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Interpolation problem for the solutions of linear elasticity equations based on monogenic functions

AIP conference Proceedings 2017年第1期1907卷

作者： Yuri Grigor’ev Klaus Gürlebeck Dmitrii Legatiuk Theoretical Physics Department North-Eastern Federal University 58 Belinsky Str. Yakutsk 677000 Russia Academy of Sciences of the Republic of Sakha Yakutia 33 Lenin Av. Yakutsk 677007 Russia Chair of Applied Mathematics Bauhaus-Universität Weimar Coudraystr. 13B Weimar 99423 Germany Research Training Group 1462 Bauhaus-Universität Weimar Marienstr. 7A Weimar 99423 Germany

Interpolation is an important tool for many practical applications, and very often it is beneficial to interpolate not only with a simple basis system, but rather with solutions of a certain differential equation, e.g. elasticity equation. A typical example for such type of interpolation are collocation methods widely used in practice. It is known, that interpolation theory is fully developed in the framework of the classical complex analysis. However, in quaternionic analysis, which shows a lot of analogies to complex analysis, the situation is more complicated due to the non-commutative multiplication. Thus, a fundamental theorem of algebra is not available, and standard tools from linear algebra cannot be applied in the usual way. To overcome these problems, a special system of monogenic polynomials the so-called Pseudo Complex Polynomials, sharing some properties of complex powers, is used. In this paper, we present an approach to deal with the interpolation problem, where solutions of elasticity equations in three dimensions are used as an interpolation basis.

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Learning HMMs with Nonparametric Emissions via Spectral Decompositions of Continuous Matrices 16

Learning HMMs with Nonparametric Emissions via Spectral Deco...

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Annual conference on Neural Information Processing Systems

作者： Kirthevasan Kandasamy Maruan Al-Shedivat Eric P. Xing Carnegie Mellon University Pittsburgh PA 15213

ISBN: (纸本)9781510838819

Recently, there has been a surge of interest in using spectral methods for estimating latent variable models. However, it is usually assumed that the distribution of the observations conditioned on the latent variables is either discrete or belongs to a parametric family. In this paper, we study the estimation of an m-state hidden Markov model (HMM) with only smoothness assumptions, such as Holderian conditions, on the emission densities. By leveraging some recent advances in continuous linear algebra and numerical analysis, we develop a computationally efficient spectral algorithm for learning nonparametric HMMs. Our technique is based on computing an SVD on nonparametric estimates of density functions by viewing them as continuous matrices. We derive sample complexity bounds via concentration results for nonparametric density estimation and novel perturbation theory results for continuous matrices. We implement our method using Chebyshev polynomial approximations. Our method is competitive with other baselines on synthetic and real problems and is also very computationally efficient.

关键词： spectral methods matrices sample complexity Hidden Markov models viewing perturbation theory Learning Emissions numerical Analysis CHEBYSHEV POLYNOMIALS

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Alinea: An Advanced linear algebra Library for Massively Parallel Computations on Graphics Processing Units

Alinea: An Advanced Linear Algebra Library for Massively Par...

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Exascale applications and Software conference (EASC)

作者： Magoules, Frederic Ahamed, Abal-Kassim Cheik Ecole Cent Paris Dept Math Dept Comp Sci F-92295 Chatenay Malabry France Ecole Cent Paris F-92295 Chatenay Malabry France

Direct and iterative methods are often used to solve linear systems in engineering. The matrices involved can be large, which leads to heavy computations on the central processing unit. A graphics processing unit can be used to accelerate these computations. In this paper, we propose a new library, named Alinea, for advanced linear algebra. This library is implemented in C++, CUDA and OpenCL. It includes several linear algebra operations and numerous algorithms for solving linear systems. For both central processing unit and graphic processing unit devices, there are different matrix storage formats, and real and complex arithmetics in single- and double-precision. The CUDA version includes a self-tuning of the grid, i.e. threading distribution, depending upon the hardware configuration and the size of the problems. numerical experiments and comparison with existing libraries illustrates the efficiency, accuracy and robustness of the proposed library.

关键词： Alinea linear algebra direct solvers iterative solvers parallel computing CPU GPU CUDA OpenCL auto-tuning energy efficiency

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Direction-cosine-matrix-based attitude control subject to actuator saturation

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IET CONTROL theory AND applications 2015年第11期9卷 1653-1661页

作者： Forbes, James Richard Univ Michigan Dept Aerosp Engn Ann Arbor MI 48109 USA

Set-point attitude control of a rigid body explicitly preventing actuator saturation is considered. The attitude control approach developed does not employ any sort of direction-cosine-matrix (DCM) parameterisation, such as Euler angles or quaternions. Rather, the DCM is used directly within the feedback control algorithm. Together a proportional control term and an angular velocity control term make up the attitude controller. The angular velocity control is composed of a strictly positive real system subject to a special input non-linearity. The specific form of the proportional control and angular velocity control ensure control torques are below the saturation level of the on-board actuators. Two controller synthesis methods are considered. The first uses the linearised system, the solution to the linear quadratic regulator problem, and the Kalman-Yakubovich-Popov lemma to design the controller. The second employs a simple low-pass filter that is guaranteed to stabilise the closed-loop system;tuning the low-pass filter is also considered. numerical simulation results demonstrate effective closed-loop control in the presence of plant disturbances and sensor noise.

