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 Geometric Variational Integrators for Multisymplectic PDEs and Adjoint Systems written by Brian Kha Tran and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Variational integrators are a class of geometric structure-preserving numerical integrators that are based on a discretization of Hamilton's variational principle. We construct, analyze and investigate the applications of variational integrators to multisymplectic partial differential equations and to adjoint systems. The variational structure of multisymplectic PDEs encodes both the conservation laws admitted by these systems via Noether's theorem and multisymplecticity, a covariant spacetime generalization of symplecticity. We develop variational integrators for these systems which preserve these properties at the discrete level, in both the Lagrangian and Hamiltonian settings. In the Lagrangian setting, we utilize compatible finite element spaces to develop these variational integrators and utilize their preservation of the de Rham complex to define discrete geometric structures associated to these integrators and naturally relate them to their continuous counterparts. In the Hamiltonian setting, we utilize a discrete Type II variational principle, based on the notion of a Type II generating functional for multisymplectic PDEs, to construct structure-preserving variational integrators for multisymplectic Hamiltonian PDEs. Adjoint systems are ubiquitous in optimization and optimal control theory since they allow for efficient computation of sensitivities of cost functionals in optimization problems and arise as necessary conditions for optimality in optimal control problems via Pontryagin's maximum principle. Adjoint systems admit a fundamental quadratic conservation law which is at the heart of the method of adjoint sensitivity analysis; this conservation law arises from the symplectic geometry of these systems. We develop a geometric theory for continuous and discrete adjoint systems associated to ordinary differential equations and differential-algebraic equations, by investigating their underlying symplectic and presymplectic structures, respectively. We develop a Type II variational principle for such systems at the continuous level. Subsequently, we discretize this variational principle to construct variational integrators for adjoint systems which preserve the quadratic conservation law at the discrete level and thus, allow for sensitivities of cost functions to be computed exactly. We further extend this framework to the Lie group setting and develop a variational integrator based on novel continuous and discrete Type II variational principles on cotangent bundles of Lie groups.

Download or read book Geometric Integrators with Application to Hamiltonian Systems written by Hebatallah Jamil Sakaji and published by . This book was released on 2015 with total page 140 pages. Available in PDF, EPUB and Kindle. Book excerpt: Geometric numerical integration is a relatively new area of numerical analysis. The aim is to preserve the geometric properties of the flow of a differential equation such as symplecticity or reversibility. A conventional numerical integrator approximates the flow of the continuous-time equations using only the information about the vector field, ignoring the physical laws and the properties of the original trajectory. In this way, small inaccuracies accumulated over long periods of time will significantly diminish the operational lifespan of such discrete solutions. Geometric integrators, on the other hand, are built in a way that preserve the structure of continuous dynamics, so maintaining the qualitative behavior of the exact flow even for long-time integration. The aim of this thesis is to design efficient geometric integrators for Hamiltonian systems and to illustrate their effectiveness. These methods are implicit for general (non-separable) Hamiltonian systems making them difficult to implement. However, we show that explicit integrators are possible in some cases. Both geometric and non-geometric integration methods are applied to several problems, then we do a comparison between these methods, in order to determine which of those quantities are preserved better by these methods. In particular, we develop explicit integrators for a special case of the restricted 3-body problem known as Hill's problem.

Download or read book Multibody Dynamics written by Zdravko Terze and published by Springer. This book was released on 2014-06-26 with total page 369 pages. Available in PDF, EPUB and Kindle. Book excerpt: By having its origin in analytical and continuum mechanics, as well as in computer science and applied mathematics, multibody dynamics provides a basis for analysis and virtual prototyping of innovative applications in many fields of contemporary engineering. With the utilization of computational models and algorithms that classically belonged to different fields of applied science, multibody dynamics delivers reliable simulation platforms for diverse highly-developed industrial products such as vehicle and railway systems, aeronautical and space vehicles, robotic manipulators, smart structures, biomechanical applications and nano-technologies. The chapters of this volume are based on the revised and extended versions of the selected scientific papers from amongst 255 original contributions that have been accepted to be presented within the program of the distinguished international ECCOMAS conference. It reflects state-of-the-art in the advances of multibody dynamics, providing excellent insight in the recent scientific developments in this prominent field of computational mechanics and contemporary engineering.

