Download or read book Generalized Polar Coordinates on Lie Groups and Numerical Integrators written by Stein Krogstad and published by . This book was released on 2003 with total page 29 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Generalized Polar Coordinates on Lie Groups and Numerical Intergrators written by Stein Krogstad and published by . This book was released on 2003 with total page 50 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Discrete Mechanics Geometric Integration and Lie Butcher Series written by Kurusch Ebrahimi-Fard and published by Springer. This book was released on 2018-11-05 with total page 361 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume resulted from presentations given at the international “Brainstorming Workshop on New Developments in Discrete Mechanics, Geometric Integration and Lie–Butcher Series”, that took place at the Instituto de Ciencias Matemáticas (ICMAT) in Madrid, Spain. It combines overview and research articles on recent and ongoing developments, as well as new research directions. Why geometric numerical integration? In their article of the same title Arieh Iserles and Reinout Quispel, two renowned experts in numerical analysis of differential equations, provide a compelling answer to this question. After this introductory chapter a collection of high-quality research articles aim at exploring recent and ongoing developments, as well as new research directions in the areas of geometric integration methods for differential equations, nonlinear systems interconnections, and discrete mechanics. One of the highlights is the unfolding of modern algebraic and combinatorial structures common to those topics, which give rise to fruitful interactions between theoretical as well as applied and computational perspectives. The volume is aimed at researchers and graduate students interested in theoretical and computational problems in geometric integration theory, nonlinear control theory, and discrete mechanics.

Download or read book Generalized Polar Decomposition for the Approximation of the Matrix Exponential written by A. Zanna and published by . This book was released on 2000 with total page 46 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book An Introduction to Lie Groups and Lie Algebras written by Alexander A. Kirillov and published by Cambridge University Press. This book was released on 2008-07-31 with total page 237 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an introduction to semisimple Lie algebras. It is concise and informal, with numerous exercises and examples.

Download or read book A Low Complexity Lie Group Method on the Stiefel Manifold written by Stein Krogstad and published by . This book was released on 2001 with total page 34 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book BIT written by and published by . This book was released on 2003 with total page 888 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Graph Searching Elimination Trees and a Generalization of Bandwidth written by Fedor V. Fomin and published by . This book was released on 2003 with total page 40 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Application of Symmetric Spaces and Lie Triple Systems in Numerical Analysis written by Hans Munthe-Kaas and published by . This book was released on 2001 with total page 42 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Broadcast Domination Algorithms for Interval Graphs Series parallel Graphs and Trees written by Jean R. S. Blair and published by . This book was released on 2003 with total page 48 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Backbone Colorings for Networks written by Hajo Broersma and published by . This book was released on 2003 with total page 44 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book The Polar Decomposition on Lie Groups with Involutive Automorphisms written by H. Munthe-Kaas and published by . This book was released on 2000 with total page 44 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Geometric Integrators for Stiff Systems Lie Groups and Control Systems written by Xuefeng Shen and published by . This book was released on 2019 with total page 158 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main idea of a geometric integrator is to adopt a geometric viewpoint of the problem and to construct integrators that preserve the geometric properties of the continuous dynamical system. For classical mechanics, both the Lagrangian and the Hamiltonian formulations can be described using the language of geometry. Due to the rich conservation properties of mechanics, it is natural to study the construction of numerical integrators that preserve some geometric properties, such as the symplectic structure, energy, and momentum maps. Such geometric structure-preserving numerical integrators exhibit nice properties compared to traditional numerical methods. This is especially true in galaxy simulations and molecular dynamics, where long time simulations are required to answer the corresponding scientific questions. Variational integrators have attracted interest in the geometric integration community as it discretizes Hamilton's principle, as opposed to the corresponding differential equation, to obtain a numerical integrator that is automatically symplectic, and which exhibits a discrete Noether's theorem. Besides classical mechanics, such an approach has also been applied to other fields, such as optimal control~\cite{junge2005discrete,leyendecker2010discrete}, partial differential equations~\cite{marsden1998multisymplectic}, stochastic differential equations~\cite{bou2009stochastic}, and so on. In this thesis, we consider generalizations of geometric integrators that are adapted to three special settings. One is the case of stiff systems of the form, $\dot{q} = Aq + f(q)$, where the coefficient matrix $A$ has a large spectral radius that is responsible for the stiffness of the system, while the nonlinear term $f(q)$ is relatively smooth. Traditionally, exponential integrators have been used to address the issue of stiffness. In Chapter~\ref{exp}, we consider a special semilinear problem with $A=JD$, $f(q)=J\nabla V(q)$, where $J^T = -J, D^T=D$, and $JD=DJ$. Then, the system is described by $\dot{q} = J(Dq+\nabla V(q))$, which naturally arises from the discretization of Hamiltonian partial differential equations. It is a constant Poisson system with Poisson structure $J_{ij}\frac{\partial}{\partial x_i}\otimes \frac{\partial}{\partial x_j}$, and Hamiltonian $H(q) = \frac{1}{2}q^TDq + V(q)$. Two types of exponential integrators are constructed, one preserves the Poisson structure, and the other preserves energy. Numerical experiments for semilinear Possion systems obtained by semi-discretizing Hamiltonian PDEs are presented. These geometric exponential integrators exhibit better long time stability properties as compared to non-geometric integrators, and are computationally more efficient than traditional symplectic integrators and energy-preserving methods based on the discrete gradient method. The other generalization is to Lie groups. When configuration manifold is a Lie group, we would like to utilize the group structure rather than simply regard it as embedded submanifold. This is particularly useful when codimension of the embedding is large. For the rigid body problem, the configuration space is $\mathbb{R}^3\rtimes SO(3)$, which is a Lie group. \citet{LeMcLe2005} were the first to directly use the Lie group structure of the rotation group to construct a Lie group variational integrator. In contrast, most prior approaches used the unit quaternion representation of the rotation group and applied symplectic integrators for constrained systems with the unit length constraint. In Chapter~\ref{quater}, we adopt the approach used in constructing Lie group variational integrators for rigid body dynamics on the rotation group and applied it to the unit quaternion representation. A Lie group variational integrator in the unit quaternion representation is derived, and it can be shown that our method is related to the RATTLE method applied to the rotation representation by the projection from unit quaternions to rotation matrices. The numerical results for our Lie group quaternion variational integrator are presented. The integrators constructed in Chapter~\ref{quater} are only second-order, and in Chapter~\ref{polar}, variational integrators of arbitrarily high-order on special orthogonal group $SO(n)$ are constructed by using the polar decomposition. It avoids the second-order derivative of the exponential map that arises in the traditional Lie group variational integrator method. Also, a reduced Lie-Poisson integrator is constructed. The resulting algorithms can naturally be implemented using fixed-point iteration. Numerical results are given for the case of $SO(3)$. The last generalization is to control systems. We studied the problem of uncertainty propagation and measurement update for systems that are partially unobservable. We construct a method that satisfies the chain property that the unobservable subspace remains perpendicular to the measurement $dh$ during propagation. We characterize the unobservable subspace in terms of the group-invariance of the control system, and obtain a reduced control system on the observable variables. By decomposing the system explicitly into unobservable and observable parts $(x_N, x_O)$, the chain property can be naturally satisfied. Also, we propose a reduced Bayesian framework, where the update from the measurement is only applied to the observable variables $x_O$. In Chapter~\ref{geometric_reduce}, we consider a planar robot model, which has one odometry sensor and one camera. Odometry is used for propagation and the camera is used for measurement. In this model, the two-dimensional position as well as the orientation are all unobservable. We applied our technique to this model and performed numerical simulations. We tested this on straight line, circle, and general trajectories and found that the reduced Kalman filter that we proposed outperforms the classical Kalman filter and modifications that were proposed in the literature. In particular, it estimates the angle quite well, and as a result, yields a better estimate of the position as well.

