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Topological Optimization And Optimal Transport

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Topological Optimization And Optimal Transport

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Author : Maïtine Bergounioux
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2017-08-07

Topological Optimization And Optimal Transport written by Maïtine Bergounioux and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-08-07 with Mathematics categories.

By discussing topics such as shape representations, relaxation theory and optimal transport, trends and synergies of mathematical tools required for optimization of geometry and topology of shapes are explored. Furthermore, applications in science and engineering, including economics, social sciences, biology, physics and image processing are covered. Contents Part I Geometric issues in PDE problems related to the infinity Laplace operator Solution of free boundary problems in the presence of geometric uncertainties Distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies High-order topological expansions for Helmholtz problems in 2D On a new phase field model for the approximation of interfacial energies of multiphase systems Optimization of eigenvalues and eigenmodes by using the adjoint method Discrete varifolds and surface approximation Part II Weak Monge–Ampere solutions of the semi-discrete optimal transportation problem Optimal transportation theory with repulsive costs Wardrop equilibria: long-term variant, degenerate anisotropic PDEs and numerical approximations On the Lagrangian branched transport model and the equivalence with its Eulerian formulation On some nonlinear evolution systems which are perturbations of Wasserstein gradient flows Pressureless Euler equations with maximal density constraint: a time-splitting scheme Convergence of a fully discrete variational scheme for a thin-film equatio Interpretation of finite volume discretization schemes for the Fokker–Planck equation as gradient flows for the discrete Wasserstein distance

Topological Optimization And Optimal Transport

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Author : Maïtine Bergounioux
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2017-08-07

Optimal Transport

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Author : Gero Friesecke
language : en
Publisher: SIAM
Release Date : 2024-12-21

Optimal Transport written by Gero Friesecke and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-12-21 with Mathematics categories.

Historically, optimal transport was about moving a pile of mortar efficiently or transferring the output of an array of steel mines optimally. This type of problem has been found to arise in many different fields of mathematics, science, and engineering—from fluid dynamics to many-electron physics to artificial intelligence—and in the last three decades interest in the subject has exploded. This accessible book begins with an elementary and self-contained chapter on optimal transport on finite state spaces that does not require measure theory or functional analysis. It builds up mathematical theory rigorously and from scratch, aided by intuitive arguments, informal discussion, and carefully selected applications. It is the first book to cover modern topics such as Wasserstein GANs and multimarginal problems and includes a discussion of numerical methods and basic MATLAB code for simulating optimal transport problems directly via linear programming or more efficiently via the Sinkhorn algorithm. Additionally, it provides classroom-tested exercises in every chapter. This book is for advanced undergraduate students, beginning graduate students, and researchers in applied mathematics. It will also be of interest to students and researchers in physics, engineering, computer science, data science, and machine learning who want to become familiar with cornerstone concepts, results, and methods. Optimal Transport: A Comprehensive Introduction to Modeling, Analysis, Simulation, Applications is appropriate for a special topics course on optimal transport. It can also be used as a supplementary text for a general course on linear optimization, convex analysis, calculus of variations, or mathematical methods in data science.

Optimal Transportation Networks

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Author : Marc Bernot
language : en
Publisher: Springer Science & Business Media
Release Date : 2009

Optimal Transportation Networks written by Marc Bernot and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Business & Economics categories.

The transportation problem can be formalized as the problem of finding the optimal way to transport a given measure into another with the same mass. In contrast to the Monge-Kantorovitch problem, recent approaches model the branched structure of such supply networks as minima of an energy functional whose essential feature is to favour wide roads. Such a branched structure is observable in ground transportation networks, in draining and irrigation systems, in electrical power supply systems and in natural counterparts such as blood vessels or the branches of trees. These lectures provide mathematical proof of several existence, structure and regularity properties empirically observed in transportation networks. The link with previous discrete physical models of irrigation and erosion models in geomorphology and with discrete telecommunication and transportation models is discussed. It will be mathematically proven that the majority fit in the simple model sketched in this volume.

Density Functional Theory

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Author : Eric Cancès
language : en
Publisher: Springer Nature
Release Date : 2023-07-18

Density Functional Theory written by Eric Cancès and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-07-18 with Mathematics categories.

Density functional theory (DFT) provides the most widely used models for simulating molecules and materials based on the fundamental laws of quantum mechanics. It plays a central role in a huge spectrum of applications in chemistry, physics, and materials science.Quantum mechanics describes a system of N interacting particles in the physical 3-dimensional space by a partial differential equation in 3N spatial variables. The standard numerical methods thus incur an exponential increase of computational effort with N, a phenomenon known as the curse of dimensionality; in practice these methods already fail beyond N=2. DFT overcomes this problem by 1) reformulating the N-body problem involving functions of 3N variables in terms of the density, a function of 3 variables, 2) approximating it by a pioneering hybrid approach which keeps important ab initio contributions and re-models the remainder in a data-driven way. This book intends to be an accessible, yet state-of-art text on DFT for graduate students and researchers in applied and computational mathematics, physics, chemistry, and materials science. It introduces and reviews the main models of DFT, covering their derivation and mathematical properties, numerical treatment, and applications.

Optimal Structural Design

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Author : Nikolay V. Banichuk
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2017-09-11

Optimal Structural Design written by Nikolay V. Banichuk and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-09-11 with Mathematics categories.

This monograph studies optimization problems for rigid punches in elastic media and for high-speed penetration of rigid strikers into deformed elastoplastic, concrete, and composite media using variational calculations, tools from functional analysis, and stochastic and min-max (guaranteed) optimization approaches with incomplete data. The book presents analytical and numerical results developed by the authors during the last ten years.

