Download or read book Optimal Processes on Manifolds written by R. Nottrot and published by Springer. This book was released on 2006-11-15 with total page 131 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Optimal Processes on Manifolds written by R Nottrot and published by Springer. This book was released on 2014-01-15 with total page 136 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Optimal Processes on Manifolds written by Roelof Nottrot and published by Springer. This book was released on 1982-01-01 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Optimal Processes on Manifolds written by Roelof Nottrot and published by . This book was released on 1982 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Mathematical Theory of Optimal Processes written by L.S. Pontryagin and published by CRC Press. This book was released on 1987-03-06 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: The fourth and final volume in this comprehensive set presents the maximum principle as a wide ranging solution to nonclassical, variational problems. This one mathematical method can be applied in a variety of situations, including linear equations with variable coefficients, optimal processes with delay, and the jump condition. As with the three preceding volumes, all the material contained with the 42 sections of this volume is made easily accessible by way of numerous examples, both concrete and abstract in nature.
Download or read book Analysis For Diffusion Processes On Riemannian Manifolds written by Feng-yu Wang and published by World Scientific. This book was released on 2013-09-23 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: Stochastic analysis on Riemannian manifolds without boundary has been well established. However, the analysis for reflecting diffusion processes and sub-elliptic diffusion processes is far from complete. This book contains recent advances in this direction along with new ideas and efficient arguments, which are crucial for further developments. Many results contained here (for example, the formula of the curvature using derivatives of the semigroup) are new among existing monographs even in the case without boundary.
Download or read book Convex Functions and Optimization Methods on Riemannian Manifolds written by C. Udriste and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 365 pages. Available in PDF, EPUB and Kindle. Book excerpt: The object of this book is to present the basic facts of convex functions, standard dynamical systems, descent numerical algorithms and some computer programs on Riemannian manifolds in a form suitable for applied mathematicians, scientists and engineers. It contains mathematical information on these subjects and applications distributed in seven chapters whose topics are close to my own areas of research: Metric properties of Riemannian manifolds, First and second variations of the p-energy of a curve; Convex functions on Riemannian manifolds; Geometric examples of convex functions; Flows, convexity and energies; Semidefinite Hessians and applications; Minimization of functions on Riemannian manifolds. All the numerical algorithms, computer programs and the appendices (Riemannian convexity of functions f:R ~ R, Descent methods on the Poincare plane, Descent methods on the sphere, Completeness and convexity on Finsler manifolds) constitute an attempt to make accesible to all users of this book some basic computational techniques and implementation of geometric structures. To further aid the readers,this book also contains a part of the folklore about Riemannian geometry, convex functions and dynamical systems because it is unfortunately "nowhere" to be found in the same context; existing textbooks on convex functions on Euclidean spaces or on dynamical systems do not mention what happens in Riemannian geometry, while the papers dealing with Riemannian manifolds usually avoid discussing elementary facts. Usually a convex function on a Riemannian manifold is a real valued function whose restriction to every geodesic arc is convex.
Download or read book The Mathematical Theory of Optimal Processes written by Lev Semenovich Pontri͡agin and published by . This book was released on 1962 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Differentiable Manifolds written by Lawrence Conlon and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is based on the full year Ph.D. qualifying course on differentiable manifolds, global calculus, differential geometry, and related topics, given by the author at Washington University several times over a twenty year period. It is addressed primarily to second year graduate students and well prepared first year students. Presupposed is a good grounding in general topology and modern algebra, especially linear algebra and the analogous theory of modules over a commutative, unitary ring. Although billed as a "first course" , the book is not intended to be an overly sketchy introduction. Mastery of this material should prepare the student for advanced topics courses and seminars in differen tial topology and geometry. There are certain basic themes of which the reader should be aware. The first concerns the role of differentiation as a process of linear approximation of non linear problems. The well understood methods of linear algebra are then applied to the resulting linear problem and, where possible, the results are reinterpreted in terms of the original nonlinear problem. The process of solving differential equations (i. e., integration) is the reverse of differentiation. It reassembles an infinite array of linear approximations, result ing from differentiation, into the original nonlinear data. This is the principal tool for the reinterpretation of the linear algebra results referred to above.
Download or read book Optimization Algorithms on Matrix Manifolds written by P.-A. Absil and published by Princeton University Press. This book was released on 2009-04-11 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.