关键词： attitude control matrix algebra feedback angular velocity control control nonlinearities control system synthesis linearisation techniques linear quadratic control closed loop systems low-pass filters aerospace robotics direction-cosine-matrix-based attitude control actuator saturation set-point attitude control rigid body direction-cosine-matrix parameterisation DCM Euler angles quaternions feedback control algorithm angular velocity control term input nonlinearity proportional control on-board actuators controller synthesis methods linearised system linear quadratic regulator problem Kalman–Yakubovich–Popov lemma low-pass filter closed-loop system plant disturbances sensor noise

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Reprint of Solution of Ambrosio-Tortorelli model for image segmentation by generalized relaxation method (REPRINTED)

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COMMUNICATIONS IN NONlinear SCIENCE AND numerical SIMULATION 2015年第1-3期21卷 225-237页

作者： D'Ambra, Pasqua Tartaglione, Gaetano CNR Inst High Performance Comp & Networking ICAR Via P Castellino 111 I-80131 Naples Italy

Image segmentation addresses the problem to partition a given image into its constituent objects and then to identify the boundaries of the objects. This problem can be formulated in terms of a variational model aimed to find optimal approximations of a bounded function by piecewise-smooth functions, minimizing a given functional. The corresponding Euler-Lagrange equations are a set of two coupled elliptic partial differential equations with varying coefficients. numerical solution of the above system often relies on alternating minimization techniques involving descent methods coupled with explicit or semi-implicit finite-difference discretization schemes, which are slowly convergent and poorly scalable with respect to image size. In this work we focus on generalized relaxation methods also coupled with multigrid linear solvers, when a finite-difference discretization is applied to the Euler-Lagrange equations of Ambrosio-Tortorelli model. We show that non-linear Gauss-Seidel, accelerated by inner linear iterations, is an effective method for large-scale image analysis as those arising from high-throughput screening platforms for stem cells targeted differentiation, where one of the main goal is segmentation of thousand of images to analyze cell colonies morphology. (C) 2014 Elsevier B.V. All rights reserved.

关键词： Image segmentation Variational models Non-linear iterative methods Multigrid solvers

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Efficient Application of the Two-Grid Technique for Solving Time-Fractional Non-linear Parabolic Problem 6th

Efficient Application of the Two-Grid Technique for Solving ...

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6th International conference on Finite Difference methods - theory and applications (FDM - T and A)

作者： Koleva, Miglena N. Univ Rousse FNSE Rousse 7017 Bulgaria

ISBN: (纸本)9783319202396;9783319202389

In this paper we present numerical methods for solving a non-linear time-fractional parabolic model. To cope with non-local in time nature of the problem, we exploit the idea of the two-grid method and develop fast numerical algorithms. Moreover, we show that suitable modifications of the standard two-grid technique lead to significant reduction of the computational time. numerical results are also discussed.

关键词： numerical methods

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Relative Gain Array of Weakly Nonlinear Systems using a Nonparametric Identification Approach

Relative Gain Array of Weakly Nonlinear Systems using a Nonp...

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IEEE conference on Control and applications (CCA)

作者： Kadhim, Ali M. H. Castano, Miguel Birk, Wolfgang Gustafsson, Thomas Lulea Univ Technol Control Grp Lulea Sweden Kufa Univ Elect Engn Dept Najaf Iraq

ISBN: (纸本)9781479977871

This article presents a procedure to estimate the relative gain array (RGA) matrix for weakly nonlinear systems by means of nonparametric identification of the frequency response matrix (FRM). Specifically, the best linear approximation of nonlinear systems and the covariance of the nonlinear distortions are used in the relative gain array estimation. For the estimation neither process model nor model structure need to be known which is an advantage over methods that require accurate knowledge of a parametric process model. The proposed approach is compared with the original RGA and a nonlinear RGA calculation using the well-known quadruple tank process as a case.

关键词： approximation theory estimation theory frequency response identification matrix algebra nonlinear systems FRM RGA matrix estimation frequency response matrix linear approximation nonlinear distortion covariance nonparametric identification relative gain array weakly nonlinear system Arrays Covariance matrices Estimation MIMO Nonlinear distortion Nonlinear systems Uncertainty Nonlinear systems Nonlinear distortion relative gain linear approximation estimation theory Approximation theory matrix algebra frequency response multiple input multiple output systems arrays variance covariance matrix residual gas analysis Covariance matrix

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H_∞ control for a class of multi-agent systems via a stochastic sampled-data method

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IET CONTROL theory AND applications 2015年第14期9卷 2057-2065页

