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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 book takes readers on a tour through modern methods in image
analysis and reconstruction based on level set and PDE techniques,
the major focus being on morphological and geometric structures in
images. The aspects covered include edge-sharpening image
reconstruction and denoising, segmentation and shape analysis in
images, and image matching. For each, the lecture notes provide
insights into the basic analysis of modern variational and
PDE-based techniques, as well as computational aspects and
applications.
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