Convex And Set Valued Analysis

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Author : Aram V. Arutyunov
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2016-12-05

Convex And Set Valued Analysis written by Aram V. Arutyunov and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-12-05 with Mathematics categories.

This textbook is devoted to a compressed and self-contained exposition of two important parts of contemporary mathematics: convex and set-valued analysis. In the first part, properties of convex sets, the theory of separation, convex functions and their differentiability, properties of convex cones in finite- and infinite-dimensional spaces are discussed. The second part covers some important parts of set-valued analysis. There the properties of the Hausdorff metric and various continuity concepts of set-valued maps are considered. The great attention is paid also to measurable set-valued functions, continuous, Lipschitz and some special types of selections, fixed point and coincidence theorems, covering set-valued maps, topological degree theory and differential inclusions. Contents: Preface Part I: Convex analysis Convex sets and their properties The convex hull of a set. The interior of convex sets The affine hull of sets. The relative interior of convex sets Separation theorems for convex sets Convex functions Closedness, boundedness, continuity, and Lipschitz property of convex functions Conjugate functions Support functions Differentiability of convex functions and the subdifferential Convex cones A little more about convex cones in infinite-dimensional spaces A problem of linear programming More about convex sets and convex hulls Part II: Set-valued analysis Introduction to the theory of topological and metric spaces The Hausdorff metric and the distance between sets Some fine properties of the Hausdorff metric Set-valued maps. Upper semicontinuous and lower semicontinuous set-valued maps A base of topology of the spaceHc(X) Measurable set-valued maps. Measurable selections and measurable choice theorems The superposition set-valued operator The Michael theorem and continuous selections. Lipschitz selections. Single-valued approximations Special selections of set-valued maps Differential inclusions Fixed points and coincidences of maps in metric spaces Stability of coincidence points and properties of covering maps Topological degree and fixed points of set-valued maps in Banach spaces Existence results for differential inclusions via the fixed point method Notation Bibliography Index

Geometric Partial Differential Equations Part 2

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Author : Andrea Bonito
language : en
Publisher: Elsevier
Release Date : 2021-01-26

Geometric Partial Differential Equations Part 2 written by Andrea Bonito and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-01-26 with Mathematics categories.

Besides their intrinsic mathematical interest, geometric partial differential equations (PDEs) are ubiquitous in many scientific, engineering and industrial applications. They represent an intellectual challenge and have received a great deal of attention recently. The purpose of this volume is to provide a missing reference consisting of self-contained and comprehensive presentations. It includes basic ideas, analysis and applications of state-of-the-art fundamental algorithms for the approximation of geometric PDEs together with their impacts in a variety of fields within mathematics, science, and engineering. - About every aspect of computational geometric PDEs is discussed in this and a companion volume. Topics in this volume include stationary and time-dependent surface PDEs for geometric flows, large deformations of nonlinearly geometric plates and rods, level set and phase field methods and applications, free boundary problems, discrete Riemannian calculus and morphing, fully nonlinear PDEs including Monge-Ampere equations, and PDE constrained optimization - Each chapter is a complete essay at the research level but accessible to junior researchers and students. The intent is to provide a comprehensive description of algorithms and their analysis for a specific geometric PDE class, starting from basic concepts and concluding with interesting applications. Each chapter is thus useful as an introduction to a research area as well as a teaching resource, and provides numerous pointers to the literature for further reading - The authors of each chapter are world leaders in their field of expertise and skillful writers. This book is thus meant to provide an invaluable, readable and enjoyable account of computational geometric PDEs

Topology Optimization Theory For Laminar Flow

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Author : Yongbo Deng
language : en
Publisher: Springer
Release Date : 2017-09-27

Topology Optimization Theory For Laminar Flow written by Yongbo Deng and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-09-27 with Technology & Engineering categories.

This book presents the topology optimization theory for laminar flows with low and moderate Reynolds numbers, based on the density method and level-set method, respectively. The density-method-based theory offers efficient convergence, while the level-set-method-based theory can provide anaccurate mathematical expression of the structural boundary. Unsteady, body-force-driven and two-phase properties are basic characteristics of the laminar flows. The book discusses these properties, which are typical of microfluidics and one of the research hotspots in the area of Micro-Electro-Mechanical Systems (MEMS), providing an efficient inverse design approach for microfluidic structures. To demonstrate the applications of this topology optimization theory in the context of microfluidics, it also investigates inverse design for the micromixer, microvalve and micropump, which are key elements in lab-on-chip devices.

Non Smooth And Complementarity Based Distributed Parameter Systems

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Author : Michael Hintermüller
language : en
Publisher: Springer Nature
Release Date : 2022-02-18

Non Smooth And Complementarity Based Distributed Parameter Systems written by Michael Hintermüller and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-02-18 with Mathematics categories.

Many of the most challenging problems in the applied sciences involve non-differentiable structures as well as partial differential operators, thus leading to non-smooth distributed parameter systems. This edited volume aims to establish a theoretical and numerical foundation and develop new algorithmic paradigms for the treatment of non-smooth phenomena and associated parameter influences. Other goals include the realization and further advancement of these concepts in the context of robust and hierarchical optimization, partial differential games, and nonlinear partial differential complementarity problems, as well as their validation in the context of complex applications. Areas for which applications are considered include optimal control of multiphase fluids and of superconductors, image processing, thermoforming, and the formation of rivers and networks. Chapters are written by leading researchers and present results obtained in the first funding phase of the DFG Special Priority Program on Nonsmooth and Complementarity Based Distributed Parameter Systems: Simulation and Hierarchical Optimization that ran from 2016 to 2019.

Topological Optimization And Optimal Transport

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