作者： Wei, Guoliang Wang, Licheng Liu, Yurong Shanghai Univ Sci & Technol Dept Control Sci & Engn Shanghai 200093 Peoples R China Yangzhou Univ Dept Math Yangzhou 225002 Peoples R China King Abdulaziz Univ Commun Syst & Networks CSN Res Grp Fac Engn Jeddah 21589 Saudi Arabia

In this study, the stochastic sampled-data approach is developed to deal with the H-infinity control problem for a class of linear multi-agent systems. For each agent, signals should be transmitted to the control input, including not only its own measurements, but also the output information received from the neighbour agents. The relationships of information communications between the whole multi-agents are totally reflected by a weighted directed graph with a fixed topology structure. To the addressed problem, the author's focus is to design an output feedback controller for each agent such that, for the external energy bounded disturbances, the exponential stability in the mean square sense and interference rejection ability under consideration can be guaranteed. By assuming that the sampling period of each agent switches between two values in a probabilistic way, the stochastic sampling phenomenon is well characterised in every sampling tasks. In virtue of Lyapunov theory and semi-definite programming method, sufficient conditions are derived to ensure the exponential stability as well as H-infinity performance, and the feedback controllers are also designed by solving some certain matrix inequalities. Finally, a numerical simulation is given to demonstrate the validness of the proposed control strategy.

关键词： linear systems multi-agent systems multi-robot systems H∞ control sampled data systems stochastic systems directed graphs mobile robots feedback control system synthesis asymptotic stability sampling methods Lyapunov methods mathematical programming matrix algebra H∞ control problem linear multiagent systems stochastic sampled-data method control input information communications weighted directed graph fixed topology structure output feedback controller design external energy bounded disturbances exponential stability mean square sense interference rejection ability sampling period stochastic sampling phenomenon sampling tasks Lyapunov theory semidefinite programming method matrix inequalities numerical simulation

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The finite and spectral cell methods for smart structure applications: transient analysis

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ACTA MECHANICA 2015年第3期226卷 845-869页

作者： Duczek, S. Liefold, S. Gabbert, U. Univ Magdeburg Inst Mech D-39106 Magdeburg Germany

This article introduces a robust and efficient numerical tool that is well suited for the simulation of ultrasonic guided waves and can be especially helpful when dealing with heterogeneous materials. The proposed method is based on a combination of high-order finite element methods (FEM) and the fictitious domain concept. If hierarchic shape functions, which are familiar from the p-version of the finite element method (p-FEM), are deployed the method is referred to as the finite cell method (FCM). Where Lagrange polynomials through Gau-Lobatto-Legendre points are used, we refer to it as the spectral cell method (SCM). The name, SCM, derives from the fact that the deployed shape functions are commonly utilized in the spectral element method. To model smart structure applications such as shape control problems, noise cancelation devices and the excitation/sensing of ultrasonic guided waves, a coupling between electrical and mechanical variables is also taken into account. In this context, we focus on including the linear theory of piezoelectricity in the variational formulation of the FCM and the SCM, respectively. Several numerical benchmark problems are then used to validate the proposed approach. The simulations show promising results with respect to the accuracy of the method and the computational effort. We observe similar convergence properties for the proposed high-order fictitious domain methods as with "conventional" high-order finite element approaches. Implementing the proposed method in existing finite element software is, moreover, a straightforward process. These properties make the method an efficient tool for practical applications in structural health monitoring problems and smart structure applications in general.

关键词： FINITE fields (algebra) SMART structures TRANSIENT analysis ROBUST control INHOMOGENEOUS materials

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17th International Workshop on Computer algebra in Scientific Computing, CASC 2015

17th International Workshop on Computer Algebra in Scientifi...

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17th International Workshop on Computer algebra in Scientific Computing, CASC 2015

ISBN: (纸本)9783319240206

The proceedings contain 35 papers. The special focus in this conference is on Computer algebra in Scientific Computing. The topics include: Hypergeometric solutions of first-order linear difference systems with rational-function coefficients;janet bases and resolutions in CoCoALib;regular chains under linear changes of coordinates and applications;a standard basis free algorithm for computing the tangent cones of a space curve;research on the stability of relative equilibria of oblate axisymmetric gyrostat by means of symbolic-numerical modelling;a new approach for computing regular solutions of linear difference systems;solving polynomial systems in the cloud with polynomial homotopy continuation;finding first integrals using normal forms modulo differential regular chains;simplification of cylindrical algebraic formulas;intuition, perturbation theory and algorithmic algebra;polynomial complexity recognizing a tropical linear variety;computing highest-order divisors for a class of quasi-linear partial differential equations;symbolic-numeric solution of boundary-value problems for the schrodinger equation using the finite element method;application of computer algebra methods to investigation of influence of constant torque on stationary motions of satellite;bounds for the condition number of polynomials systems with integer coefficients;on invariant manifolds and their stability in the problem of motion of a rigid body under the influence of two force fields;homotopy analysis method for stochastic differential equations with maxima;on the topology and visualization of plane algebraic curves;piecewise-quadratics and reparameterizations for interpolating reduced data and parametric solvable polynomial rings and applications.

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