Download or read book Journal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis written by Society for Industrial and Applied Mathematics and published by . This book was released on 2005 with total page 944 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Encyclopaedia of Mathematics written by Michiel Hazewinkel and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 743 pages. Available in PDF, EPUB and Kindle. Book excerpt: This ENCYCLOPAEDIA OF MATHEMATICS aims to be a reference work for all parts of mathe matics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-1985. The annotated translation consists of ten volumes including a special index volume. There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type articles dealing with the various main directions in mathematics (where a rather fine subdivi sion has been used). The main requirement for these articles has been that they should give a reasonably complete up-to-date account of the current state of affairs in these areas and that they should be maximally accessible. On the whole, these articles should be understandable to mathematics students in their first specialization years, to graduates from other mathematical areas and, depending on the specific subject, to specialists in other domains of science, en gineers and teachers of mathematics. These articles treat their material at a fairly general level and aim to give an idea of the kind of problems, techniques and concepts involved in the area in question. They also contain background and motivation rather than precise statements of precise theorems with detailed definitions and technical details on how to carry out proofs and constructions. The second kind of article, of medium length, contains more detailed concrete problems, results and techniques.

Download or read book Lie Groups and Algebraic Groups written by Arkadij L. Onishchik and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 347 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is based on the notes of the authors' seminar on algebraic and Lie groups held at the Department of Mechanics and Mathematics of Moscow University in 1967/68. Our guiding idea was to present in the most economic way the theory of semisimple Lie groups on the basis of the theory of algebraic groups. Our main sources were A. Borel's paper [34], C. ChevalIey's seminar [14], seminar "Sophus Lie" [15] and monographs by C. Chevalley [4], N. Jacobson [9] and J-P. Serre [16, 17]. In preparing this book we have completely rearranged these notes and added two new chapters: "Lie groups" and "Real semisimple Lie groups". Several traditional topics of Lie algebra theory, however, are left entirely disregarded, e.g. universal enveloping algebras, characters of linear representations and (co)homology of Lie algebras. A distinctive feature of this book is that almost all the material is presented as a sequence of problems, as it had been in the first draft of the seminar's notes. We believe that solving these problems may help the reader to feel the seminar's atmosphere and master the theory. Nevertheless, all the non-trivial ideas, and sometimes solutions, are contained in hints given at the end of each section. The proofs of certain theorems, which we consider more difficult, are given directly in the main text. The book also contains exercises, the majority of which are an essential complement to the main contents.

Download or read book Langlands Correspondence for Loop Groups written by Edward Frenkel and published by Cambridge University Press. This book was released on 2007-06-28 with total page 5 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first account of local geometric Langlands Correspondence, a new area of mathematical physics developed by the author.