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Topological Optimization and Optimal Transport

Book Description

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

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Contents
  5. Part I
    1. Geometric issues in PDE problems related to the infinity Laplace operator
      1. 1.1 Introduction
      2. 1.2 On the Dirichlet problem
      3. 1.3 On the overdetermined problem: the simple (web) case
      4. 1.4 On stadium-like domains
      5. 1.5 On the overdetermined problem: the general (non-web) case
      6. 1.6 Open problems
    2. Solution of free boundary problems in the presence of geometric uncertainties
      1. 2.1 Introduction
      2. 2.2 Modelling uncertain domains
        1. 2.2.1 Notation
        2. 2.2.2 Random interior boundary
        3. 2.2.3 Random exterior boundary
        4. 2.2.4 Expectation and variance of the domain
        5. 2.2.5 Stochastic quadrature method
        6. 2.2.6 Analytical example
      3. 2.3 Computing free boundaries
        1. 2.3.1 Trial method
        2. 2.3.2 Discretizing the free boundary
        3. 2.3.3 Boundary integral equations
        4. 2.3.4 Expectation and variance of the potential
      4. 2.4 Numerical results
        1. 2.4.1 First example
        2. 2.4.2 Second example
        3. 2.4.3 Third example
        4. 2.4.4 Fourth example
      5. 2.5 Conclusion
    3. Distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies
      1. 3.1 Introduction
      2. 3.2 Optimal control problem for the time discretization
      3. 3.3 Yosida approximation and gradient method
        1. 3.3.1 Sequential Yosida approximation
        2. 3.3.2 Steepest descent method with expansive line search
        3. 3.3.3 Newton’s method for the primal system
      4. 3.4 Finite element approximation
      5. 3.5 Numerical results
        1. 3.5.1 Disk to a ring segment
        2. 3.5.2 Ring to disks
        3. 3.5.3 Grid pattern of disks
        4. 3.5.4 Grid pattern of finger-like regions
      6. 3.6 Conclusions
    4. High-order topological expansions for Helmholtz problems in 2D
      1. 4.1 Introduction
      2. 4.2 Background Helmholtz problem
        1. 4.2.1 Inner asymptotic expansion by Fourier series in near field
      3. 4.3 Helmholtz problems for geometric objects under Neumann (sound hard) boundary condition
        1. 4.3.1 Outer asymptotic expansion by Fourier series in far field
        2. 4.3.2 Uniform asymptotic expansion of solution of the Neumann problem
        3. 4.3.3 Inverse Helmholtz problem under Neumann boundary condition
      4. 4.4 Helmholtz problems for geometric objects under Dirichlet (sound soft) boundary condition
        1. 4.4.1 Outer and inner asymptotic expansions by Fourier series
        2. 4.4.2 High-order uniform asymptotic expansion of the Dirichlet problem
        3. 4.4.3 Inverse Helmholtz problem under Dirichlet boundary condition
      5. 4.5 Helmholtz problems for geometric objects under Robin (impedance) boundary condition
        1. 4.5.1 Outer asymptotic expansion by Fourier series in far field
        2. 4.5.2 Combined uniform asymptotic expansion of the Robin problem
        3. 4.5.3 Inverse Helmholtz problem under Robin boundary condition
        4. 4.5.4 Necessary optimality condition for the topology optimization
    5. On a new phase field model for the approximation of interfacial energies of multiphase systems
      1. 5.1 Introduction
      2. 5.2 Derivation of the phase-field model
        1. 5.2.1 The classical constant case σi,j = 1
        2. 5.2.2 Additive surface tensions
        3. 5.2.3 ℓ1-embeddable surface tensions
