Download or read book Differential Equations Methods for the Monge Kantorovich Mass Transfer Problem written by Lawrence C. Evans and published by American Mathematical Soc.. This book was released on 1999 with total page 81 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this volume, the authors demonstrate under some assumptions on $f $, $f $ that a solution to the classical Monge-Kantorovich problem of optimally rearranging the measure $\mu{ }=f dx$ onto $\mu =f dy$ can be constructed by studying the $p$-Laplacian equation $- \roman{div}(\vert DU_p\vert p-2}Du_p)=f -f $ in the limit as $p\rightarrow\infty$. The idea is to show $u_p\rightarrow u$, where $u$ satisfies $\vert Du\vert\leq 1, -\roman{div}(aDu)=f -f $ for some density $a\geq0$, and then to build a flow by solving a nonautonomous ODE involving $a, Du, f $ and $f $
Download or read book Gradient Flows written by Luigi Ambrosio and published by Springer Science & Business Media. This book was released on 2006-03-30 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to a theory of gradient ?ows in spaces which are not nec- sarily endowed with a natural linear or di?erentiable structure. It is made of two parts, the ?rst one concerning gradient ?ows in metric spaces and the second one 2 1 devoted to gradient ?ows in the L -Wasserstein space of probability measures on p a separable Hilbert space X (we consider the L -Wasserstein distance, p? (1,?), as well). The two parts have some connections, due to the fact that the Wasserstein space of probability measures provides an important model to which the “metric” theory applies, but the book is conceived in such a way that the two parts can be read independently, the ?rst one by the reader more interested to Non-Smooth Analysis and Analysis in Metric Spaces, and the second one by the reader more oriented to theapplications in Partial Di?erential Equations, Measure Theory and Probability.
Download or read book Advanced Courses Of Mathematical Analysis V Proceedings Of The Fifth International School written by Juan Carlos Navarro Pascual and published by World Scientific. This book was released on 2016-06-24 with total page 319 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains recent papers by several specialists in different fields of mathematical analysis. It offers a reasonably wide perspective of the current state of research, and new trends, in areas related to measure theory, harmonic analysis, non-associative structures in functional analysis and summability in locally convex spaces.Those interested in researching any areas of mathematical analysis will find here numerous suggestions on possible topics with an important impact today. Often, the contributions are presented in an expository nature and this makes the discussed topics accessible to a more general audience.
Download or read book Game Theory and Partial Differential Equations written by Pablo Blanc and published by Walter de Gruyter GmbH & Co KG. This book was released on 2019-07-22 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt: Extending the well-known connection between classical linear potential theory and probability theory (through the interplay between harmonic functions and martingales) to the nonlinear case of tug-of-war games and their related partial differential equations, this unique book collects several results in this direction and puts them in an elementary perspective in a lucid and self-contained fashion.
Download or read book Optimal Transportation Networks written by Marc Bernot and published by Springer. This book was released on 2008-10-23 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides mathematical proof of several existence, structure and regularity properties, empirically observed in transportation networks.
Download or read book Stochastic Optimal Transportation written by Toshio Mikami and published by Springer Nature. This book was released on 2021-06-15 with total page 129 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, the optimal transportation problem (OT) is described as a variational problem for absolutely continuous stochastic processes with fixed initial and terminal distributions. Also described is Schrödinger’s problem, which is originally a variational problem for one-step random walks with fixed initial and terminal distributions. The stochastic optimal transportation problem (SOT) is then introduced as a generalization of the OT, i.e., as a variational problem for semimartingales with fixed initial and terminal distributions. An interpretation of the SOT is also stated as a generalization of Schrödinger’s problem. After the brief introduction above, the fundamental results on the SOT are described: duality theorem, a sufficient condition for the problem to be finite, forward–backward stochastic differential equations (SDE) for the minimizer, and so on. The recent development of the superposition principle plays a crucial role in the SOT. A systematic method is introduced to consider two problems: one with fixed initial and terminal distributions and one with fixed marginal distributions for all times. By the zero-noise limit of the SOT, the probabilistic proofs to Monge’s problem with a quadratic cost and the duality theorem for the OT are described. Also described are the Lipschitz continuity and the semiconcavity of Schrödinger’s problem in marginal distributions and random variables with given marginals, respectively. As well, there is an explanation of the regularity result for the solution to Schrödinger’s functional equation when the space of Borel probability measures is endowed with a strong or a weak topology, and it is shown that Schrödinger’s problem can be considered a class of mean field games. The construction of stochastic processes with given marginals, called the marginal problem for stochastic processes, is discussed as an application of the SOT and the OT.
