Course on Optimal Transport

1. Mathématiques - Informatique Bruno Lévy ALICE Géométrie & Lumière CENTRE INRIA Nancy Grand-Est Symposium on Geometry Processing – Paris – 2018 Course on Numerical Optimal Transport – Bruno Lévy

2. Part. 1. Goals and Motivations Part. 2. Introduction to Optimal Transport Part. 3. Semi-Discrete Optimal Transport Part. 4. Applications in Computational Physics OVERVIEW

3. Goals and Motivations 1

4. Part. 1 Optimal Transport Goal #1: “Understanding”

5. Part. 1 Optimal Transport Goal #1: “Understanding” What I can’t create I don’t understand Richard Feynman

6. Part. 1 Optimal Transport Goal #1: “Understanding” Jos Stam, Stable Fluids, 1999 The art of fluid sim. Understand fluids Explain Program

7. Part. 1 Optimal Transport Goal #1: “Understanding” I wrote code

8. Part. 1 Optimal Transport Goal #1: “Understanding” Your mission statement: 1. Understand the stuff 2. Explain it in simple terms Be a good teacher, to others and to yourself Know what you know and what you don’t know Try to know what you don’t know 3. Program it

9. Part. 1 Optimal Transport Measuring distances between functions

12. Part. 1 Optimal Transport Measuring distances between function

13. Part. 1 Optimal Transport Interpolating functions

25. Part. 1 Optimal Transport Gaspard Monge - 1784

26. Part. 1 Optimal Transport Gaspard Monge – geometry and light ANR TOMMI Workshop

27. Cédric Villani Optimal Transport Old & New Topics on Optimal Transport Yann Brenier The polar factorization theorem (Brenier Transport) Part. 1 Optimal Transport Monge-Brenier-Villani, the french connection

28. Part. 1 Optimal Transport [Castro, Merigot, Thibert 2014] Optimal transport geometry and light [Caffarelli, Kochengin, and Oliker 1999]

29. Part. 1 Optimal Transport – Image Processing Barycenters / mixing textures Video-style transfer, A.I., “data sciences” [Nicolas Bonneel, Julien Rabin, Gabriel Peyŕe, Hanspeter Pfister] [Nicolas Bonneel, Kalyan Sunkavalli, Sylvain Paris, Hanspeter Pfister] [Marco Cuturi, Gabriel Peyré]

30. Part. 1 Optimal Transport Optimal transport - geometry and light [Chwartzburg, Testuz, Tagliasacchi, Pauly, SIGGRAPH 2014]

31. Part. 1. Motivations Discretization of functionals involving the Monge-Ampère operator, Benamou, Carlier, Mérigot, Oudet arXiv:1408.4536 The variational formulation of the Fokker-Planck equation Jordan, Kinderlehrer and Otto SIAM J. on Mathematical Analysis Geostrophic current

32. Part. 1 Optimal Transport How to “morph” a shape into another one of same mass while minimizing the “effort” ? ? T

33. Part. 1 Optimal Transport How to “morph” a shape into another one of same mass while minimizing the “effort” ? ? T The “effort” of the best T defines a distance between the shapes

34. Part. 1 Optimal Transport How to “morph” a shape into another one while preserving mass and minimizing the effort ? ? T

35. Part. 1 Optimal Transport

36. Part. 1 Optimal Transport How to “morph” a shape into another one while preserving mass and minimizing the effort ? ? T “minimum action principle”

37. Part. 1 Optimal Transport How to “morph” a shape into another one while preserving mass and minimizing the effort ? ? T “minimum action principle”“conservation law”

38. Part. 1 Optimal Transport “minimum action principle subject to conservation law” OT= Yann Brenier: “Each time the Laplace operator is used in a PDE, it can be replaced with the Monge-Ampère operator”

39. Part. 1 Optimal Transport “minimum action principle subject to conservation law” OT= Yann Brenier: “Each time the Laplace operator is used in a PDE, it can be replaced with the Monge-Ampère operator” New ways of simulating physics with a computer

40. Part. 1 Optimal Transport “minimum action principle subject to conservation law” OT= Yann Brenier: “Each time the Laplace operator is used in a PDE, it can be replaced with the Monge-Ampère operator” New ways of simulating physics with a computer Fast Fourier Transform

41. Part. 1 Optimal Transport “minimum action principle subject to conservation law” OT= Yann Brenier: “Each time the Laplace operator is used in a PDE, it can be replaced with the Monge-Ampère operator” New ways of simulating physics with a computer Fast Fourier Transform Fast OT algo. ???

42. Optimal Transport an elementary introduction 2

43. Part. 2 Optimal Transport – Monge’s problem (X;μ) (Y;ν) Two measures μ, ν such that ∫dμ(x) = ∫dν(x) X Y

44. Part. 2 Optimal Transport – Monge’s problem A map T is a transport map between μ and ν if μ(T-1(B)) = ν(B) for any Borel subset B of Y (X;μ) (Y;ν) (Borel subset = subset that can be measured)

45. Part. 2 Optimal Transport – Monge’s problem A map T is a transport map between μ and ν if μ(T-1(B)) = ν(B) for any Borel subset B of Y B (X;μ) (Y;ν)

46. Part. 2 Optimal Transport – Monge’s problem A map T is a transport map between μ and ν if μ(T-1(B)) = ν(B) for any Borel subset B of Y B T-1(B) (X;μ) (Y;ν)

47. Part. 2 Optimal Transport – Monge’s problem A map T is a transport map between μ and ν if μ(T-1(B)) = ν(B) for any Borel subset B of Y Notation: if T is a transport map between μ and ν then one writes ν = T#μ (ν is the pushforward of μ) (X;μ) (Y;ν)

48. Part. 2 Optimal Transport – Monge’s problem Monge’s problem: Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) (X;μ) (Y;ν)

49. Part. 2 Optimal Transport – Monge’s problem Monge’s problem: Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) • Difficult to study • If μ has an atom (isolated Dirac), it can only be mapped to another Dirac (T needs to be a map)

50. Part. 2 Optimal Transport – Monge’s problem Monge’s problem: Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) Transport from a measure concentrated on L1 onto another one concentrated on L2 and L3

53. Part. 2 Optimal Transport – Monge’s problem Monge’s problem: Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) Transport from a measure concentrated on L1 onto another one concentrated on L2 and L3 The infimum is never realized by a map, need for a relaxation

54. Part. 2 Optimal Transport – Kantorovich Monge’s problem: Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) Kantorovich’s problem (1942): Find a measure γ defined on X x Y such that ∫x in X dγ(x,y) = dν(y) and ∫y in Y dγ(x,y) = dμ(x) that minimizes ∫∫X x Y || x – y ||2 dγ(x,y)

55. Part. 2 Optimal Transport – Kantorovich Monge’s problem: Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) Kantorovich’s problem: Find a measure γ defined on X x Y such that ∫x in X dγ(x,y) = dν(y) and ∫y in Y dγ(x,y) = dμ(x) that minimizes ∫∫X x Y || x – y ||2 dγ(x,y) “γ(x,y)” : How much sand goes from x to y

56. Part. 2 Optimal Transport – Kantorovich Monge’s problem: Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) Kantorovich’s problem: Find a measure γ defined on X x Y such that ∫x in X dγ(x,y) = dν(y) and ∫y in Y dγ(x,y) = dμ(x) that minimizes ∫∫X x Y || x – y ||2 dγ(x,y) Everything that is transported from x sums to “μ(x)”

