3,067 research outputs found

    A novel iterative method to approximate structured singular values

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    A novel method for approximating structured singular values (also known as mu-values) is proposed and investigated. These quantities constitute an important tool in the stability analysis of uncertain linear control systems as well as in structured eigenvalue perturbation theory. Our approach consists of an inner-outer iteration. In the outer iteration, a Newton method is used to adjust the perturbation level. The inner iteration solves a gradient system associated with an optimization problem on the manifold induced by the structure. Numerical results and comparison with the well-known Matlab function mussv, implemented in the Matlab Control Toolbox, illustrate the behavior of the method

    Lattice Boltzmann Methods for Partial Differential Equations

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    Lattice Boltzmann methods provide a robust and highly scalable numerical technique in modern computational fluid dynamics. Besides the discretization procedure, the relaxation principles form the basis of any lattice Boltzmann scheme and render the method a bottom-up approach, which obstructs its development for approximating broad classes of partial differential equations. This work introduces a novel coherent mathematical path to jointly approach the topics of constructability, stability, and limit consistency for lattice Boltzmann methods. A new constructive ansatz for lattice Boltzmann equations is introduced, which highlights the concept of relaxation in a top-down procedure starting at the targeted partial differential equation. Modular convergence proofs are used at each step to identify the key ingredients of relaxation frequencies, equilibria, and moment bases in the ansatz, which determine linear and nonlinear stability as well as consistency orders of relaxation and space-time discretization. For the latter, conventional techniques are employed and extended to determine the impact of the kinetic limit at the very foundation of lattice Boltzmann methods. To computationally analyze nonlinear stability, extensive numerical tests are enabled by combining the intrinsic parallelizability of lattice Boltzmann methods with the platform-agnostic and scalable open-source framework OpenLB. Through upscaling the number and quality of computations, large variations in the parameter spaces of classical benchmark problems are considered for the exploratory indication of methodological insights. Finally, the introduced mathematical and computational techniques are applied for the proposal and analysis of new lattice Boltzmann methods. Based on stabilized relaxation, limit consistent discretizations, and consistent temporal filters, novel numerical schemes are developed for approximating initial value problems and initial boundary value problems as well as coupled systems thereof. In particular, lattice Boltzmann methods are proposed and analyzed for temporal large eddy simulation, for simulating homogenized nonstationary fluid flow through porous media, for binary fluid flow simulations with higher order free energy models, and for the combination with Monte Carlo sampling to approximate statistical solutions of the incompressible Euler equations in three dimensions

    Entanglement, randomness and chaos

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    Entanglement is not only the most intriguing feature of quantum mechanics, but also a key resource in quantum information science. The entanglement content of random pure quantum states is almost maximal; such states find applications in various quantum information protocols. The preparation of a random state or, equivalently, the implementation of a random unitary operator, requires a number of elementary one- and two-qubit gates that is exponential in the number n_q of qubits, thus becoming rapidly unfeasible when increasing n_q. On the other hand, pseudo-random states approximating to the desired accuracy the entanglement properties of true random states may be generated efficiently, that is, polynomially in n_q. In particular, quantum chaotic maps are efficient generators of multipartite entanglement among the qubits, close to that expected for random states. This review discusses several aspects of the relationship between entanglement, randomness and chaos. In particular, I will focus on the following items: (i) the robustness of the entanglement generated by quantum chaotic maps when taking into account the unavoidable noise sources affecting a quantum computer; (ii) the detection of the entanglement of high-dimensional (mixtures of) random states, an issue also related to the question of the emergence of classicality in coarse grained quantum chaotic dynamics; (iii) the decoherence induced by the coupling of a system to a chaotic environment, that is, by the entanglement established between the system and the environment.Comment: Review paper, 40 pages, 7 figures, added reference

    Wavelet-based density estimation for noise reduction in plasma simulations using particles