Download or read book Mathematical Analysis Approximation Theory and Their Applications written by Themistocles M. Rassias and published by Springer. This book was released on 2016-06-03 with total page 745 pages. Available in PDF, EPUB and Kindle. Book excerpt: Designed for graduate students, researchers, and engineers in mathematics, optimization, and economics, this self-contained volume presents theory, methods, and applications in mathematical analysis and approximation theory. Specific topics include: approximation of functions by linear positive operators with applications to computer aided geometric design, numerical analysis, optimization theory, and solutions of differential equations. Recent and significant developments in approximation theory, special functions and q-calculus along with their applications to mathematics, engineering, and social sciences are discussed and analyzed. Each chapter enriches the understanding of current research problems and theories in pure and applied research.
Download or read book Second Order Analysis on mathscr P 2 M W 2 written by Nicola Gigli and published by American Mathematical Soc.. This book was released on 2012-02-22 with total page 173 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author develops a rigorous second order analysis on the space of probability measures on a Riemannian manifold endowed with the quadratic optimal transport distance $W_2$. The discussion includes: definition of covariant derivative, discussion of the problem of existence of parallel transport, calculus of the Riemannian curvature tensor, differentiability of the exponential map and existence of Jacobi fields. This approach does not require any smoothness assumption on the measures considered.
Download or read book Asymptotics for Solutions of Linear Differential Equations Having Turning Points with Applications written by Shlomo Strelitz and published by American Mathematical Soc.. This book was released on 1999 with total page 105 pages. Available in PDF, EPUB and Kindle. Book excerpt: Asymptotics are built for the solutions $y_j(x, \lambda)$, $y_j DEGREES{(k)}(0, \lambda)=\delta_{j\, n-k}$, $0\le j, k+1\le n$ of the equation $L(y)=\lambda p(x)y, \quad x\in [0,1], $ where $L(y)$ is a linear differential operator of whatever order $n\ge 2$ and $p(x)$ is assumed to possess a finite number of turning points. The established asymptotics are afterwards applied to the study of: 1) the existence of infinite eigenvalue sequences for various multipoint boundary problems posed on $L(y)=\lambda p(x)y, \quad x\in [0,1], $, especially as $n=2$ and $n=3$ (let us be aware that the same method can be successfully applied on many occasions in case $n>3$ too) and 2) asymptotical distribution of the corresponding eigenvalue sequences on the
Download or read book Topics in Optimal Transportation written by Cédric Villani and published by American Mathematical Soc.. This book was released on 2021-08-25 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first comprehensive introduction to the theory of mass transportation with its many—and sometimes unexpected—applications. In a novel approach to the subject, the book both surveys the topic and includes a chapter of problems, making it a particularly useful graduate textbook. In 1781, Gaspard Monge defined the problem of “optimal transportation” (or the transferring of mass with the least possible amount of work), with applications to engineering in mind. In 1942, Leonid Kantorovich applied the newborn machinery of linear programming to Monge's problem, with applications to economics in mind. In 1987, Yann Brenier used optimal transportation to prove a new projection theorem on the set of measure preserving maps, with applications to fluid mechanics in mind. Each of these contributions marked the beginning of a whole mathematical theory, with many unexpected ramifications. Nowadays, the Monge-Kantorovich problem is used and studied by researchers from extremely diverse horizons, including probability theory, functional analysis, isoperimetry, partial differential equations, and even meteorology. Originating from a graduate course, the present volume is intended for graduate students and researchers, covering both theory and applications. Readers are only assumed to be familiar with the basics of measure theory and functional analysis.
Download or read book Optimal Transport written by Yann Ollivier and published by Cambridge University Press. This book was released on 2014-08-07 with total page 317 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of optimal transportation has its origins in the eighteenth century when the problem of transporting resources at a minimal cost was first formalised. Through subsequent developments, particularly in recent decades, it has become a powerful modern theory. This book contains the proceedings of the summer school 'Optimal Transportation: Theory and Applications' held at the Fourier Institute in Grenoble. The event brought together mathematicians from pure and applied mathematics, astrophysics, economics and computer science. Part I of this book is devoted to introductory lecture notes accessible to graduate students, while Part II contains research papers. Together, they represent a valuable resource on both fundamental and advanced aspects of optimal transportation, its applications, and its interactions with analysis, geometry, PDE and probability, urban planning and economics. Topics covered include Ricci flow, the Euler equations, functional inequalities, curvature-dimension conditions, and traffic congestion.
