänna egenskaper hos tvåfas QDs. Begreppet tvåfas QDs, är mer komplicerat än enafasmotsvarigheten och introducerar därmed intressanta öppna problem. Vi presenterar två numeriska metoder för att approximera enfas QDs i andra artikeln. Den första metoden är baserad på egenskaperna hos den fria randen och nivå mängdmetoden. Vi använder forsoptimeringmanalys för att konstruera den andra metoden. Båda metoderna är testade i olika numeriska simuleringar. I det tredje artikeln vi approximera flerafas QDs med konstruktionen tvåmetoder finita differens. Vi visar att den andra metoden har monotonicitat, konsistens och stabilitet och följaktligen är metoden konvergent tack vare Barles-Souganidis sats. Vi presenterar också olika numeriska simuleringar i fallet med Diracmåt. Vi introducerar QDs i en delmängd av Rn och studerar existens och entydighet jämte en numerisk metod baserad på nivå mängdmetoden i fjärde pappret. I det sista pappret studerar vi den tangentiella touchen för ett semilinjärt problem. Vi visar att det enbart är enafasrandpunkter på den platta delen av den fixerade randen. Vi visar också att den fria randen är en likformig C1-graf upp till den delen av den fixerade randen.

Numerous approaches have been proposed for solving partial differential equations; all these methods have their own advantages and disadvantages depending on the problems being treated. In recent years there has been much development of particle methods for mechanical problems. Among these are the Smoothed Particle Hydrodynamics (SPH), Reproducing Kernel Particle Method (RKPM), Element Free Galerkin (EFG) and Moving Least Squares (MLS) methods. This development is motivated by the extension of their applications to mechanical and engineering problems. Since numerical experiments are one of the basic tools used in computational mechanics, in physics, in biology etc, a robust spatial discretization would be a significant contribution towards solutions of a number of problems. Even a well-defined stable and convergent formulation of a continuous model does not guarantee a perfect numerical solution to the problem under investigation. Particle methods especially SPH and RKPM have advantages over meshed methods for problems, in which large distortions and high discontinuities occur, such as high velocity impact, fragmentation, hydrodynamic ram. These methods are also convenient for open problems. Recently, SPH and its family have grown into a successful simulation tools and the extension of these methods to initial boundary value problems requires further research in numerical fields. In this thesis, several problem areas of the SPH formulation were examined. Firstly, a new approach based on ‘Hamilton’s variational principle’ is used to derive the equations of motion in the SPH form. Secondly, the application of a complex Von Neumann analysis to SPH method reveals the existence of a number of physical mechanisms accountable for the stability of the method. Finally, the notion of the amplification matrix is used to detect how numerical errors propagate permits the identification of the mechanisms responsible for the delimitation of the domain of numerical stability. By doing so, we were able to erect a link between the physics and the numerics that govern the SPH formulation.

The vector-valued coarea formula provides a relationship between the integral of the Jacobian of a map from high dimensions down to low dimensions with the integral over the measure of the fibers of this map. We explore minimizers of this functional, proving existence using both a variational approach and an approach with currents. Additionally, we consider what properties these minimizers will have and provide examples. Finally, this problem is considered in metric spaces, where a third existence proof is given.

The source and nature, as well as the history of ray-effects, is described. A benchmark code, using piecewise constant functions in angle and diamond differencing in space, is derived in order to analyze four sample problems. The results of this analysis are presented showing the ray effects and how increasing the resolution (number of angles) eliminates them. The theory of wavelets is introduced and the use of wavelets in multiresolution analysis is discussed. This multiresolution analysis is applied to the transport equation, and equations that can be solved to calculate the coefficients in the wavelet expansion for the angular flux are derived. The use of thresholding to eliminate wavelet coefficients that are not required to adequately solve a problem is then discussed. An iterative sweeping algorithm, called the SN-WN method, is derived to solve the wavelet-based equations. The convergence of the SN-WN method is discussed. An algorithm for solving the equations is derived, by solving a matrix within each cell directly for the expansion coefficients. This algorithm is called the CWWN method. The results of applying the CW-WN method to the benchmark problems are presented. These results show that more research is needed to improve the convergence of the SN-WN method, and that the CW-WN method is computationally too costly to be seriously considered.

Indian Institute of Science / A large number of mathematical models in engineering and physical sciences employ Partial Differential Equations (PDEs). The sheer number of operations required in numerically integrating PDEs in these applications has motivated the search for faster methods of computing. The conventional uniprocessor computers are often unable to fulfill the performance requirements for these computation intensive problems. In this dissertation, a cost-effective message-based multiprocessor system which we call the Broadcast Cube System (BCS) has been proposed for solving important computation intensive problems such as, systems of linear algebraic equations and PDEs. A simulator for performance evaluation of parallel algorithms to be executed on the BCS has been implemented. A strategy (task assignment . algorithm) for assigning program tasks with precedence and communication constraints to the Processing Elements (PEs) in the BCS has been developed and its effectiveness demonstrated. This task assignment algorithm has been shown to produce optimal assignments for PDE problems. Optimal partitioning of the problems, solving systems of linear algebraic equations and PDEs, into tasks and their assignment to the PEs in the BCS have been given. Efficient parallel algorithms for solving these problems on the BCS have been designed. The performance of the parallel algorithms has been evaluated by both analytical and simulation methods. The results indicate that the BCS is highly effective in solving systems of linear algebraic equations and PDEs. The performance of these algorithms on the BCS has also been compared with that of their implementations on popular hypercube machines. The results show that the performance of the BCS is better than that of the hypercubes for linear algebraic equations and compares very well for PDEs, with a modest number of PEs despite the constant PE connectivity of three in the BCS. Finally, the effectiveness of the BCS in solving non-linear PDEs occurring in two important practical problems, (i) heat transfer and fluid flow simulation and (ii) global weather modeling, has been demonstrated.

The Boussinesq approximation is a set of fluids equations utilized in the atmospheric and oceanographic sciences. They may be thought of as inhomogeneous, incompressible Euler or Navier-Stokes equations, where the inhomogeneous term is a scalar quantity, typically representing density or temperature, governed by a convection-diffusion equation. In this thesis, we prove local-in-time existence and uniqueness of an inviscid Boussinesq system. Furthermore, we show that under stronger assumptions, the local-in-time results can be extended to global-in-time existence and uniqueness as well. We assume the density equation contains nonzero diffusion and that our initial vorticity and density belong to a space of borderline Besov-type. We use paradifferential calculus and properties of the Besov-type spaces to control the growth of vorticity via an a priori estimate on the growth of density. This result is motivated by work of M. Vishik demonstrating local-in-time existence and uniqueness for 2D Euler equations in borderline Besov-type spaces, and by work of R. Danchin and M. Paicu showing the global well-posedness of the 2D Boussinesq system with initial data in critical Besov and Lp-spaces. / text