Download or read book Computational Science ICCS 2006 written by and published by Springer Science & Business Media. This book was released on 2006 with total page 1128 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Stochastic Models Information Theory and Lie Groups Volume 2 written by Gregory S. Chirikjian and published by Springer Science & Business Media. This book was released on 2011-11-15 with total page 460 pages. Available in PDF, EPUB and Kindle. Book excerpt: This unique two-volume set presents the subjects of stochastic processes, information theory, and Lie groups in a unified setting, thereby building bridges between fields that are rarely studied by the same people. Unlike the many excellent formal treatments available for each of these subjects individually, the emphasis in both of these volumes is on the use of stochastic, geometric, and group-theoretic concepts in the modeling of physical phenomena. Stochastic Models, Information Theory, and Lie Groups will be of interest to advanced undergraduate and graduate students, researchers, and practitioners working in applied mathematics, the physical sciences, and engineering. Extensive exercises, motivating examples, and real-world applications make the work suitable as a textbook for use in courses that emphasize applied stochastic processes or differential geometry.

Download or read book Scientific and Technical Aerospace Reports written by and published by . This book was released on 1995 with total page 702 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Computational Methods in Physics written by Simon Širca and published by Springer. This book was released on 2018-06-21 with total page 894 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is intended to help advanced undergraduate, graduate, and postdoctoral students in their daily work by offering them a compendium of numerical methods. The choice of methods pays significant attention to error estimates, stability and convergence issues, as well as optimization of program execution speeds. Numerous examples are given throughout the chapters, followed by comprehensive end-of-chapter problems with a more pronounced physics background, while less stress is given to the explanation of individual algorithms. The readers are encouraged to develop a certain amount of skepticism and scrutiny instead of blindly following readily available commercial tools. The second edition has been enriched by a chapter on inverse problems dealing with the solution of integral equations, inverse Sturm-Liouville problems, as well as retrospective and recovery problems for partial differential equations. The revised text now includes an introduction to sparse matrix methods, the solution of matrix equations, and pseudospectra of matrices; it discusses the sparse Fourier, non-uniform Fourier and discrete wavelet transformations, the basics of non-linear regression and the Kolmogorov-Smirnov test; it demonstrates the key concepts in solving stiff differential equations and the asymptotics of Sturm-Liouville eigenvalues and eigenfunctions. Among other updates, it also presents the techniques of state-space reconstruction, methods to calculate the matrix exponential, generate random permutations and compute stable derivatives.

Download or read book Research in Progress written by and published by . This book was released on 1980 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Geometric Numerical Integration written by Ernst Hairer and published by Springer Science & Business Media. This book was released on 2006-05-18 with total page 660 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book covers numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions. It presents a theory of symplectic and symmetric methods, which include various specially designed integrators, as well as discusses their construction and practical merits. The long-time behavior of the numerical solutions is studied using a backward error analysis combined with KAM theory.

Download or read book Research in Progress Between and written by United States. Army Research Office and published by . This book was released on 1978 with total page 458 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Highly Oscillatory Problems written by Bjorn Engquist and published by Cambridge University Press. This book was released on 2009-07-02 with total page 254 pages. Available in PDF, EPUB and Kindle. Book excerpt: Review papers from experts in areas of active research into highly oscillatory problems, with an emphasis on computation.

Download or read book Physics Briefs written by and published by . This book was released on 1981-07 with total page 1542 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Geometric Numerical Integration written by Ernst Hairer and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 526 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book deals with numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions. A complete self-contained theory of symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators, is presented and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory. The book is illustrated by numerous figures, treats applications from physics and astronomy, and contains many numerical experiments and comparisons of different approaches.