        4. 5.2.4 Derivation of the approximation perimeter for ℓ1-embeddable surface tensions
      3. 5.3 Convergence of the approximating multiphase perimeter
      4. 5.4 L2-gradient flow and some extensions
        1. 5.4.1 Additional volume constraints
        2. 5.4.2 Application to the wetting of multiphase droplets on solid surfaces
      5. 5.5 Numerical experiments
        1. 5.5.1 Evolution of partitions
        2. 5.5.2 Wetting of multiphase droplets on solid surfaces
    6. Optimization of eigenvalues and eigenmodes by using the adjoint method
      1. 6.1 Introduction
      2. 6.2 Setting of the problem and the objective functionals
      3. 6.3 The derivatives of the eigenvalues and eigenmodes of vibration
      4. 6.4 The derivative of the objective functional by the adjoint method
      5. 6.5 Multiple eigenvalues
      6. 6.6 The adjoint method in the framework of Bloch waves
    7. Discrete varifolds and surface approximation
      1. 7.1 Introduction
      2. 7.2 Varifolds
      3. 7.3 Discrete varifolds
      4. 7.4 Approximation of rectifiable varifolds by discrete varifolds
      5. 7.5 Curvature of a varifold: a new convolution approach
        1. 7.5.1 Regularization of the first variation and conditions of bounded first variation
        2. 7.5.2 ϵ-Approximation of the mean curvature vector
      6. 7.6 Mean curvature of point-cloud varifolds
  6. Part II
    1. Weak Monge–Ampère solutions of the semi-discrete optimal transportation problem
      1. 8.1 Introduction
      2. 8.2 Duality of Aleksandrov and Pogorelov solutions
        1. 8.2.1 Properties of the Legendre–Fenchel dual
        2. 8.2.2 Aleksandrov solutions
        3. 8.2.3 Geometric characterisation
      3. 8.3 Mixed Aleksandrov–viscosity formulation
        1. 8.3.1 Viscosity solutions
        2. 8.3.2 Mixed Aleksandrov–viscosity solutions
        3. 8.3.3 Characterisation of subgradient measure
      4. 8.4 Approximation scheme
        1. 8.4.1 Monge–Ampère operator and boundary conditions
        2. 8.4.2 Discretization of subgradient measure
        3. 8.4.3 Properties of the approximation scheme
      5. 8.5 Numerical results
        1. 8.5.1 Comparison to viscosity solver
        2. 8.5.2 Comparison to exact solver
        3. 8.5.3 Multiple Diracs
      6. 8.6 Conclusions
    2. A Convexity
    3. B Discretization of Monge–Ampère
    4. C Extension to non-constant densities
    5. Optimal transportation theory with repulsive costs
      1. 9.1 Why multi-marginal transport theory for repulsive costs?
        1. 9.1.1 Brief introduction to quantum mechanics of N-body systems
        2. 9.1.2 Probabilistic Interpretation and Marginals
        3. 9.1.3 Density functional theory (DFT)
        4. 9.1.4 ‘Semi-classical limit’ and optimal transport problem
      2. 9.2 DFT meets optimal transportation theory
        1. 9.2.1 Couplings and multi-marginal optimal transportation problem
        2. 9.2.2 Multi-marginal optimal transportation problem
        3. 9.2.3 Dual formulation
        4. 9.2.4 Geometry of the Optimal Transport sets
        5. 9.2.5 Symmetric Case
      3. 9.3 Multi-marginal OT with Coulomb cost
        1. 9.3.1 General theory: duality, equivalent formulations, and many particles limit
        2. 9.3.2 The Monge problem: deterministic examples and counterexamples
      4. 9.4 Multi-marginal OT with repulsive harmonic cost
      5. 9.5 Multi-marginal OT for the determinant
      6. 9.6 Numerics
        1. 9.6.1 The regularized problem and the iterative proportional fitting procedure
        2. 9.6.2 Numerical experiments: Coulomb cost
        3. 9.6.3 Numerical experiments: repulsive Harmonic cost
        4. 9.6.4 Numerical experiments: Determinant cost