57. Part. 2 Optimal Transport – Kantorovich Monge’s problem: Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) Kantorovich’s problem: Find a measure γ defined on X x Y such that ∫x in X dγ(x,y) = dν(y) and ∫y in Y dγ(x,y) = dμ(x) that minimizes ∫∫X x Y || x – y ||2 dγ(x,y) Everything that is transported to y sums to “ν(y)”

58. Part. 2 Optimal Transport – Kantorovich Transport plan – example 1/4 : translation of a segment

59. Part. 2 Optimal Transport – Kantorovich Transport plan – example 1/4 : translation of a segment

60. Part. 2 Optimal Transport – Kantorovich Transport plan – example 2/4 : spitting a segment

61. Part. 2 Optimal Transport – Kantorovich

64. Part. 2 Optimal Transport – Kantorovich Transport plan – example 3/4 : splitting a Dirac into two Diracs

65. Part. 2 Optimal Transport – Kantorovich Transport plan – example 3/4 : splitting a Dirac into two Diracs (No transport map)

66. Part. 2 Optimal Transport – Kantorovich Transport plan – example 4/4 : splitting a Dirac into two segments

67. Part. 2 Optimal Transport – Kantorovich Transport plan – example 4/4 : splitting a Dirac into two segments (No transport map)

68. Part. 2 Optimal Transport – Duality

69. Part. 2 Optimal Transport – Duality Duality is easier to understand with a discrete version Then we’ll go back to the continuous setting.

70. Part. 2 Optimal Transport – Duality (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0

71. Part. 2 Optimal Transport – Duality (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0 γ = γ11 γ12 … γ1n γ22 … γ2n … … γmn Є IRmn

72. Part. 2 Optimal Transport – Duality (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0 γ = γ11 γ12 … γ1n γ22 … γ2n … … γmn c = c11 c12 … c1n c22 … c2n … … cmn Є IRmnЄ IRmn

73. Part. 2 Optimal Transport – Duality (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0 γ = γ11 γ12 … γ1n γ22 … γ2n … … γmn c = c11 c12 … c1n c22 … c2n … … cmn cij = || xi – yj ||2 Є IRmnЄ IRmn

74. Part. 2 Optimal Transport – Duality (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0 γ = γ11 γ12 … γ1n γ22 … γ2n … … γmn c = c11 c12 … c1n c22 … c2n … … cmn cij = || xi – yj ||2 mn X m Є IRmnЄ IRmn

75. Part. 2 Optimal Transport – Duality (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0 γ = γ11 γ12 … γ1n γ22 … γ2n … … γmn c = c11 c12 … c1n c22 … c2n … … cmn cij = || xi – yj ||2 mn X n mn X m Є IRmnЄ IRmn

76. Part. 2 Optimal Transport – Duality (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0 γ = γ11 γ12 … γ1n γ22 … γ2n … … γmn c = c11 c12 … c1n c22 … c2n … … cmn Є IRmnЄ IRmn mn X n mn X m

77. Part. 2 Optimal Transport – Duality Consider L(φ,ψ) = <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0 < u, v > denotes the dot product between u and v

78. Part. 2 Optimal Transport – Duality Consider L(φ,ψ) = <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> Remark: Sup[ L(φ,ψ) ] = < c, γ> if P1 γ = u and P2 γ = v φ Є IRm ψ Є IRn (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0

79. Part. 2 Optimal Transport – Duality Consider L(φ,ψ) = <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> Remark: Sup[ L(φ,ψ) ] = < c, γ> if P1 γ = u and P2 γ = v φ Є IRm ψ Є IRn = +∞ otherwise (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0

80. Part. 2 Optimal Transport – Duality Consider L(φ,ψ) = <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> Remark: Sup[ L(φ,ψ) ] = < c, γ> if P1 γ = u and P2 γ = v φ Є IRm ψ Є IRn = +∞ otherwise Consider now: Inf [ Sup[ L(φ,ψ) ] ] φ Є IRm ψ Є IRn γ ≥ 0 (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0

81. Part. 2 Optimal Transport – Duality Consider L(φ,ψ) = <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> Remark: Sup[ L(φ,ψ) ] = < c, γ> if P1 γ = u and P2 γ = v φ Є IRm ψ Є IRn = +∞ otherwise Consider now: Inf [ Sup[ L(φ,ψ) ] ] = Inf [ < c, γ> ] φ Є IRm ψ Є IRn γ ≥ 0 γ ≥ 0 P1 γ = u P2 γ = v (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0

82. Part. 2 Optimal Transport – Duality Consider L(φ,ψ) = <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> Remark: Sup[ L(φ,ψ) ] = < c, γ> if P1 γ = u and P2 γ = v φ Є IRm ψ Є IRn = +∞ otherwise Consider now: Inf [ Sup[ L(φ,ψ) ] ] = Inf [ < c, γ> ] (DMK) φ Є IRm ψ Є IRn γ ≥ 0 γ ≥ 0 P1 γ = u P2 γ = v (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0

83. Part. 2 Optimal Transport – Duality φ Є IRm ψ Є IRn γ ≥ 0 Inf [ Sup[ <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> ] ] (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0

84. Part. 2 Optimal Transport – Duality φ Є IRm ψ Є IRn γ ≥ 0 Inf [ Sup[ <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> ] ] φ Є IRm ψ Є IRn γ ≥ 0 Sup[ Inf[ <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> ] ] Exchange Inf Sup (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0

85. Part. 2 Optimal Transport – Duality φ Є IRm ψ Є IRn γ ≥ 0 Inf [ Sup[ <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> ] ] φ Є IRm ψ Є IRn γ ≥ 0 Sup[ Inf[ <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> ] ] φ Є IRm ψ Є IRn γ ≥ 0 Sup[ Inf[ < γ,c-P1 t φ – P2 t ψ > + <φ,u> + <ψ, v> ] ] Exchange Inf Sup Expand/Reorder/Collect (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0

86. Part. 2 Optimal Transport – Duality φ Є IRm ψ Є IRn γ ≥ 0 Inf [ Sup[ <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> ] ] φ Є IRm ψ Є IRn γ ≥ 0 Sup[ Inf[ <c, γ> - <φ, P1 γ - u> - <ψ, P2 γ - v> ] ] φ Є IRm ψ Є IRn γ ≥ 0 Sup[ Inf[ < γ,c-P1 t φ – P2 t ψ > + <φ,u> + <ψ, v> ] ] Exchange Inf Sup Expand/Reorder/Collect Interpret (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0

87. Part. 2 Optimal Transport – Duality φ Є IRm ψ Є IRn γ ≥ 0 Sup[ Inf[ < γ,c-P1 t φ – P2 t ψ > + <φ,u> + <ψ, v> ] ] Interpret Sup[ <φ,u> + <ψ, v> ] φ Є IRm ψ Є IRn P1 t φ + P2 t ψ ≤ c (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0 (DDMK)