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    For given computational resources, the accuracy of plasma simulations using particles is mainly held back by the noise due to limited statistical sampling in the reconstruction of the particle distribution function. A method based on wavelet analysis is proposed and tested to reduce this noise. The method, known as wavelet based density estimation (WBDE), was previously introduced in the statistical literature to estimate probability densities given a finite number of independent measurements. Its novel application to plasma simulations can be viewed as a natural extension of the finite size particles (FSP) approach, with the advantage of estimating more accurately distribution functions that have localized sharp features. The proposed method preserves the moments of the particle distribution function to a good level of accuracy, has no constraints on the dimensionality of the system, does not require an a priori selection of a global smoothing scale, and its able to adapt locally to the smoothness of the density based on the given discrete particle data. Most importantly, the computational cost of the denoising stage is of the same order as one time step of a FSP simulation. The method is compared with a recently proposed proper orthogonal decomposition based method, and it is tested with three particle data sets that involve different levels of collisionality and interaction with external and self-consistent fields

    Unified control/structure design and modeling research

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    To demonstrate the applicability of the control theory for distributed systems to large flexible space structures, research was focused on a model of a space antenna which consists of a rigid hub, flexible ribs, and a mesh reflecting surface. The space antenna model used is discussed along with the finite element approximation of the distributed model. The basic control problem is to design an optimal or near-optimal compensator to suppress the linear vibrations and rigid-body displacements of the structure. The application of an infinite dimensional Linear Quadratic Gaussian (LQG) control theory to flexible structure is discussed. Two basic approaches for robustness enhancement were investigated: loop transfer recovery and sensitivity optimization. A third approach synthesized from elements of these two basic approaches is currently under development. The control driven finite element approximation of flexible structures is discussed. Three sets of finite element basic vectors for computing functional control gains are compared. The possibility of constructing a finite element scheme to approximate the infinite dimensional Hamiltonian system directly, instead of indirectly is discussed

    Split representation of adaptively compressed polarizability operator

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    The polarizability operator plays a central role in density functional perturbation theory and other perturbative treatment of first principle electronic structure theories. The cost of computing the polarizability operator generally scales as O(Ne4)\mathcal{O}(N_{e}^4) where NeN_e is the number of electrons in the system. The recently developed adaptively compressed polarizability operator (ACP) formulation [L. Lin, Z. Xu and L. Ying, Multiscale Model. Simul. 2017] reduces such complexity to O(Ne3)\mathcal{O}(N_{e}^3) in the context of phonon calculations with a large basis set for the first time, and demonstrates its effectiveness for model problems. In this paper, we improve the performance of the ACP formulation by splitting the polarizability into a near singular component that is statically compressed, and a smooth component that is adaptively compressed. The new split representation maintains the O(Ne3)\mathcal{O}(N_e^3) complexity, and accelerates nearly all components of the ACP formulation, including Chebyshev interpolation of energy levels, iterative solution of Sternheimer equations, and convergence of the Dyson equations. For simulation of real materials, we discuss how to incorporate nonlocal pseudopotentials and finite temperature effects. We demonstrate the effectiveness of our method using one-dimensional model problem in insulating and metallic regimes, as well as its accuracy for real molecules and solids.Comment: 32 pages, 8 figures. arXiv admin note: text overlap with arXiv:1605.0802

    A mass-conserving sparse grid combination technique with biorthogonal hierarchical basis functions for kinetic simulations

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    The exact numerical simulation of plasma turbulence is one of the assets and challenges in fusion research. For grid-based solvers, sufficiently fine resolutions are often unattainable due to the curse of dimensionality. The sparse grid combination technique provides the means to alleviate the curse of dimensionality for kinetic simulations. However, the hierarchical representation for the combination step with the state-of-the-art hat functions suffers from poor conservation properties and numerical instability. The present work introduces two new variants of hierarchical multiscale basis functions for use with the combination technique: the biorthogonal and full weighting bases. The new basis functions conserve the total mass and are shown to significantly increase accuracy for a finite-volume solution of constant advection. Further numerical experiments based on the combination technique applied to a semi-Lagrangian Vlasov--Poisson solver show a stabilizing effect of the new bases on the simulations
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