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On The Relation Between Gradient Flows And The Large Deviation Principle With Applications To Markov Chains And Diffusion
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Book Synopsis On the Relation Between Gradient Flows and the Large-deviation Principle, with Applications to Markov Chains and Diffusion by : Alexander Mielke
Download or read book On the Relation Between Gradient Flows and the Large-deviation Principle, with Applications to Markov Chains and Diffusion written by Alexander Mielke and published by . This book was released on 2013 with total page 36 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis From Particle Systems to Partial Differential Equations by : Patrícia Gonçalves
Download or read book From Particle Systems to Partial Differential Equations written by Patrícia Gonçalves and published by Springer. This book was released on 2018-12-29 with total page 167 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the proceedings of the international conference Particle Systems and Partial Differential Equations V, which was held at the University of Minho, Braga, Portugal, from the 28th to 30th November 2016. It includes papers on mathematical problems motivated by various applications in physics, engineering, economics, chemistry, and biology. The purpose of the conference was to bring together prominent researchers working in the fields of particle systems and partial differential equations, providing a venue for them to present their latest findings and discuss their areas of expertise. Further, it was intended to introduce a vast and varied public, including young researchers, to the subject of interacting particle systems, its underlying motivation, and its relation to partial differential equations. The book appeals to probabilists, analysts and also to mathematicians in general whose work focuses on topics in mathematical physics, stochastic processes and differential equations, as well as to physicists working in the area of statistical mechanics and kinetic theory.
Book Synopsis Analysis and Computation of Microstructure in Finite Plasticity by : Sergio Conti
Download or read book Analysis and Computation of Microstructure in Finite Plasticity written by Sergio Conti and published by Springer. This book was released on 2015-04-23 with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book addresses the need for a fundamental understanding of the physical origin, the mathematical behavior and the numerical treatment of models which include microstructure. Leading scientists present their efforts involving mathematical analysis, numerical analysis, computational mechanics, material modelling and experiment. The mathematical analyses are based on methods from the calculus of variations, while in the numerical implementation global optimization algorithms play a central role. The modeling covers all length scales, from the atomic structure up to macroscopic samples. The development of the models ware guided by experiments on single and polycrystals and results will be checked against experimental data.
Book Synopsis Control of Self-Organizing Nonlinear Systems by : Eckehard Schöll
Download or read book Control of Self-Organizing Nonlinear Systems written by Eckehard Schöll and published by Springer. This book was released on 2016-01-22 with total page 478 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book summarizes the state-of-the-art of research on control of self-organizing nonlinear systems with contributions from leading international experts in the field. The first focus concerns recent methodological developments including control of networks and of noisy and time-delayed systems. As a second focus, the book features emerging concepts of application including control of quantum systems, soft condensed matter, and biological systems. Special topics reflecting the active research in the field are the analysis and control of chimera states in classical networks and in quantum systems, the mathematical treatment of multiscale systems, the control of colloidal and quantum transport, the control of epidemics and of neural network dynamics.
Book Synopsis Patterns of Dynamics by : Pavel Gurevich
Download or read book Patterns of Dynamics written by Pavel Gurevich and published by Springer. This book was released on 2018-02-07 with total page 411 pages. Available in PDF, EPUB and Kindle. Book excerpt: Theoretical advances in dynamical-systems theory and their applications to pattern-forming processes in the sciences and engineering are discussed in this volume that resulted from the conference Patterns in Dynamics held in honor of Bernold Fiedler, in Berlin, July 25-29, 2016.The contributions build and develop mathematical techniques, and use mathematical approaches for prediction and control of complex systems. The underlying mathematical theories help extract structures from experimental observations and, conversely, shed light on the formation, dynamics, and control of spatio-temporal patterns in applications. Theoretical areas covered include geometric analysis, spatial dynamics, spectral theory, traveling-wave theory, and topological data analysis; also discussed are their applications to chemotaxis, self-organization at interfaces, neuroscience, and transport processes.