88. Part. 2 Optimal Transport – Duality φ Є IRm ψ Є IRn γ ≥ 0 Sup[ Inf[ < γ,c-P1 t φ – P2 t ψ > + <φ,u> + <ψ, v> ] ] Interpret Sup[ <φ,u> + <ψ, v> ] φ Є IRm ψ Є IRn P1 t φ + P2 t ψ ≤ c φi + ψj ≤ cij (i,j) A (DMK): Min <c, γ> P1 γ = u s.t. P2 γ = v γ ≥ 0 (DDMK)

89. Part. 2 Optimal Transport – Kantorovich dual Kantorovich’s problem: Find a measure γ defined on X x Y such that ∫x in X dγ(x,y) = dμ(x) and ∫y in Y dγ(x,y) = dν(x) that minimizes ∫∫X x Y || x – y ||2 dγ(x,y) Dual formulation of Kantorovich’s problem (Continuous): Find two functions φ in L1(μ) and ψ in L1(ν) Such that for all x,y, φ(x) + ψ(y) ≤ ½|| x – y ||2 that maximize ∫X φdμ + ∫Y ψdν

90. Part. 2 Optimal Transport – Kantorovich dual Kantorovich’s problem: Find a measure γ defined on X x Y such that ∫x in X dγ(x,y) = dμ(x) and ∫y in Y dγ(x,y) = dν(x) that minimizes ∫∫X x Y || x – y ||2 dγ(x,y) Dual formulation of Kantorovich’s problem: Find two functions φ in L1(μ) and ψ in L1(ν) Such that for all x,y, φ(x) + ψ(y) ≤ ½|| x – y ||2 that maximize ∫X φdμ + ∫Y ψdν Your point of view: Try to minimize transport cost

91. Part. 2 Optimal Transport – Kantorovich dual Kantorovich’s problem: Find a measure γ defined on X x Y such that ∫x in X dγ(x,y) = dμ(x) and ∫y in Y dγ(x,y) = dν(x) that minimizes ∫∫X x Y || x – y ||2 dγ(x,y) Dual formulation of Kantorovich’s problem: Find two functions φ in L1(μ) and ψ in L1(ν) Such that for all x,y, φ(x) + ψ(y) ≤ ½|| x – y ||2 that maximize ∫X φdμ + ∫Y ψdν Point of view of a “transport company”: Try to maximize transport price Your point of view: Try to minimize transport cost

92. Part. 2 Optimal Transport – Kantorovich dual Kantorovich’s problem: Find a measure γ defined on X x Y such that ∫x in X dγ(x,y) = dμ(x) and ∫y in Y dγ(x,y) = dν(x) that minimizes ∫∫X x Y || x – y ||2 dγ(x,y) Dual formulation of Kantorovich’s problem: Find two functions φ in L1(μ) and ψ in L1(ν) Such that for all x,y, φ(x) + ψ(y) ≤ ½|| x – y ||2 that maximize ∫X φ(x)dμ + ∫Y ψ(y)dν What they charge for loading at x Your point of view: Try to minimize transport cost

93. Part. 2 Optimal Transport – Kantorovich dual Kantorovich’s problem: Find a measure γ defined on X x Y such that ∫x in X dγ(x,y) = dμ(x) and ∫y in Y dγ(x,y) = dν(x) that minimizes ∫∫X x Y || x – y ||2 dγ(x,y) Dual formulation of Kantorovich’s problem: Find two functions φ in L1(μ) and ψ in L1(ν) Such that for all x,y, φ(x) + ψ(y) ≤ ½|| x – y ||2 that maximize ∫X φ(x)dμ + ∫Y ψ(y)dν What they charge for loading at x What they charge for unloading at y Your point of view: Try to minimize transport cost

94. Part. 2 Optimal Transport – Kantorovich dual Kantorovich’s problem: Find a measure γ defined on X x Y such that ∫x in X dγ(x,y) = dμ(x) and ∫y in Y dγ(x,y) = dν(x) that minimizes ∫∫X x Y || x – y ||2 dγ(x,y) Dual formulation of Kantorovich’s problem: Find two functions φ in L1(μ) and ψ in L1(ν) Such that for all x,y, φ(x) + ψ(y) ≤ ½|| x – y ||2 that maximize ∫X φ(x)dμ + ∫Y ψ(y)dν What they charge for loading at x What they charge for unloading at y Price (loading + unloading) cannot be greater than transport cost (else you do the job yourself) Your point of view: Try to minimize transport cost

95. Part. 2 Optimal Transport – c-conjugate functions Dual formulation of Kantorovich’s problem: Find two functions φ in L1(μ) and ψ in L1(ν) Such that for all x,y, φ(x) + ψ(y) ≤ ½|| x – y ||2 that maximize ∫X φ(x)dμ + ∫Y ψ(y)dν

96. Part. 2 Optimal Transport – c-conjugate functions Dual formulation of Kantorovich’s problem: Find two functions φ in L1(μ) and ψ in L1(ν) Such that for all x,y, φ(x) + ψ(y) ≤ ½|| x – y ||2 that maximize ∫X φ(x)dμ + ∫Y ψ(y)dν If we got two functions φ and ψ that satisfy the constraint Then it is possible to obtain a better solution by replacing ψ with φc defined by: For all y, φc(y) = inf x in X ½|| x – y ||2 - φ(y)

97. Part. 2 Optimal Transport – c-conjugate functions Dual formulation of Kantorovich’s problem: Find two functions φ in L1(μ) and ψ in L1(ν) Such that for all x,y, φ(x) + ψ(y) ≤ ½|| x – y ||2 that maximize ∫X φ(x)dμ + ∫Y ψ(y)dν If we got two functions φ and ψ that satisfy the constraint Then it is possible to obtain a better solution by replacing ψ with φc defined by: For all y, φc(y) = inf x in X ½|| x – y ||2 - φ(y) • φc is called the c-conjugate function of φ • If there is a function φ such that ψ = φc then ψ is said to be c-concave • If ψ is c-concave, then ψ cc = ψ

98. Part. 2 Optimal Transport – c-conjugate functions Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν

99. Part. 2 Optimal Transport – c-conjugate functions Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν ψ is called a “Kantorovich potential”

100. Part. 2 Optimal Transport – c-subdifferential What about our initial problem ? ψ is called a “Kantorovich potential” Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν

101. Part. 2 Optimal Transport – c-subdifferential What about our initial problem ? (i.e., this is T() that we want to find …) ψ is called a “Kantorovich potential” Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν

102. Part. 2 Optimal Transport – c-subdifferential

103. Part. 2 Optimal Transport – c-subdifferential Proof: see OTON, chap. 10.

104. Part. 2 Optimal Transport – c-subdifferential Proof: see OTON, chap. 10. Heuristic argument (at the beginning of the same chapter):

110. Part. 2 Optimal Transport – c-subdifferential Proof: see OTON, chap. 10. Heuristic argument (at the beginning of the same chapter): w

111. Part. 2 Optimal Transport – c-subdifferential Proof: see OTON, chap. 10. Heuristic argument (at the beginning of the same chapter): -w

112. Part. 2 Optimal Transport – c-subdifferential Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν

113. Part. 2 Optimal Transport – c-subdifferential Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν { grad ψ(x) with ψ(x) := (½ x2- ψ(x))

114. Part. 2 Optimal Transport – convexity Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν { grad ψ(x) with ψ(x) := (½ x2- ψ(x)) ψ is convex

117. Part. 2 Optimal Transport – no collision If T(.) exists, then T(x) = x – grad ψ(x) = grad (½ x2- ψ(x) ) {grad ψ (x) Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν ψ is convex Two transported particles cannot “collide”

120. Part. 2 Optimal Transport – Monge-Ampere What about our initial problem ? If T(.) exists, then one can show that: T(x) = x – grad ψ(x) = grad (½ x2- ψ(x) ) { for all borel set A, ∫A dμ = ∫T(A) (|JT|) dν (change of variable) Jacobian of T (1st order derivatives) Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν grad ψ(x) with ψ(x) := (½ x2- ψ(x))

121. Part. 2 Optimal Transport – Monge-Ampere What about our initial problem ? If T(.) exists, then one can show that: T(x) = x – grad ψ(x) = grad (½ x2- ψ(x) ) { for all borel set A, ∫A dμ = ∫T(A) (|JT|) dν = ∫T(A) (H ψ ) dν Det. of the Hessian of ψ (2nd order derivatives) Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν grad ψ(x) with ψ(x) := (½ x2- ψ(x))

122. Part. 2 Optimal Transport – Monge-Ampere What about our initial problem ? T(x) = x – grad ψ(x) = grad (½ x2- ψ(x) ) { When μ and ν have a density u and v, (H ψ(x)). v(grad ψ(x)) = u(x) Monge-Ampère equation for all borel set A, ∫A dμ = ∫T(A) (|JT|) dν = ∫T(A) (H ψ ) dν Dual formulation of Kantorovich’s problem: Find a c-concave function ψ that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν {grad ψ(x) with ψ(x) := (½ x2- ψ(x))

123. Part. 2 Optimal Transport – summary Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x)

124. Part. 2 Optimal Transport – summary Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) After several rewrites and under some conditions…. (Kantorovich formulation, dual, c-convex functions)

125. Part. 2 Optimal Transport – summary Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) After several rewrites and under some conditions…. (Kantorovich formulation, dual, c-convex functions) Monge-Ampère equation (When μ and ν have a density u and v resp.) Solve (H ψ(x)). v(grad ψ(x)) = u(x)

126. Part. 2 Optimal Transport – summary Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x) After several rewrites and under some conditions…. (Kantorovich formulation, dual, c-convex functions) Brenier, Mc Cann, Trudinger: The optimal transport map is then given by: T(x) = grad ψ(x) Solve (H ψ(x)). v(grad ψ(x)) = u(x) Monge-Ampère equation (When μ and ν have a density u and v resp.)

127. Semi-Discrete Optimal Transport 3

128. Part. 3 Optimal Transport – how to program ? Continuous (X;μ) (Y;ν)

129. Part. 3 Optimal Transport – how to program ? Continuous Semi-discrete (X;μ) (Y;ν)

130. Part. 3 Optimal Transport – how to program ? Continuous Semi-discrete Discrete (X;μ) (Y;ν)

131. Part. 3 Optimal Transport – how to program ? Continuous Semi-discrete Discrete (X;μ) (Y;ν)

132. Part. 3 Optimal Transport – semi-discrete ∫X ψc (x)dμ + ∫Y ψ(y)dν Sup ψ Є ψc (DMK) (X;μ) (Y;ν)

133. Part. 3 Optimal Transport – semi-discrete (X;μ) (Y;ν) ∑j ψ(yj) vj ∫X ψc (x)dμ + ∫Y ψ(y)dν Sup ψ Є ψc (DMK)

134. Part. 3 Optimal Transport – semi-discrete ∫X ψc (x)dμ + ∫Y ψ(y)dν Sup ψ Є ψc (DMK) ∑j ψ(yj) vj

135. Part. 3 Optimal Transport – semi-discrete ∫X ψc (x)dμ + ∫Y ψ(y)dν Sup ψ Є ψc (DMK) ∫X inf yj ЄY [ || x – yj ||2 - ψ(yj) ] dμ ∑j ψ(yj) vj

136. Part. 3 Optimal Transport – semi-discrete ∫X ψc (x)dμ + ∫Y ψ(y)dν Sup ψ Є ψc (DMK) ∑j ∫Lagψ(yj) || x – yj ||2 - ψ(yj) dμ ∫X inf yj ЄY [ || x – yj ||2 - ψ(yj) ] dμ ∑j ψ(yj) vj

137. Part. 3 Optimal Transport – semi-discrete G(ψ) =∑j ∫Lag ψ(yj) || x – yj ||2 - ψ(yj) dμ + ∑j ψ(yj) vj Sup ψ Є ψc (DMK) Where: Lag ψ(yj) = { x | || x – yj ||2 – ψ(yj) < || x – yj ||2 - ψ(yj’) } for all j’ ≠ j

138. Part. 3 Optimal Transport – semi-discrete G(ψ) =∑j ∫Lag ψ(yj) || x – yj ||2 - ψ(yj) dμ + ∑j ψ(yj) vj Sup ψ Є ψc (DMK) Where: Lag ψ(yj) = { x | || x – yj ||2 – ψ(yj) < || x – yj ||2 - ψ(yj’) } for all j’ ≠ j Laguerre diagram of the yj’s (with the L2 cost || x – y ||2 used here, Power diagram)

139. Part. 3 Optimal Transport – semi-discrete G(ψ) =∑j ∫Lag ψ(yj) || x – yj ||2 - ψ(yj) dμ + ∑j ψ(yj) vj Sup ψ Є ψc (DMK) Where: Lag ψ(yj) = { x | || x – yj ||2 – ψ(yj) < || x – yj ||2 - ψ(yj’) } for all j’ ≠ j Laguerre diagram of the yj’s (with the L2 cost || x – y ||2 used here, Power diagram) Weight of yj in the power diagram

140. Part. 3 Optimal Transport – semi-discrete G(ψ) =∑j ∫Lag ψ(yj) || x – yj ||2 - ψ(yj) dμ + ∑j ψ(yj) vj Sup ψ Є ψc (DMK) Where: Lag ψ(yj) = { x | || x – yj ||2 – ψ(yj) < || x – yj ||2 - ψ(yj’) } for all j’ ≠ j Laguerre diagram of the yj’s (with the L2 cost || x – y ||2 used here, Power diagram) Weight of yj in the power diagram ψ is determined by the weight vector [ψ(y1) ψ(y2) … ψ(ym)]

141. Voronoi diagram: Vor(xi) = { x | d2(x,xi) < d2(x,xj) } Part. 3 Power Diagrams

142. Power diagram: Pow(xi) = { x | d2(x,xi) – ψi < d2(x,xj) – ψj } Voronoi diagram: Vor(xi) = { x | d2(x,xi) < d2(x,xj) } Part. 3 Power Diagrams

143. Part. 3 Power Diagrams

144. Part. 3 Optimal Transport Theorem: (direct consequence of MK duality alternative proof in [Aurenhammer, Hoffmann, Aronov 98] ): Given a measure μ with density, a set of points (yj), a set of positive coefficients vj such that ∑ vj = ∫ dμ(x), it is possible to find the weights W = [ψ(y1) ψ(y2) … ψ(ym)] such that the map TS W is the unique optimal transport map between μ and ν = ∑ vj δ(yj)

145. Part. 3 Optimal Transport Theorem: (direct consequence of MK duality alternative proof in [Aurenhammer, Hoffmann, Aronov 98] ): Given a measure μ with density, a set of points (yj), a set of positive coefficients vj such that ∑ vj = ∫ dμ(x), it is possible to find the weights W = [ψ(y1) ψ(y2) … ψ(ym)] such that the map TS W is the unique optimal transport map between μ and ν = ∑ vj δ(yj) Proof: G(ψ) =∑j ∫Lag ψ(yj) || x – yj ||2 - ψ(yj) dμ + ∑j ψ(yj) vj Is a concave function of the weight vector [ψ(y1) ψ(y2) … ψ(ym)]

146. Part. 3 Optimal Transport Theorem: (direct consequence of MK duality alternative proof in [Aurenhammer, Hoffmann, Aronov 98] ): Given a measure μ with density, a set of points (yj), a set of positive coefficients vj such that ∑ vj = ∫ dμ(x), it is possible to find the weights W = [ψ(y1) ψ(y2) … ψ(ym)] such that the map TS W is the unique optimal transport map between μ and ν = ∑ vj δ(yj) Proof: G(ψ) =∑j ∫Lag ψ(yj) || x – yj ||2 - ψ(yj) dμ + ∑j ψ(yj) vj Is a concave function of the weight vector [ψ(y1) ψ(y2) … ψ(ym)]

147. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) The (unknown) weights W = [ψ(y1) ψ(y2) … ψ(ym)]

148. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) T : an arbitrary but fixed assignment.

152. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) W fT(W) fT(W)

153. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) W fT(W) fT(W)

154. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) fT(W) is linear in W fT(W) W fT(W) Fixed T

155. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) fT(W) is linear in W f TW (W) : defined by Laguerre diagram fT(W) fixed W W

156. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) fT(W) is linear in W f TW (W) = minT fT(W) fT(W) fixed W W

157. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) fT(W) is linear in W f: W → f TW (W) is concave !! (because its graph is the lower enveloppe of linear functions) fT(W) W fT(W) f TW (W) = minT fT(W)

158. Part. 3 Optimal Transport – the AHA paper Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) fT(W) is linear in W f: W → f TW (W) is concave (because its graph is the lower enveloppe of linear functions) fT(W) W fT(W) f TW (W) = minT fT(W) Consider g(W) = f TW (W) + ∑vj ψj

159. Part. 3 Optimal Transport – the AHA paper fT(W) W fT(W) f TW (W) = minT fT(W) Consider g(W) = f TW (W) + ∑vj ψj vj -∫Lag ψ (yj) dμ(x) and g is concave.∂ g / ∂ ψ j = Idea of the proof Consider the function fT(W) = ∫(|| x– T(x) ||2 – ψ(T(X)))dμ(x) fT(W) is linear in W f: W → f TW (W) is concave (because its graph is the lower enveloppe of linear functions)

160. Part. 3 Optimal Transport – the algorithm Semi-discrete OT Summary: ∫X ψc (x)dμ + ∫Y ψ(y)dν Sup ψ Є ψc (DMK) G(ψ) =

161. Part. 3 Optimal Transport – the algorithm Semi-discrete OT Summary: G(ψ) = g(W) =∑j ∫Lag ψ (yj) || x – yj ||2 - ψ(yj) dμ + ∑j ψ(yj) vj is concave ∫X ψc (x)dμ + ∫Y ψ(y)dν Sup ψ Є ψc (DMK) G(ψ) =

162. Part. 3 Optimal Transport – the algorithm Semi-discrete OT Summary: G(ψ) = g(W) =∑j ∫Lag ψ (yj) || x – yj ||2 - ψ(yj) dμ + ∑j ψ(yj) vj is concave vj -∫Lag(yj)dμ(x) (= 0 at the maximum)∂ G / ∂ ψ j = ∫X ψc (x)dμ + ∫Y ψ(y)dν Sup ψ Є ψc (DMK) G(ψ) =

163. Part. 3 Optimal Transport – the algorithm Semi-discrete OT Summary: G(ψ) = g(W) =∑j ∫Lag ψ (yj) || x – yj ||2 - ψ(yj) dμ + ∑j ψ(yj) vj is concave vj -∫Lag(yj)dμ(x) (= 0 at the maximum)∂ G / ∂ ψ j = ∫X ψc (x)dμ + ∫Y ψ(y)dν Sup ψ Є ψc (DMK) G(ψ) = Desired mass at yj Mass transported to yj

164. Part. 3 Optimal Transport – the Hessian vj -∫Lag(yj)dμ(x)∂ G / ∂ ψ j =

165. Part. 3 Optimal Transport – the Hessian vj -∫Lag(yj)dμ(x)∂ G / ∂ ψ j = ∂ 2 G / ∂ ψ i ψ j = - ∂ / ∂ ψ j ∫Lag(yj)dμ(x)

166. Part. 3 Optimal Transport – the Hessian vj -∫Lag(yj)dμ(x)∂ G / ∂ ψ j = ∂ 2 G / ∂ ψ i ψ j = - ∂ / ∂ ψ j ∫Lag(yj)dμ(x) yi yj

167. Part. 3 Optimal Transport – the Hessian vj -∫Lag(yj)dμ(x)∂ G / ∂ ψ j = ∂ 2 G / ∂ ψ i ψ j = - ∂ / ∂ ψ j ∫Lag(yj)dμ(x) yi yj ψ j ψ j + δψ j

168. Part. 3 Optimal Transport – the Hessian vj -∫Lag(yj)dμ(x)∂ G / ∂ ψ j = ∂ 2 G / ∂ ψ i ψ j = - ∂ / ∂ ψ j ∫Lag(yj)dμ(x) yi yj ψ j ψ j + δψ j Reynold’s thm: ∂ / ∂ ψ j ∫Lag(yj)dμ(x) = ∫∂ Lag(yj) v.n dμ(x)

169. Part. 3 Optimal Transport – the Hessian yi yj Reynold’s thm: ∂ / ∂ ψ j ∫Lag(yj)dμ(x) = ∫∂ Lag(yj) v.n dμ(x) fij(x) = 0

170. Part. 3 Optimal Transport – the Hessian yi yj Reynold’s thm: ∂ / ∂ ψ j ∫Lag(yj)dμ(x) = ∫∂ Lag(yj) v.n dμ(x) fij(x) = 0 c(x,yi) - c(x,yj) + ψ j - ψ i = 0

171. Part. 3 the Hessian yi yj Reynold’s thm: ∂ / ∂ ψ j ∫Lag(yj)dμ(x) = ∫∂ Lag(yj) v.n dμ(x) fij(x) = 0 c(x,yi) - c(x,yj) + ψ j - ψ i = 0 δx δψ j dfij = gradx(c(x,yi) - c(x,yj))dx + dψ j n

172. Part. 3 the Hessian yi yj Reynold’s thm: ∂ / ∂ ψ j ∫Lag(yj)dμ(x) = ∫∂ Lag(yj) v.n dμ(x) fij(x) = 0 c(x,yi) - c(x,yj) + ψ j - ψ i = 0 δx δψ j dfij = gradx(c(x,yi) - c(x,yj))dx + dψ j n δx = δh n = δh gradx fij(x) / || gradx fij(x)||

173. Part. 3 the Hessian yi yj Reynold’s thm: ∂ / ∂ ψ j ∫Lag(yj)dμ(x) = ∫∂ Lag(yj) v.n dμ(x) fij(x) = 0 c(x,yi) - c(x,yj) + ψ j - ψ i = 0 δx δψ j dfij = gradx(c(x,yi) - c(x,yj))dx + dψ j n δx = δh n = δh gradx fij(x) / || gradx fij(x)|| δh

174. Part. 3 the Hessian yi yj Reynold’s thm: ∂ / ∂ ψ j ∫Lag(yj)dμ(x) = ∫∂ Lag(yj) v.n dμ(x) fij(x) = 0 c(x,yi) - c(x,yj) + ψ j - ψ i = 0 δx δψ j dfij = gradx(c(x,yi) - c(x,yj))dx + dψ j n δx = δh n = δh gradx fij(x) / || gradx fij(x)|| δh ∂ h / ∂ ψ j = -1/ || gradx c(x,yi) – gradx c(x,yj) ||

175. Part. 3 the Hessian yi yj Reynold’s thm: ∂ / ∂ ψ j ∫Lag(yj)dμ(x) = ∫∂ Lag(yj) v.n dμ(x) fij(x) = 0 c(x,yi) - c(x,yj) + ψ j - ψ i = 0 δx δψ j dfij = gradx(c(x,yi) - c(x,yj))dx + dψ j n δx = δh n = δh gradx fij(x) / || gradx fij(x)|| δh ∂ h / ∂ ψ j = -1/ || gradx c(x,yi) – gradx c(x,yj) || ∂ / ∂ ψ j ∫Lag(yj)dμ(x) = ∫Lag(yi) ∩Lag(yj) -1/|| gradx c(x,yi) – gradx c(x,yj) ||dμ(x)

176. Part. 3 the Hessian ∂ 2 / ∂ ψ i ∂ ψ j F = ∫Lag(yi) ∩ Lag(yj) -1/|| gradx c(x,yi) – gradx c(x,yj) ||dμ(x) ∂ 2 / ∂ ψ i 2 F = - ∑ ∂ 2 / ∂ ψ i ∂ ψ j

177. Part. 3 the Hessian ∂ 2 / ∂ ψ i ∂ ψ j F = ∫Lag(yi) ∩ Lag(yj) -1/|| gradx c(x,yi) – gradx c(x,yj) ||dμ(x) ∂ 2 / ∂ ψ i 2 F = - ∑ ∂ 2 / ∂ ψ i ∂ ψ j c(x,y) = || x – y ||2 ∂ 2 / ∂ ψ i ∂ ψ j F = ∫Lag(yi) ∩ Lag(yj) 1 / || xj – xi || dμ(x)

178. Part. 3 the Hessian ∂ 2 / ∂ ψ i ∂ ψ j F = ∫Lag(yi) ∩ Lag(yj) -1/|| gradx c(x,yi) – gradx c(x,yj) ||dμ(x) ∂ 2 / ∂ ψ i 2 F = - ∑ ∂ 2 / ∂ ψ i ∂ ψ j c(x,y) = || x – y ||2 ∂ 2 / ∂ ψ i ∂ ψ j F = ∫Lag(yi) ∩ Lag(yj) 1 / || xj – xi || dμ(x) IP1 FEM Laplacian (not a big surprise)

179. Part. 3 Optimal Transport Let’s program it ! Hierarchical algorithm [Mérigot] Geometry, 3D [L], [L, Schwindt] Damped Newton algorithm, [Kitagawa, Mérigot, Thibert]

180. Optimal Transport applications in computational physics 4

181. Euler Lagrange The Least Action Principle Hamilton, Legendre, Maupertuis Axiom 1: There exists a function L(x,x,t) that describes the state of a physical system Short summary of the 1st chapter of Landau,Lifshitz Course of Theoretical Physics

182. Euler Lagrange The Least Action Principle Hamilton, Legendre, Maupertuis Axiom 1: There exists a function L(x,x,t) that describes the state of a physical system position

183. Euler Lagrange The Least Action Principle Hamilton, Legendre, Maupertuis Axiom 1: There exists a function L(x,x,t) that describes the state of a physical system position speed

184. Euler Lagrange The Least Action Principle Hamilton, Legendre, Maupertuis Axiom 1: There exists a function L(x,x,t) that describes the state of a physical system position speed time

185. Euler Lagrange ∫t1 t2 L(x,x,t) dt The Least Action Principle Hamilton, Legendre, Maupertuis Axiom 1: There exists a function L(x,x,t) that describes the state of a physical system Axiom 2: The movement (time evolution) of the physical system minimizes the following integral

186. ∫t1 t2 L(x,x,t) dt The Least Action Principle Axiom 1: There exists a function L(x,x,t) that describes the state of a physical system Axiom 2: The movement (time evolution) of the physical system minimizes the following integral

187. ∫t1 t2 L(x,x,t) dt The Least Action Principle Axiom 1: There exists a function L(x,x,t) that describes the state of a physical system Axiom 2: The movement (time evolution) of the physical system minimizes the following integral Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = t=0 t=1

188. ∫t1 t2 L(x,x,t) dt The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t

189. ∫t1 t2 L(x,x,t) dt The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Theorem 2: ∂L ∂x x - L = cte

190. ∫t1 t2 L(x,x,t) dt The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Theorem 2: ∂L ∂x x - L = cte Homogeneity of time → Preservation of energy

191. ∫t1 t2 L(x,x,t) dt The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Theorem 2: ∂L ∂x x - L = cte Homogeneity of time → Preservation of energy Homogeneity of space → Preservation of momentum

192. ∫t1 t2 L(x,x,t) dt The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Theorem 2: ∂L ∂x x - L = cte Homogeneity of time → Preservation of energy Homogeneity of space → Preservation of momentum Isotropy of space → Preservation of angular momentum

193. ∫t1 t2 L(x,x,t) dt The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Theorem 2: ∂L ∂x x - L = cte Homogeneity of time → Preservation of energy Homogeneity of space → Preservation of momentum Isotropy of space → Preservation of angular momentum Preserved quantities “Integrals of Motion” Noether’s theorem

194. The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes ∫ L Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Theorem 2: ∂L ∂x x - L = cte Homogeneity of time → Preservation of energy Homogeneity of space → Preservation of momentum Isotropy of space → Preservation of angular momentum Free particle: Theorem 3: v = cte (Newton’s law I)

195. The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes ∫ L Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Theorem 2: ∂L ∂x x - L = cte Homogeneity of time → Preservation of energy Homogeneity of space → Preservation of momentum Isotropy of space → Preservation of angular momentum Free particle: Theorem 3: v = cte (Newton’s law I) Expression of the Lagrangian: L = ½ m v2

196. The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes ∫ L Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Theorem 2: ∂L ∂x x - L = cte Homogeneity of time → Preservation of energy Homogeneity of space → Preservation of momentum Isotropy of space → Preservation of angular momentum Free particle: Theorem 3: v = cte (Newton’s law I) Expression of the Lagrangian: L = ½ m v2 Expression of the Energy: E = ½ m v2

197. The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes ∫ L Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Particle in a field: Expression of the Lagrangian: L = ½ m v2 – U(x) Free particle: Theorem 3: v = cte (Newton’s law I) Expression of the Lagrangian: L = ½ m v2 Expression of the Energy: E = ½ m v2