Download or read book The Mathematical Theory of Optimal Processes written by Lev Semenovich Pontri︠a︡gin and published by Pergamon. This book was released on 1964 with total page 356 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Control and Optimization of Hybrid Systems on Riemannian Manifolds written by Farzin Taringoo and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: "The fundamental motivation for the work in this thesis is the analysis of the optimal control of hybrid systems on Riemannian manifolds using the language of differential geometry. Hybrid systems theory constitutes one of the major frameworks within which one may model and analyze the behaviour of large and complex systems; in particular, the optimal control of hybrid systems has been a focus of research over the last decades resulting in the important generalization of Minimum (Maximum) Principle of classic optimal control to hybrid systems. In the work of Shaikh and Caines (2007) and their predecessors, a formulation for a class of optimal control problems for general hybrid systems with nonlinear dynamics and autonomous or controlled switchings at switching states and times is proposed. In this thesis we extend the framework of Shaikh and Caines (2007) to a general class of hybrid systems defined on Riemannian manifolds. Due to the formulation generality, this class of hybrid systems covers a vast range of practical examples arising in such different areas as mechanical systems, chemical processes, air traffic control systems and cooperative robotic manipulator systems. In this thesis, a formulation for general hybrid systems on differentiable Riemannian manifolds is first presented. In the case of autonomous switchings, switching manifolds are modelled by embedded orientable submanifolds of the ambient state manifold and consequently hybrid optimal control problems are defined for hybrid systems in this general setting. Second, the classic Minimum Principle is extended to the Hybrid Minimum Principle (HMP) yielding the optimality necessary conditions for hybrid systems at the optimal switching states and times. The HMP statement in this thesis is obtained by employing the so-called needle control variation in the control value space. This class of control variations results in state trajectory variations along the nominal state trajectory in the ambient state manifold where the optimality conditions are derived by analyzing the cost function variation with respect to state variations. Third, in order to optimize switching states and times, numerical optimization algorithms (Gradient Geodesic-HMP, Newton Geodesic-HMP) are formulated by employing the HMP equations on general Riemannian state manifolds. The convergence analysis for the proposed algorithms is based upon the LaSalle Invariance Theorem. Technically these algorithms generalize the standard steepest descent and Newton methods in Euclidean spaces to Reimannian manifolds by employing the notion of Levi-Civita connections. Fourth, the derivation of the HMP results for hybrid systems on Riemannian manifolds is carried out for hybrid systems on Lie groups. The group structure of the ambient state manifold gives rise to a special form for the adjoint processes and Hamiltonian functions as the solutions for the optimality equations. In this thesis hybrid optimal control problems on Lie groups are only considered for the class of left invariant systems, however, the analysis can be easily modified to right invariant systems. In the setting of left invariant hybrid systems on Lie groups, the Gradient Geodesic-HMP and Newton Geodesic-HMP algorithm are modified into algorithms called the Gradient Exponential-HMP and Newton Exponential-HMP algorithms. The fifth and last part of the thesis focuses on the problem of optimization of autonomous hybrid optimal control problems with respect to the geometrical features of switching manifolds. Such features include first order and second order information on the switching manifolds such as curvature tensors and normal differential forms. " --
Download or read book Instantons and Four Manifolds written by D. S. Freed and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the outcome of a seminar organized by Michael Freedman and Karen Uhlenbeck (the senior author) at the Mathematical Sciences Research Institute in Berkeley during its first few months of existence. Dan Freed (the junior author) was originally appointed as notetaker. The express purpose of the seminar was to go through a proof of Simon Donaldson's Theorem, which had been announced the previous spring. Donaldson proved the nonsmoothability of certain topological four-manifolds; a year earlier Freedman had constructed these manifolds as part of his solution to the four dimensional ; Poincare conjecture. The spectacular application of Donaldson's and Freedman's theorems to the existence of fake 1R4,s made headlines (insofar as mathematics ever makes headlines). Moreover, Donaldson proved his theorem in topology by studying the solution space of equations the Yang-Mills equations which come from ultra-modern physics. The philosophical implications are unavoidable: we mathematicians need physics! The seminar was initially very well attended. Unfortunately, we found after three months that we had covered most of the published material, but had made little real progress towards giving a complete, detailed proof. Mter joint work extending over three cities and 3000 miles, this book now provides such a proof. The seminar bogged down in the hard analysis (56 59), which also takes up most of Donaldson's paper (in less detail). As we proceeded it became clear to us that the techniques in partial differential equations used in the proof differ strikingly from the geometric and topological material.
Download or read book Population Based Optimization on Riemannian Manifolds written by Robert Simon Fong and published by Springer Nature. This book was released on 2022-05-17 with total page 171 pages. Available in PDF, EPUB and Kindle. Book excerpt: Manifold optimization is an emerging field of contemporary optimization that constructs efficient and robust algorithms by exploiting the specific geometrical structure of the search space. In our case the search space takes the form of a manifold. Manifold optimization methods mainly focus on adapting existing optimization methods from the usual “easy-to-deal-with” Euclidean search spaces to manifolds whose local geometry can be defined e.g. by a Riemannian structure. In this way the form of the adapted algorithms can stay unchanged. However, to accommodate the adaptation process, assumptions on the search space manifold often have to be made. In addition, the computations and estimations are confined by the local geometry. This book presents a framework for population-based optimization on Riemannian manifolds that overcomes both the constraints of locality and additional assumptions. Multi-modal, black-box manifold optimization problems on Riemannian manifolds can be tackled using zero-order stochastic optimization methods from a geometrical perspective, utilizing both the statistical geometry of the decision space and Riemannian geometry of the search space. This monograph presents in a self-contained manner both theoretical and empirical aspects of stochastic population-based optimization on abstract Riemannian manifolds.
Download or read book Diffusion Processes for Stochastic Global Optimization on a Manifold with Applications in Image Processing written by Paula Andersen Shorter and published by . This book was released on 1996 with total page 340 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Manifolds II written by Paul Bracken and published by BoD – Books on Demand. This book was released on 2019-05-22 with total page 148 pages. Available in PDF, EPUB and Kindle. Book excerpt: Differential geometry is a very active field of research and has many applications to areas such as physics, in particular gravity. The chapters in this book cover a number of subjects that will be of interest to workers in these areas. It is hoped that these chapters will be able to provide a useful resource for researchers with regard to current fields of research in this important area.
Download or read book Hamiltonian and Gradient Flows Algorithms and Control written by Anthony Bloch and published by American Mathematical Soc.. This book was released on with total page 172 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the proceedings of a conference held at the Fields Insitute and designed to bring together traditionally disparate fields of mathematical research. On such key interraction occurs between dynamical systems and algorithms. This volume explores many such interractions as well as related work in optimal control and partial differential equations.