Book Synopsis Multiscale Thermo-Dynamics by : Michal Pavelka
Download or read book Multiscale Thermo-Dynamics written by Michal Pavelka and published by Walter de Gruyter GmbH & Co KG. This book was released on 2018-08-06 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: One common feature of new emerging technologies is the fusion of the very small (nano) scale and the large scale engineering. The classical environment provided by single scale theories, as for instance by the classical hydrodynamics, is not anymore satisfactory. The main challenge is to keep the important details while still be able to keep the overall picture and simplicity. It is the thermodynamics that addresses this challenge. Our main reason for writing this book is to explain such general viewpoint of thermodynamics and to illustrate it on a very wide range of examples. Contents Levels of description Hamiltonian mechanics Irreversible evolution Reversible and irreversible evolution Multicomponent systems Contact geometry Appendix: Mathematical aspects
Book Synopsis Finite Volumes for Complex Applications X—Volume 1, Elliptic and Parabolic Problems by : Emmanuel Franck
Download or read book Finite Volumes for Complex Applications X—Volume 1, Elliptic and Parabolic Problems written by Emmanuel Franck and published by Springer Nature. This book was released on 2023-11-01 with total page 381 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume comprises the first part of the proceedings of the 10th International Conference on Finite Volumes for Complex Applications, FVCA, held in Strasbourg, France, during October 30 to November 3, 2023. The Finite Volume method, and several of its variants, is a spatial discretization technique for partial differential equations based on the fundamental physical principle of conservation. Recent decades have brought significant success in the theoretical understanding of the method. Many finite volume methods are also built to preserve some properties of the continuous equations, including maximum principles, dissipativity, monotone decay of the free energy, asymptotic stability, or stationary solutions. Due to these properties, finite volume methods belong to the wider class of compatible discretization methods, which preserve qualitative properties of continuous problems at the discrete level. This structural approach to the discretization of partial differential equations becomes particularly important for multiphysics and multiscale applications. In recent years, the efficient implementation of these methods in numerical software packages, more specifically to be used in supercomputers, has drawn some attention. This volume contains all invited papers, as well as the contributed papers focusing on finite volume schemes for elliptic and parabolic problems. They include structure-preserving schemes, convergence proofs, and error estimates for problems governed by elliptic and parabolic partial differential equations. The second volume is focused on finite volume methods for hyperbolic and related problems, such as methods compatible with the low Mach number limit or able to exactly preserve steady solutions, the development and analysis of high order methods, or the discretization of kinetic equations.
Download or read book Gradient Flows written by Luigi Ambrosio and published by Springer Science & Business Media. This book was released on 2008-10-29 with total page 333 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book is devoted to the theory of gradient flows in the general framework of metric spaces, and in the more specific setting of the space of probability measures, which provide a surprising link between optimal transportation theory and many evolutionary PDE's related to (non)linear diffusion. Particular emphasis is given to the convergence of the implicit time discretization method and to the error estimates for this discretization, extending the well established theory in Hilbert spaces. The book is split in two main parts that can be read independently of each other.
Book Synopsis Generalized Normalizing Flows via Markov Chains by : Paul Lyonel Hagemann
Download or read book Generalized Normalizing Flows via Markov Chains written by Paul Lyonel Hagemann and published by Cambridge University Press. This book was released on 2023-01-31 with total page 78 pages. Available in PDF, EPUB and Kindle. Book excerpt: Normalizing flows, diffusion normalizing flows and variational autoencoders are powerful generative models. This Element provides a unified framework to handle these approaches via Markov chains. The authors consider stochastic normalizing flows as a pair of Markov chains fulfilling some properties, and show how many state-of-the-art models for data generation fit into this framework. Indeed numerical simulations show that including stochastic layers improves the expressivity of the network and allows for generating multimodal distributions from unimodal ones. The Markov chains point of view enables the coupling of both deterministic layers as invertible neural networks and stochastic layers as Metropolis-Hasting layers, Langevin layers, variational autoencoders and diffusion normalizing flows in a mathematically sound way. The authors' framework establishes a useful mathematical tool to combine the various approaches.
Book Synopsis Multidimensional Diffusion Processes by : Daniel W. Stroock
Download or read book Multidimensional Diffusion Processes written by Daniel W. Stroock and published by Springer. This book was released on 2007-02-03 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the reviews: "This book is an excellent presentation of the application of martingale theory to the theory of Markov processes, especially multidimensional diffusions. [...] This monograph can be recommended to graduate students and research workers but also to all interested in Markov processes from a more theoretical point of view." Mathematische Operationsforschung und Statistik
Book Synopsis Limit Theorems for Markov-modulated and Reflected Diffusion Processes by :
Download or read book Limit Theorems for Markov-modulated and Reflected Diffusion Processes written by and published by . This book was released on 2015 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt: "In this thesis, asymptotic properties of two variants of one-dimensional diffusion processes, which are Markov-modulated and reflected Ornstein-Uhlenbeck processes, are studied. Besides the random term of the Brownian motion, the Markov-modulated diffusion process evolves in an extra random environment, namely the finite-state Markov chain. The reflected Ornstein-Uhlenbeck process behaves as an Ornstein-Uhlenbeck process which has instantaneous reflection at boundaries. They are widely used in modeling due to above mentioned their distinctive features and great analytical tractability. We obtain four limit theorems from the perspectives of weak convergence and large deviations. Firstly, we prove weak convergence of a sequence of Markov-modulated diffusion processes with rapid switching to an ordinary diffusion process by verifying its tightness property. Secondly, a sample-path large deviations principle for the coupling of the Markov-modulated diffusion process with small noise and the occupation measure of the rapid switching Markov chain is obtained. The large deviations principles for each individual term are derived by the contraction principle. Those results reveal interesting behavior of Markov-modulated diffusion process when the modulating Markov chain switches fast. Thirdly, transient asymptotics of large deviations type are acquired for reflected and doubly reflected Ornstein-Uhlenbeck processes. Fourthly, we prove central limit theorems and functional central limit theorems for the centered and scaled loss and idle processes of doubly reflected Ornstein-Uhlenbeck processes."--Samenvatting auteur.