198. The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes ∫ L Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Particle in a field: Expression of the Lagrangian: L = ½ m v2 – U(x) Expression of the Energy: E = ½ m v2 + U(x) Free particle: Theorem 3: v = cte (Newton’s law I) Expression of the Lagrangian: L = ½ m v2 Expression of the Energy: E = ½ m v2

199. The Least Action Principle Axiom 1: There exists L Axiom 2: The movement minimizes ∫ L Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. change of Gallileo frame + hom. + isotrop. : x’ t’ = x+vt t Particle in a field: Expression of the Lagrangian: L = ½ m v2 – U(x) Expression of the Energy: E = ½ m v2 + U(x) Theorem 4: mx = - U (Newton’s law II) Free particle: Theorem 3: v = cte (Newton’s law I) Expression of the Lagrangian: L = ½ m v2 Expression of the Energy: E = ½ m v2

200. Axiom 1: There exists L Axiom 2: The movement minimizes ∫ L Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. Lorentz change of frame x’ t’ = (x-vt) x γ (t – vx/c2) x γ γ = 1 / √ ( 1 – v2 / c2) The Least Action Principle (relativistic setting – just for fun…)

201. The Least Action Principle (relativistic setting – just for fun…) Axiom 1: There exists L Axiom 2: The movement minimizes ∫ L Theorem 1: (Lagrange equation): ∂L ∂x d dt ∂L ∂x = Axiom 3: Invariance w.r.t. Lorentz change of frame x’ t’ = (x-vt) x γ (t – vx/c2) x γ γ = 1 / √ ( 1 – v2 / c2) Theorem 5: E = ½ γ m v2 + mc2

202. The Least Action Principle (quantum physics setting – just for fun…)

203. ? T Fluids – Benamou Brenier ρ1 ρ2

204. ? T Fluids – Benamou Brenier Minimize A(ρ,v) = ∫t1 t2 (t2-t1) ∫Ω ρ(x,t) ||v(t,x)||2 dxdt s.t. ρ(t1,.) = ρ1 ; ρ(t2,.) = ρ2 ; d ρ dt = - div(ρv) ρ1 ρ2

205. ? T Fluids – Benamou Brenier ∫t1 t2 (t2-t1) ∫Ω ρ(x,t) ||v(t,x)||2 dxdt s.t. ρ(t1,.) = ρ1 ; ρ(t2,.) = ρ2 ; d ρ dt = - div(ρv) ρ1 ρ2 Minimize C(T) = ∫Ω || x – T(x) ||2 dx s.t. T is measure-preserving ρ1(x) Minimize A(ρ,v) =

206. Le schéma [Mérigot-Gallouet] Applications en graphisme: [De Goes et.al] (power particles) Part. 4 Optimal Transport – Fluids

207. Part. 4 Optimal Transport – Fluids Let’s code it !

208. Part. 4 Optimal Transport – Fluids

211. Part. 4 To infinity and beyond…

212. René Descartes - 1663 Vortices in “ether” ? Part. 4 To infinity and beyond

213. The Data: #1: the Cosmic Microwave Background COBE 1992Part. 4 To infinity and beyond…

214. WMAP 2003 2006 2008 2010 Part. 4 To infinity and beyond… The Data: #1 the Cosmic Microwave Background

215. Part. 4 To infinity and beyond… The Data: #2 redshift acquisition surveys

218. The millenium simulation project, Max Planck Institute fur Astrophysik pc/h : parsec (= 3.2 années lumières)Part. 4 To infinity and beyond…

219. Part. 4 To infinity and beyond… The universal swimming pool

220. Coop. with MOKAPLAN & Institut d’Astrophysique de Paris & Observatoire de Paris Time = Now Early Universe Reconstruction

221. Coop. with MOKAPLAN & Institut d’Astrophysique de Paris & Observatoire de Paris . . . . . Early Universe Reconstruction

222. Coop. with MOKAPLAN & Institut d’Astrophysique de Paris & Observatoire de Paris Time = BigBang (- 13.7 billion Y) Early Universe Reconstruction

223. Coop. with MOKAPLAN & Institut d’Astrophysique de Paris & Observatoire de Paris “Time-warped” map of the universe Early Universe Reconstruction

224. Coop. with MOKAPLAN & Institut d’Astrophysique de Paris & Observatoire de Paris Cosmic Microware Background: “Fossil light” emitted 380 000 Y after BigBang and measured now “Time-warped” map of the universe Early Universe Reconstruction

225. Coop. with MOKAPLAN & Institut d’Astrophysique de Paris & Observatoire de Paris Cosmic Microware Background: “Fossil light” emitted 380 000 Y after BigBang and measured now Do they match ? “Time-warped” map of the universe Early Universe Reconstruction

226. Conclusions Open Questions References Online resources

227. Conclusions – Open questions * Connections with physics, Legendre transform and entropy ? [Cuturi & Peyré] – regularized discrete optimal transport – why does it work ? Hint 1: Minimum action principle subject to conservation laws Hint 2: Entropy = dual of temperature ; Legendre = Fourier[(+,*) → (Max,+)]… * More continuous numerical algorithms ? [Benamou & Brenier] fluid dynamics point of view – very elegant, but 4D problem !! FEM-type adaptive discretization of the subdifferential (graph of T) ? * Can we characterize OT in other semi-discrete settings ? measures supported on unions of spheres piecewise linear densities * Connections with computational geometry ? Singularity set [Figalli] = set of points where T is discontinuous Looks like a “mutual power diagram”, anisotropic Voronoi diagrams

228. Conclusions - References A Multiscale Approach to Optimal Transport, Quentin Mérigot, Computer Graphics Forum, 2011 Variational Principles for Minkowski Type Problems, Discrete Optimal Transport, and Discrete Monge-Ampere Equations Xianfeng Gu, Feng Luo, Jian Sun, S.-T. Yau, ArXiv 2013 Minkowski-type theorems and least-squares clustering AHA! (Aurenhammer, Hoffmann, and Aronov), SIAM J. on math. ana. 1998 Topics on Optimal Transportation, 2003 Optimal Transport Old and New, 2008 Cédric Villani

229. Conclusions - References Polar factorization and monotone rearrangement of vector-valued functions Yann Brenier, Comm. On Pure and Applied Mathematics, June 1991 A computational fluid mechanics solution of the Monge-Kantorovich mass transfer problem, J.-D. Benamou, Y. Brenier, Numer. Math. 84 (2000), pp. 375-393 Pogorelov, Alexandrov – Gradient maps, Minkovsky problem (older than AHA paper, some overlap, in slightly different context, formalism used by Gu & Yau) Rockafeller – Convex optimization – Theorem to switch inf(sup()) – sup(inf()) with convex functions (used to justify Kantorovich duality) Filippo Santambrogio – Optimal Transport for Applied Mathematician, Calculus of Variations, PDEs and Modeling – Jan 15, 2015 Gabriel Peyré, Marco Cuturi, Computational Optimal Transport, 2018

230. Online resources All the sourcecode/documentation available from: http://alice.loria.fr/software/geogram Demo: www.loria.fr/~levy/GLUP/vorpaview * L., A numerical algorithm for semi-discrete L2 OT in 3D, ESAIM Math. Modeling and Analysis, 2015 * L. and E. Schwindt, Notions of OT and how to implement them on a computer, Computer and Graphics, 2018. J. Lévy – 1936-2018 - To you Dad, I miss you so much.

