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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
This monograph presents a rigorous mathematical introduction to
optimal transport as a variational problem, its use in modeling
various phenomena, and its connections with partial differential
equations. Its main goal is to provide the reader with the
techniques necessary to understand the current research in optimal
transport and the tools which are most useful for its applications.
Full proofs are used to illustrate mathematical concepts and each
chapter includes a section that discusses applications of optimal
transport to various areas, such as economics, finance, potential
games, image processing and fluid dynamics. Several topics are
covered that have never been previously in books on this subject,
such as the Knothe transport, the properties of functionals on
measures, the Dacorogna-Moser flow, the formulation through minimal
flows with prescribed divergence formulation, the case of the
supremal cost, and the most classical numerical methods. Graduate
students and researchers in both pure and applied mathematics
interested in the problems and applications of optimal transport
will find this to be an invaluable resource.
This volume provides an introduction to the theory of Mean Field
Games, suggested by J.-M. Lasry and P.-L. Lions in 2006 as a
mean-field model for Nash equilibria in the strategic interaction
of a large number of agents. Besides giving an accessible
presentation of the main features of mean-field game theory, the
volume offers an overview of recent developments which explore
several important directions: from partial differential equations
to stochastic analysis, from the calculus of variations to modeling
and aspects related to numerical methods. Arising from the CIME
Summer School "Mean Field Games" held in Cetraro in 2019, this book
collects together lecture notes prepared by Y. Achdou (with M.
Lauriere), P. Cardaliaguet, F. Delarue, A. Porretta and F.
Santambrogio. These notes will be valuable for researchers and
advanced graduate students who wish to approach this theory and
explore its connections with several different fields in
mathematics.
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