Book Synopsis Large Deviations of Specific Empirical Fluxes of Independent Markov Chains, with Implications for Macroscopic Fluctuation Theory by : D. R. Michiel Renger
Download or read book Large Deviations of Specific Empirical Fluxes of Independent Markov Chains, with Implications for Macroscopic Fluctuation Theory written by D. R. Michiel Renger and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider a system of independent particles on a finite state space, and prove a dynamic large-deviation principle for the empirical measure-empirical flux pair, taking the specific fluxes rather than net fluxes into account. We prove the large deviations under deterministic initial conditions, and under random initial conditions satisfying a large-deviation principle. We then show how to use this result to generalise a number of principles from Macroscopic Fluctuation Theory to the finite-space setting.
Book Synopsis Analysis and Geometry of Markov Diffusion Operators by : Dominique Bakry
Download or read book Analysis and Geometry of Markov Diffusion Operators written by Dominique Bakry and published by Springer Science & Business Media. This book was released on 2013-11-18 with total page 555 pages. Available in PDF, EPUB and Kindle. Book excerpt: The present volume is an extensive monograph on the analytic and geometric aspects of Markov diffusion operators. It focuses on the geometric curvature properties of the underlying structure in order to study convergence to equilibrium, spectral bounds, functional inequalities such as Poincaré, Sobolev or logarithmic Sobolev inequalities, and various bounds on solutions of evolution equations. At the same time, it covers a large class of evolution and partial differential equations. The book is intended to serve as an introduction to the subject and to be accessible for beginning and advanced scientists and non-specialists. Simultaneously, it covers a wide range of results and techniques from the early developments in the mid-eighties to the latest achievements. As such, students and researchers interested in the modern aspects of Markov diffusion operators and semigroups and their connections to analytic functional inequalities, probabilistic convergence to equilibrium and geometric curvature will find it especially useful. Selected chapters can also be used for advanced courses on the topic.
Book Synopsis Diffusions, Markov Processes, and Martingales: Volume 1, Foundations by : L. C. G. Rogers
Download or read book Diffusions, Markov Processes, and Martingales: Volume 1, Foundations written by L. C. G. Rogers and published by Cambridge University Press. This book was released on 2000-04-13 with total page 412 pages. Available in PDF, EPUB and Kindle. Book excerpt: Now available in paperback for the first time; essential reading for all students of probability theory.
Book Synopsis From Large Deviations to Wasserstein Gradient Flows in Multiple Dimensions by : Matthias Erbar
Download or read book From Large Deviations to Wasserstein Gradient Flows in Multiple Dimensions written by Matthias Erbar and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We study the large deviation rate functional for the empirical measure of independent Brownian particles with drift. In one dimension, it has been shown by Adams, Dirr, Peletier and Zimmer [ADPZ11] that this functional is asymptotically equivalent (in the sense of -convergence) to the Jordan-Kinderlehrer-Otto functional arising in the Wasserstein gradient flow structure of the Fokker-Planck equation. In higher dimensions, part of this statement (the lower bound) has been recently proved by Duong, Laschos and Renger, but the upper bound remained open, since the proof in [DLR13] relies on regularity properties of optimal transport maps that are restricted to one dimension. In this note we present a new proof of the upper bound, thereby generalising the result of [ADPZ11] to arbitrary dimensions.
Book Synopsis Generalized Diffusion Processes by : Nikola_ Ivanovich Portenko
Download or read book Generalized Diffusion Processes written by Nikola_ Ivanovich Portenko and published by American Mathematical Soc.. This book was released on 1990-12-21 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: Diffusion processes serve as a mathematical model for the physical phenomenon of diffusion. One of the most important problems in the theory of diffusion processes is the development of methods for constructing these processes from a given diffusion matrix and a given drift vector. Focusing on the investigation of this problem, this book is intended for specialists in the theory of random processes and its applications. A generalized diffusion process (that is, a continuous Markov process for which the Kolmogorov local characteristics exist in the generalized sense) can serve as a model for diffusion in a medium moving in a nonregular way. The author constructs generalized diffusion processes under two assumptions: first, that the diffusion matrix is sufficiently regular; and second, that the drift vector is a function integrable to some power, or is a generalized function of the type of the derivative of a measure.
Book Synopsis Gradient Flows Vs. Large Deviations by : Kaveh Bashiri
Download or read book Gradient Flows Vs. Large Deviations written by Kaveh Bashiri and published by . This book was released on 2015 with total page 89 pages. Available in PDF, EPUB and Kindle. Book excerpt:
