Differentiating discretized metrics and applications Filippo - PowerPoint PPT Presentation

The continuous framework Applications Discretization Results Differentiating discretized metrics and applications Filippo Santambrogio Laboratoire de Math´ ematiques d’Orsay, Universit´ e Paris-Sud http://www.math.u-psud.fr/ ∼ santambr/ PICOF – April 3rd, 2012, ´ Ecole Polytechnique logo Filippo Santambrogio Differentiating discretized metrics and applications

The continuous framework Applications Discretization Results The continuous framework 1 Distances and Eikonal Equation Geodesics appear when differentiating Applications 2 Slow down the opposant Travel-time tomography Traffic congestion equilibria Discretization 3 FMM Derivative computation Results 4 Slow down the opposant Travel-time tomography Traffic congestion equilibria logo Filippo Santambrogio Differentiating discretized metrics and applications

The continuous framework Applications Discretization Results The continuous framework 1 Distances and Eikonal Equation Geodesics appear when differentiating Applications 2 Slow down the opposant Travel-time tomography Traffic congestion equilibria Discretization 3 FMM Derivative computation Results 4 Slow down the opposant Travel-time tomography Traffic congestion equilibria logo Filippo Santambrogio Differentiating discretized metrics and applications

The continuous framework Applications Discretization Results The continuous framework 1 Distances and Eikonal Equation Geodesics appear when differentiating Applications 2 Slow down the opposant Travel-time tomography Traffic congestion equilibria Discretization 3 FMM Derivative computation Results 4 Slow down the opposant Travel-time tomography Traffic congestion equilibria logo Filippo Santambrogio Differentiating discretized metrics and applications

The continuous framework Applications Discretization Results The continuous framework 1 Distances and Eikonal Equation Geodesics appear when differentiating Applications 2 Slow down the opposant Travel-time tomography Traffic congestion equilibria Discretization 3 FMM Derivative computation Results 4 Slow down the opposant Travel-time tomography Traffic congestion equilibria logo Filippo Santambrogio Differentiating discretized metrics and applications

The continuous framework Applications Discretization Results The continuous framework Riemannian distances and geodesics logo Filippo Santambrogio Differentiating discretized metrics and applications

The continuous framework Applications Discretization Results Geodesic distances Let ξ : Ω → R + be a given (regular) function. The distance d ξ is defined through � 1 ξ ( ω ( t )) | ω ′ ( t ) | dt , d ξ ( x , y ) := ω (0)= x , ω (1)= y L ξ ( ω ) := inf 0 exactly as for a conformal Riemannian Metric (completely isotropic). If x = x 0 is fixed, let us denote U x 0 ,ξ ( y ) := d ξ ( x 0 , y ) : this function is a (viscosity) solution of the Eikonal Equation |∇U| = ξ, U ( x 0 ) = 0 . Question : how does U ξ depend on ξ ? logo Filippo Santambrogio Differentiating discretized metrics and applications

The continuous framework Applications Discretization Results Geodesic distances Let ξ : Ω → R + be a given (regular) function. The distance d ξ is defined through � 1 ξ ( ω ( t )) | ω ′ ( t ) | dt , d ξ ( x , y ) := ω (0)= x , ω (1)= y L ξ ( ω ) := inf 0 exactly as for a conformal Riemannian Metric (completely isotropic). If x = x 0 is fixed, let us denote U x 0 ,ξ ( y ) := d ξ ( x 0 , y ) : this function is a (viscosity) solution of the Eikonal Equation |∇U| = ξ, U ( x 0 ) = 0 . Question : how does U ξ depend on ξ ? logo Filippo Santambrogio Differentiating discretized metrics and applications

The continuous framework Applications Discretization Results Geodesic distances Let ξ : Ω → R + be a given (regular) function. The distance d ξ is defined through � 1 ξ ( ω ( t )) | ω ′ ( t ) | dt , d ξ ( x , y ) := ω (0)= x , ω (1)= y L ξ ( ω ) := inf 0 exactly as for a conformal Riemannian Metric (completely isotropic). If x = x 0 is fixed, let us denote U x 0 ,ξ ( y ) := d ξ ( x 0 , y ) : this function is a (viscosity) solution of the Eikonal Equation |∇U| = ξ, U ( x 0 ) = 0 . Question : how does U ξ depend on ξ ? logo Filippo Santambrogio Differentiating discretized metrics and applications

The continuous framework Applications Discretization Results Concavity, derivatives and subgradients As an infimum of linear quantities (linear in ξ , depending on the curve), ξ �→ d ξ ( x 0 , y 0 ) is obviously a concave function. If we replace ξ with ξ + ε h , we get � 1 d d ε U x 0 ,ξ + ε h ( y ) = h ( ω x 0 , y ( t )) | ω ′ x 0 , y ( t ) | dt = L h ( ω x 0 , y ) , 0 for a geodesic ω x 0 , y (geodesic for the metric ξ ). Which one ? the one minimizing this integral of h . Anyway, for all geodesic curve ω connecting x 0 to y , we have � 1 h ( ω x 0 , y ( t )) | ω ′ U x 0 ,ξ + ε h ( y ) ≤ U x 0 ,ξ ( y ) + ε x 0 , y ( t ) | dt , 0 and the derivatives we computed allow to define an element of the subdifferential (super-differential, actually) : � � h �→ L h ( ω x 0 , y ) ∈ ∂ − ξ U x 0 ,ξ ( y ) . logo Filippo Santambrogio Differentiating discretized metrics and applications