      7. 9.7 Conclusion
    6. Wardrop equilibria: long-term variant, degenerate anisotropic PDEs and numerical approximations
      1. 10.1 Introduction
        1. 10.1.1 Presentation of the general discrete model
        2. 10.1.2 Assumptions and preliminary results
      2. 10.2 Equivalence with Beckmann problem
      3. 10.3 Characterization of minimizers via anisotropic elliptic PDEs
      4. 10.4 Regularity when the 𝑣k’s and ck’s are constant
      5. 10.5 Numerical simulations
        1. 10.5.1 Description of the algorithm
        2. 10.5.2 Numerical schemes and convergence study
    7. On the Lagrangian branched transport model and the equivalence with its Eulerian formulation
      1. 11.1 The Lagrangian model: irrigation plans
        1. 11.1.1 Notation and general framework
        2. 11.1.2 The Lagrangian irrigation problem
        3. 11.1.3 Existence of minimizers
      2. 11.2 The energy formula
        1. 11.2.1 Rectifiable irrigation plans
        2. 11.2.2 Proof of the energy formula
        3. 11.2.3 Optimal irrigation plans are simple
      3. 11.3 The Eulerian model: irrigation flows
        1. 11.3.1 The discrete model
        2. 11.3.2 The continuous model
      4. 11.4 Equivalence between models
        1. 11.4.1 From Lagrangian to Eulerian
        2. 11.4.2 From Eulerian to Lagrangian
        3. 11.4.3 The equivalence theorem
    8. On some nonlinear evolution systems which are perturbations of Wasserstein gradient flows
      1. 12.1 Introduction
      2. 12.2 Wasserstein space and main result
        1. 12.2.1 The Wasserstein distance
        2. 12.2.2 Main result
      3. 12.3 Semi-implicit JKO scheme
      4. 12.4 κ-flows and gradient estimate
        1. 12.4.1 κ-flows
        2. 12.4.2 Gradient estimate
      5. 12.5 Passage to the limit
        1. 12.5.1 Weak and strong convergences
        2. 12.5.2 Limit of the discrete system
      6. 12.6 The case of a bounded domain Ω
      7. 12.7 Uniqueness of solutions
    9. Pressureless Euler equations with maximal density constraint: a time-splitting scheme
      1. 13.1 Introduction
      2. 13.2 Time-stepping scheme
        1. 13.2.1 Time discretization strategy
        2. 13.2.2 The scheme
      3. 13.3 Numerical illustrations
        1. 13.3.1 Space discretization scheme
        2. 13.3.2 Numerical tests
      4. 13.4 Conclusive remarks
    10. Convergence of a fully discrete variational scheme for a thin-film equation
      1. 14.1 Introduction
        1. 14.1.1 The equation and its properties
        2. 14.1.2 Definition of the discretization
        3. 14.1.3 Main results
        4. 14.1.4 Relation to the literature
        5. 14.1.5 Key estimates
        6. 14.1.6 Structure of the paper
      2. 14.2 Definition of the fully discrete scheme
        1. 14.2.1 Ansatz space and discrete entropy/information functionals
        2. 14.2.2 Discretization in time
        3. 14.2.3 Spatial interpolations
        4. 14.2.4 A discrete Sobolev-type estimate
      3. 14.3 A priori estimates and compactness
        1. 14.3.1 Energy and entropy dissipation
        2. 14.3.2 Compactness
        3. 14.3.3 Convergence of time interpolants
      4. 14.4 Weak formulation of the limit equation
      5. 14.5 Numerical results
        1. 14.5.1 Nonuniform meshes
        2. 14.5.2 Implementation
        3. 14.5.3 Numerical experiments
    11. A Appendix
    12. Interpretation of finite volume discretization schemes for the Fokker–Planck equation as gradient flows for the discrete Wasserstein distance
      1. 15.1 Introduction
      2. 15.2 Preliminaries
      3. 15.3 Discretization of the Fokker–Planck equation
      4. 15.4 Conclusive remarks, perspectives
  7. Index