Download or read book On Sudakov s Type Decomposition of Transference Plans with Norm Costs written by Stefano Bianchini and published by American Mathematical Soc.. This book was released on 2018-02-23 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm cost with , probability measures in and absolutely continuous w.r.t. . The key idea in this approach is to decompose (via disintegration of measures) the Kantorovich optimal transportation problem into a family of transportation problems in , where are disjoint regions such that the construction of an optimal map is simpler than in the original problem, and then to obtain by piecing together the maps . When the norm is strictly convex, the sets are a family of -dimensional segments determined by the Kantorovich potential called optimal rays, while the existence of the map is straightforward provided one can show that the disintegration of (and thus of ) on such segments is absolutely continuous w.r.t. the -dimensional Hausdorff measure. When the norm is not strictly convex, the main problems in this kind of approach are two: first, to identify a suitable family of regions on which the transport problem decomposes into simpler ones, and then to prove the existence of optimal maps. In this paper the authors show how these difficulties can be overcome, and that the original idea of Sudakov can be successfully implemented. The results yield a complete characterization of the Kantorovich optimal transportation problem, whose straightforward corollary is the solution of the Monge problem in each set and then in . The strategy is sufficiently powerful to be applied to other optimal transportation problems.
Download or read book Transport Equations and Multi D Hyperbolic Conservation Laws written by Luigi Ambrosio and published by Springer Science & Business Media. This book was released on 2008-02-17 with total page 141 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of nonlinear hyperbolic equations in several space dimensions has recently obtained remarkable achievements. This volume provides an up-to-date overview of the status and perspectives of two areas of research in PDEs, related to hyperbolic conservation laws. The captivating volume contains surveys of recent deep results and provides an overview of further developments and related open problems. Readers should have basic knowledge of PDE and measure theory.
Download or read book Nonlinear Conservation Laws and Applications written by Alberto Bressan and published by Springer Science & Business Media. This book was released on 2011-04-19 with total page 487 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the Summer Program on Nonlinear Conservation Laws and Applications held at the IMA on July 13--31, 2009. Hyperbolic conservation laws is a classical subject, which has experienced vigorous growth in recent years. The present collection provides a timely survey of the state of the art in this exciting field, and a comprehensive outlook on open problems. Contributions of more theoretical nature cover the following topics: global existence and uniqueness theory of one-dimensional systems, multidimensional conservation laws in several space variables and approximations of their solutions, mathematical analysis of fluid motion, stability and dynamics of viscous shock waves, singular limits for viscous systems, basic principles in the modeling of turbulent mixing, transonic flows past an obstacle and a fluid dynamic approach for isometric embedding in geometry, models of nonlinear elasticity, the Monge problem, and transport equations with rough coefficients. In addition, there are a number of papers devoted to applications. These include: models of blood flow, self-gravitating compressible fluids, granular flow, charge transport in fluids, and the modeling and control of traffic flow on networks.
Download or read book Data driven Models in Inverse Problems written by Tatiana A. Bubba and published by Walter de Gruyter GmbH & Co KG. This book was released on 2024-11-18 with total page 508 pages. Available in PDF, EPUB and Kindle. Book excerpt: Advances in learning-based methods are revolutionizing several fields in applied mathematics, including inverse problems, resulting in a major paradigm shift towards data-driven approaches. This volume, which is inspired by this cutting-edge area of research, brings together contributors from the inverse problem community and shows how to successfully combine model- and data-driven approaches to gain insight into practical and theoretical issues.
Download or read book Needle Decompositions in Riemannian Geometry written by Bo’az Klartag and published by American Mathematical Soc.. This book was released on 2017-09-25 with total page 90 pages. Available in PDF, EPUB and Kindle. Book excerpt: The localization technique from convex geometry is generalized to the setting of Riemannian manifolds whose Ricci curvature is bounded from below. In a nutshell, the author's method is based on the following observation: When the Ricci curvature is non-negative, log-concave measures are obtained when conditioning the Riemannian volume measure with respect to a geodesic foliation that is orthogonal to the level sets of a Lipschitz function. The Monge mass transfer problem plays an important role in the author's analysis.
Download or read book The Riemann Problem for the Transportation Equations in Gas Dynamics written by Wancheng Sheng and published by American Mathematical Soc.. This book was released on 1999 with total page 93 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this volume, the one-dimensional and two-dimensional Riemann problems for the transportation equations in gas dynamics are solved constructively. In either the 1-D or 2-D case, there are only two kinds of solutions: one involves Dirac delta waves, and the other involves vacuums, which has been merely discussed so far. The generalized Rankine-Hugoniot and entropy conditions for Dirac delta waves are clarified with viscous vanishing method. All of the existence, uniqueness and stability for viscous perturbations are proved analytically