Download or read book Proceedings of 8th GACM Colloquium on Computational Mechanics written by Tobias Gleim and published by kassel university press GmbH. This book was released on 2019-09-04 with total page 493 pages. Available in PDF, EPUB and Kindle. Book excerpt: This conference book contains papers presented at the 8th GACM Colloquium on Computational Mechanics for Young Scientists from Academia and Industry. The conference was held from August 28th – 30th, 2019 in Kassel, hosted by the Institute of Mechanics and Dynamics of the department for civil and environmental engineering and by the chair of Engineering Mechanics / Continuum Mechanics of the department for mechanical engineering of the University of Kassel. The aim of the conference is, to bring together young scientits who are engaged in academic and industrial research on Computational Mechanics and Computer Methods in Applied Sciences. It provides a plattform to present and discuss recent results from research efforts and industrial applications. In more than 150 presentations, given by young scientists, current scientific developments and advances in engineering practice in this field are presented and discussed. The contributions of the young researchers are supplemented by a poster session and plenary talks from four senior scientists from academia and industry as well as from the GACM Best PhD Award winners 2017 and 2018.

Download or read book Feedback Systems written by Karl Johan Åström and published by Princeton University Press. This book was released on 2021-02-02 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The essential introduction to the principles and applications of feedback systems—now fully revised and expanded This textbook covers the mathematics needed to model, analyze, and design feedback systems. Now more user-friendly than ever, this revised and expanded edition of Feedback Systems is a one-volume resource for students and researchers in mathematics and engineering. It has applications across a range of disciplines that utilize feedback in physical, biological, information, and economic systems. Karl Åström and Richard Murray use techniques from physics, computer science, and operations research to introduce control-oriented modeling. They begin with state space tools for analysis and design, including stability of solutions, Lyapunov functions, reachability, state feedback observability, and estimators. The matrix exponential plays a central role in the analysis of linear control systems, allowing a concise development of many of the key concepts for this class of models. Åström and Murray then develop and explain tools in the frequency domain, including transfer functions, Nyquist analysis, PID control, frequency domain design, and robustness. Features a new chapter on design principles and tools, illustrating the types of problems that can be solved using feedback Includes a new chapter on fundamental limits and new material on the Routh-Hurwitz criterion and root locus plots Provides exercises at the end of every chapter Comes with an electronic solutions manual An ideal textbook for undergraduate and graduate students Indispensable for researchers seeking a self-contained resource on control theory

Download or read book Handbook of Variational Methods for Nonlinear Geometric Data written by Philipp Grohs and published by Springer Nature. This book was released on 2020-04-03 with total page 701 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book covers different, current research directions in the context of variational methods for non-linear geometric data. Each chapter is authored by leading experts in the respective discipline and provides an introduction, an overview and a description of the current state of the art. Non-linear geometric data arises in various applications in science and engineering. Examples of nonlinear data spaces are diverse and include, for instance, nonlinear spaces of matrices, spaces of curves, shapes as well as manifolds of probability measures. Applications can be found in biology, medicine, product engineering, geography and computer vision for instance. Variational methods on the other hand have evolved to being amongst the most powerful tools for applied mathematics. They involve techniques from various branches of mathematics such as statistics, modeling, optimization, numerical mathematics and analysis. The vast majority of research on variational methods, however, is focused on data in linear spaces. Variational methods for non-linear data is currently an emerging research topic. As a result, and since such methods involve various branches of mathematics, there is a plethora of different, recent approaches dealing with different aspects of variational methods for nonlinear geometric data. Research results are rather scattered and appear in journals of different mathematical communities. The main purpose of the book is to account for that by providing, for the first time, a comprehensive collection of different research directions and existing approaches in this context. It is organized in a way that leading researchers from the different fields provide an introductory overview of recent research directions in their respective discipline. As such, the book is a unique reference work for both newcomers in the field of variational methods for non-linear geometric data, as well as for established experts that aim at to exploit new research directions or collaborations. Chapter 9 of this book is available open access under a CC BY 4.0 license at link.springer.com.