231. Bonus Slides The Isoperimetric Inequality

232. The isoperimetric inequality For a given volume, ball is the shape that minimizes border area

233. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: Given f: IRn → IR sufficiently regular Explanation in [Dario Cordero Erauquin] course notes The isoperimetric inequality

236. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: Given f: IRn → IR sufficiently regular Consider a compact set Ω such that Vol(Ω) = Vol(B2 3) and f = the indicatrix function of Ω The isoperimetric inequality

237. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: Given f: IRn → IR sufficiently regular Consider a compact set Ω such that Vol(Ω) = Vol(B2 3) and f = the indicatrix function of Ω Vol(∂Ω) ≥ n Vol(B2 3)1/3 Vol(B2 3)2/3 The isoperimetric inequality

238. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: Given f: IRn → IR sufficiently regular Consider a compact set Ω such that Vol(Ω) = Vol(B2 3) and f = the indicatrix function of Ω Vol(∂Ω) ≥ n Vol(B2 3)1/3 Vol(B2 3)2/3 Vol(∂Ω) ≥ 4 π = Vol(∂B2 3) The isoperimetric inequality

239. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] The isoperimetric inequality

240. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] We suppose w.l.o.g. that ∫f n/(n-1) = 1 The isoperimetric inequality

241. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] There exists an optimal transport T = gradΨ between f n/(n-1)(x)dx and 1B2 n/Vol(B2 n)dx We suppose w.l.o.g. that ∫f n/(n-1) = 1 The isoperimetric inequality

242. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] There exists an optimal transport T = gradΨ between f n/(n-1)(x)dx and 1B2 n/Vol(B2 n)dx We suppose w.l.o.g. that ∫f n/(n-1) = 1 Monge-Ampère equation: Vol(B2 n) fn/(n-1)(x) = det Hess Ψ The isoperimetric inequality

243. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] There exists an optimal transport T = gradΨ between f n/(n-1)(x)dx and 1B2 n/Vol(B2 n)dx We suppose w.l.o.g. that ∫f n/(n-1) = 1 Monge-Ampère equation: Vol(B2 n) fn/(n-1)(x) = det Hess Ψ Arithmetico-geometric ineq: det (H) 1/n ≤ trace(H)/n if H positive The isoperimetric inequality

244. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] There exists an optimal transport T = gradΨ between f n/(n-1)(x)dx and 1B2 n/Vol(B2 n)dx We suppose w.l.o.g. that ∫f n/(n-1) = 1 Monge-Ampère equation: Vol(B2 n) fn/(n-1)(x) = det Hess Ψ Arithmetico-geometric ineq: det (H) 1/n ≤ trace(H)/n if H positive det (Hess Ψ) 1/n ≤ trace(Hess Ψ)/n The isoperimetric inequality

245. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] There exists an optimal transport T = gradΨ between f n/(n-1)(x)dx and 1B2 n/Vol(B2 n)dx We suppose w.l.o.g. that ∫f n/(n-1) = 1 Monge-Ampère equation: Vol(B2 n) fn/(n-1)(x) = det Hess Ψ Arithmetico-geometric ineq: det (H) 1/n ≤ trace(H)/n if H positive det (Hess Ψ) 1/n ≤ trace(Hess Ψ)/n det (Hess Ψ) 1/n ≤ ∆ Ψ / n The isoperimetric inequality

246. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] We suppose w.l.o.g. that ∫f n/(n-1) = 1 det (Hess Ψ) 1/n ≤ (∆ Ψ)/n Monge-Ampère equation: Vol(B2 n) fn/(n-1)(x) = det Hess Ψ The isoperimetric inequality

247. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] We suppose w.l.o.g. that ∫f n/(n-1) = 1 Vol(B2 n) = Vol(B2 n) ∫f n/(n-1) = ∫f Vol(B2 n) f 1/(n-1) det (Hess Ψ) 1/n ≤ (∆ Ψ)/n Monge-Ampère equation: Vol(B2 n) fn/(n-1)(x) = det Hess Ψ The isoperimetric inequality

248. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] We suppose w.l.o.g. that ∫f n/(n-1) = 1 Vol(B2 n) = Vol(B2 n) ∫f n/(n-1) = ∫f Vol(B2 n) f 1/(n-1) ≤ 1/n ∫ f ∆ Ψ det (Hess Ψ) 1/n ≤ (∆ Ψ)/n Monge-Ampère equation: Vol(B2 n) fn/(n-1)(x) = det Hess Ψ The isoperimetric inequality

249. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] We suppose w.l.o.g. that ∫f n/(n-1) = 1 Vol(B2 n) = Vol(B2 n) ∫f n/(n-1) = ∫f Vol(B2 n) f 1/(n-1) ≤ 1/n ∫ f ∆ Ψ ∫f ∆ Ψ = - ∫ grad f . grad Ψ det (Hess Ψ) 1/n ≤ (∆ Ψ)/n Monge-Ampère equation: Vol(B2 n) fn/(n-1)(x) = det Hess Ψ The isoperimetric inequality

250. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] We suppose w.l.o.g. that ∫f n/(n-1) = 1 Vol(B2 n) = Vol(B2 n) ∫f n/(n-1) = ∫f Vol(B2 n) f 1/(n-1) ≤ 1/n ∫ f ∆ Ψ ∫f ∆ Ψ = - ∫ grad f . grad Ψ ≤ ∫ | grad f | (T = grad Ψ Є B2 n ) det (Hess Ψ) 1/n ≤ (∆ Ψ)/n Monge-Ampère equation: Vol(B2 n) fn/(n-1)(x) = det Hess Ψ The isoperimetric inequality

251. ∫| grad f | ≥ n Vol(B2 n)1/n (∫f n/(n-1))(n-1)/n L1 Sobolev inegality: a proof with OT [Gromov] We suppose w.l.o.g. that ∫f n/(n-1) = 1 Vol(B2 n) = Vol(B2 n) ∫f n/(n-1) = ∫f Vol(B2 n) f 1/(n-1) ≤ 1/n ∫ f ∆ Ψ ∫f ∆ Ψ = - ∫ grad f . grad Ψ ≤ ∫ | grad f | (T = grad Ψ Є B2 n ) ∫ | grad f | ≥ n Vol(B2 n)1/n ■ det (Hess Ψ) 1/n ≤ (∆ Ψ)/n Monge-Ampère equation: Vol(B2 n) fn/(n-1)(x) = det Hess Ψ The isoperimetric inequality

252. Bonus Slides Plotting the potential & optics

253. The [AHA] paper summary: • The optimal weights minimize a convex function • The gradient and Hessian of this convex function is easy to compute Note: the weight w(s) correspond to the Kantorovich potential ψ(x) (solves a “discrete Monge-Ampere” equation) The algorithm: Summary: The algorithm computes the weights wi such that the power cells associated with the Diracs correspond to the preimages of the Diracs. μ ν Plotting the potential, “optics”

258. Plotting the potential, “optics”

264. hi Plotting the potential, “optics”

270. Numerical Experiment: A disk becomes two disks Plotting the potential, “optics”

271. End Of Line.

Course on Optimal Transport

Course on Optimal Transport

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