The continuous framework Applications Discretization Results Concavity, derivatives and subgradients As an infimum of linear quantities (linear in ξ , depending on the curve), ξ �→ d ξ ( x 0 , y 0 ) is obviously a concave function. If we replace ξ with ξ + ε h , we get � 1 d d ε U x 0 ,ξ + ε h ( y ) = h ( ω x 0 , y ( t )) | ω ′ x 0 , y ( t ) | dt = L h ( ω x 0 , y ) , 0 for a geodesic ω x 0 , y (geodesic for the metric ξ ). Which one ? the one minimizing this integral of h . Anyway, for all geodesic curve ω connecting x 0 to y , we have � 1 h ( ω x 0 , y ( t )) | ω ′ U x 0 ,ξ + ε h ( y ) ≤ U x 0 ,ξ ( y ) + ε x 0 , y ( t ) | dt , 0 and the derivatives we computed allow to define an element of the subdifferential (super-differential, actually) : � � h �→ L h ( ω x 0 , y ) ∈ ∂ − ξ U x 0 ,ξ ( y ) . logo Filippo Santambrogio Differentiating discretized metrics and applications

The continuous framework Applications Discretization Results Concavity, derivatives and subgradients As an infimum of linear quantities (linear in ξ , depending on the curve), ξ �→ d ξ ( x 0 , y 0 ) is obviously a concave function. If we replace ξ with ξ + ε h , we get � 1 d d ε U x 0 ,ξ + ε h ( y ) = h ( ω x 0 , y ( t )) | ω ′ x 0 , y ( t ) | dt = L h ( ω x 0 , y ) , 0 for a geodesic ω x 0 , y (geodesic for the metric ξ ). Which one ? the one minimizing this integral of h . Anyway, for all geodesic curve ω connecting x 0 to y , we have � 1 h ( ω x 0 , y ( t )) | ω ′ U x 0 ,ξ + ε h ( y ) ≤ U x 0 ,ξ ( y ) + ε x 0 , y ( t ) | dt , 0 and the derivatives we computed allow to define an element of the subdifferential (super-differential, actually) : � � h �→ L h ( ω x 0 , y ) ∈ ∂ − ξ U x 0 ,ξ ( y ) . logo Filippo Santambrogio Differentiating discretized metrics and applications

The continuous framework Applications Discretization Results Concavity, derivatives and subgradients As an infimum of linear quantities (linear in ξ , depending on the curve), ξ �→ d ξ ( x 0 , y 0 ) is obviously a concave function. If we replace ξ with ξ + ε h , we get � 1 d d ε U x 0 ,ξ + ε h ( y ) = h ( ω x 0 , y ( t )) | ω ′ x 0 , y ( t ) | dt = L h ( ω x 0 , y ) , 0 for a geodesic ω x 0 , y (geodesic for the metric ξ ). Which one ? the one minimizing this integral of h . Anyway, for all geodesic curve ω connecting x 0 to y , we have � 1 h ( ω x 0 , y ( t )) | ω ′ U x 0 ,ξ + ε h ( y ) ≤ U x 0 ,ξ ( y ) + ε x 0 , y ( t ) | dt , 0 and the derivatives we computed allow to define an element of the subdifferential (super-differential, actually) : � � h �→ L h ( ω x 0 , y ) ∈ ∂ − ξ U x 0 ,ξ ( y ) . logo Filippo Santambrogio Differentiating discretized metrics and applications

The continuous framework Applications Discretization Results Concavity, derivatives and subgradients As an infimum of linear quantities (linear in ξ , depending on the curve), ξ �→ d ξ ( x 0 , y 0 ) is obviously a concave function. If we replace ξ with ξ + ε h , we get � 1 d d ε U x 0 ,ξ + ε h ( y ) = h ( ω x 0 , y ( t )) | ω ′ x 0 , y ( t ) | dt = L h ( ω x 0 , y ) , 0 for a geodesic ω x 0 , y (geodesic for the metric ξ ). Which one ? the one minimizing this integral of h . Anyway, for all geodesic curve ω connecting x 0 to y , we have � 1 h ( ω x 0 , y ( t )) | ω ′ U x 0 ,ξ + ε h ( y ) ≤ U x 0 ,ξ ( y ) + ε x 0 , y ( t ) | dt , 0 and the derivatives we computed allow to define an element of the subdifferential (super-differential, actually) : � � h �→ L h ( ω x 0 , y ) ∈ ∂ − ξ U x 0 ,ξ ( y ) . logo Filippo Santambrogio Differentiating discretized metrics and applications

The continuous framework Applications Discretization Results Applications Optimization involving d ξ logo Filippo Santambrogio Differentiating discretized metrics and applications