447 research outputs found

    Stable continuous orthonormalization techniques for linear boundary value problems

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    An investigation is made of a hybrid method inspired by Riccati transformations and marching algorithms employing (parts of) orthogonal matrices, both being decoupling algorithms. It is shown that this so-called continuous orthonormalisation is stable and practical as well. Nevertheless, if the problem is stiff and many output points are required the method does not give much gain over, say, multiple shootin

    Stable Continuous Orthonormalization Techniques for Linear Boundary Value Problems

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    Indirect methods for the numerical solution of ordinary linear boundary value problems

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    PhD thesisThis thesis is mainly concerned with indirect numerical solution methods for linear two point boundary value problems. We concentrate particularly on problems with separated boundary conditions which have a 'dichotomy' property. We investigate the inter-relationship of various methods including some which have first appeared since the work for this thesis began. We examine the stability of these methods and in particular we consider circumstances in which the methods discussed give rise to well conditioned decoupling transformations. Empirical comparisons of some of the methods are described using a set of test problems including a number of ill conditioned problams. 'stiff' and marginally In the past the main method of error estimation has been to repaat the whole calculation. Here an alternative error estimation technique is proposed and a related iterative improvement method is considered. Although results for this are not completely conclusive we think they justify the need for further research on the method as it shows promise of being a novel and reliable practical method of solving both well conditioned and ill conditioned problems

    Evaluation of (unstable) non-causal systems applied to iterative learning control

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    Abstract This paper presents a new approach towards the design of iterative learning control (Moore, 199

    Unitary Integrators and Applications to Continuous Orthonormalization Techniques

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    This is the published version, also available here: http://dx.doi.org/10.1137/0731014.In this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed. Such systems have skew-Hermitian coefficient matrices in the linear case and a related structure in the nonlinear case. These skew systems arise in a number of applications, and interest originates from application to continuous orthogonal decoupling techniques. In this case, the matrix system has a cubic nonlinearity. Numerical integration schemes that compute a unitary approximate solution for all stepsizes are studied. These schemes can be characterized as being of two classes: automatic and projected unitary schemes. In the former class, there belong those standard finite difference schemes which give a unitary solution; the only ones are in fact the Gauss–Legendre point Runge–Kutta (Gauss RK) schemes. The second class of schemes is created by projecting approximations computed by an arbitrary scheme into the set of unitary matrices. In the analysis of these unitary schemes, the stability considerations are guided by the skew-Hermitian character of the problem. Various error and implementation issues are considered, and the methods are tested on a number of examples

    The Conformal Bootstrap: Theory, Numerical Techniques, and Applications

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    Conformal field theories have been long known to describe the fascinating universal physics of scale invariant critical points. They describe continuous phase transitions in fluids, magnets, and numerous other materials, while at the same time sit at the heart of our modern understanding of quantum field theory. For decades it has been a dream to study these intricate strongly coupled theories nonperturbatively using symmetries and other consistency conditions. This idea, called the conformal bootstrap, saw some successes in two dimensions but it is only in the last ten years that it has been fully realized in three, four, and other dimensions of interest. This renaissance has been possible both due to significant analytical progress in understanding how to set up the bootstrap equations and the development of numerical techniques for finding or constraining their solutions. These developments have led to a number of groundbreaking results, including world record determinations of critical exponents and correlation function coefficients in the Ising and O(N)O(N) models in three dimensions. This article will review these exciting developments for newcomers to the bootstrap, giving an introduction to conformal field theories and the theory of conformal blocks, describing numerical techniques for the bootstrap based on convex optimization, and summarizing in detail their applications to fixed points in three and four dimensions with no or minimal supersymmetry.Comment: 81 pages, double column, 58 figures; v3: updated references, minor typos correcte

    Aspects of solving non-linear boundary value problems numerically

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    Mott physics and spin fluctuations: a functional viewpoint

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    We present a formalism for strongly correlated systems with fermions coupled to bosonic modes. We construct the three-particle irreducible functional K\mathcal{K} by successive Legendre transformations of the free energy of the system. We derive a closed set of equations for the fermionic and bosonic self-energies for a given K\mathcal{K}. We then introduce a local approximation for K\mathcal{K}, which extends the idea of dynamical mean field theory (DMFT) approaches from two- to three-particle irreducibility. This approximation entails the locality of the three-leg electron-boson vertex Λ(iω,iΩ)\Lambda(i\omega,i\Omega), which is self-consistently computed using a quantum impurity model with dynamical charge and spin interactions. This local vertex is used to construct frequency- and momentum-dependent electronic self-energies and polarizations. By construction, the method interpolates between the spin-fluctuation or GW approximations at weak coupling and the atomic limit at strong coupling. We apply it to the Hubbard model on two-dimensional square and triangular lattices. We complement the results of Phys.Rev. B 92, 115109 by (i) showing that, at half-filling, as DMFT, the method describes the Fermi-liquid metallic state and the Mott insulator, separated by a first-order interacting-driven Mott transition at low temperatures, (ii) investigating the influence of frustration and (iii) discussing the influence of the bosonic decoupling channel.Comment: 29 pages, 14 figure
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