Algorithms and methods in scientific computing based on Voronoi diagrams. Mesh generation, optimal transport and early universe reconstruction.

1. Remeshing
Variational methods
Bruno Lévy
ALICE Géométrie & Lumière
CENTRE INRIA Nancy Grand-Est

2. OVERVIEW
Part. 1. Introduction - Motivations
Part. 2. Blowing Bubbles
Part. 3. Least Action (the power of power diagrams)
Conclusions

3. Introduction - Motivations
1

4. 1. Motivations
[Martinez, Dumas, Lefebvre]

5. 1. Motivations

6. 1. Motivations

7. 1. Motivations

8. 1. Motivations
René Descartes
Wonder
Shapes
Structures
Symmetry

9. Blowing Bubbles
Centroidal Voronoi Tesselations
2

10. Voronoi diagram: Vor(xi) = { x | d2(x,xi) < d2(x,xj) }
Chap. 2. Centroidal Voronoi Tesselation

11. Chap. 2.
Centroidal
Voronoi
Tessellation

12. Chap. 2. Centroidal Voronoi Tesselation
The classical method:
Lloyd algorithm = gradient descent

13. Chap. 2. Centroidal Voronoi Tesselation
The classical method:
Lloyd algorithm = gradient descent
Which objective function ?

14. Chap. 2. Centroidal Voronoi Tesselation
The classical method:
Lloyd algorithm = gradient descent
Which objective function ? This one (quantization noise power)
F=
∫Vor(i)
2
dxxi - x
i

15. Chap. 2. Centroidal Voronoi Tesselation
F=
∫Vor(i)
2
dxxi - x
i
The classical method:
Lloyd algorithm = gradient descent
Which objective function ? This one (quantization noise power)
Why Lloyd works ? What is the gradient of F ?

16. Chap. 2. Centroidal Voronoi Tesselation
F=
∫Vor(i)
2
dxxi - x
i
=
∫Inf i
2
dxxi - x
The classical method:
Lloyd algorithm = gradient descent
Which objective function ? This one (quantization noise power)
Why Lloyd works ? What is the gradient of F ?

17. Chap. 2. Centroidal Voronoi Tesselation
F=
∫Vor(i)
2
dxxi - x
i
=
∫Inf i
2
dxxi - x
The classical method:
Lloyd algorithm = gradient descent
Which objective function ? This one (quantization noise power)
Why Lloyd works ? What is the gradient of F ?

18. Chap. 2. Centroidal Voronoi Tesselation
F|xi = 2 mi (xi - gi) [Iri et.al], [Du et.al]
F=
∫Vor(i)
2
dxxi - x
i
=
∫Inf i
2
dxxi - x
The classical method:
Lloyd algorithm = gradient descent
Which objective function ? This one (quantization noise power)
Why Lloyd works ? What is the gradient of F ?

19. Chap. 2. Centroidal Voronoi Tesselation
F|xi = 2 mi (xi - gi) [Iri et.al], [Du et.al]
F=
∫Vor(i)
2
dxxi - x
i
=
∫Inf i
2
dxxi - x
The classical method:
Lloyd algorithm = gradient descent
Which objective function ? This one (quantization noise power)
Why Lloyd works ? What is the gradient of F ?
Volume of Vor(i)

20. Chap. 2. Centroidal Voronoi Tesselation
F|xi = 2 mi (xi - gi) [Iri et.al], [Du et.al]
F=
∫Vor(i)
2
dxxi - x
i
=
∫Inf i
2
dxxi - x
The classical method:
Lloyd algorithm = gradient descent
Which objective function ? This one (quantization noise power)
Why Lloyd works ? What is the gradient of F ?
Volume of Vor(i) Centroid of Vor(i)

21. Chap. 2. Centroidal Voronoi Tesselation
The classical method:
Lloyd algorithm = gradient descent
F=
∫Vor(i)
2
dxxi - x
i
F is smooth (C2) [Liu, Wang, L, Sun, Yan, Lu, Yang 2009]

22. Chap. 2. Centroidal Voronoi Tesselation
The classical method:
Lloyd algorithm = gradient descent
F=
∫Vor(i)
2
dxxi - x
i
F is smooth (C2) [Liu, Wang, L, Sun, Yan, Lu, Yang 2009]

23. [Yan, L, Liu, Sun and Wang 2009]

24. Chap. 2. Centroidal Voronoi Tesselation
F=
∫Vor(i)
2
dxxi - x
i
Blowing square bubbles

25. Chap. 2. Centroidal Voronoi Tesselation
p=2 p=4 p=8 ….
F=
∫Vor(i)
2
dxxi - x
i p

26. Chap. 2.

27. Chap. 2. Centroidal Voronoi Tesselation
[L, Liu, 2010]

28. Chap. 2. Centroidal Voronoi Tesselation
[L, Liu, 2010] [Ray, Sokolov, L, 2015, 2016]
(completely different method:
direction fields + global parameterization)

29. Least Action
Power Diagrams and Optimal Transport
3

30. Yann Brenier
The polar factorization theorem
(Brenier Transport)
Each time the Laplace operator is used in a PDE, it can be replaced with the
Monte-Ampère operator, and then interesting things occur

31. 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

32. 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

33. 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

34. 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

35. 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

36. ∫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

37. ∫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

38. ∫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

39. ∫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

40. ∫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

41. ∫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

42. ∫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

43. ∫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
Noeter theorem

44. 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 law I)

45. 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 law I)
Expression of the Lagrangian:
L = ½ m v2

46. 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 law I)
Expression of the Lagrangian:
L = ½ m v2
Expression of the Energy:
E = ½ m v2

47. 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 law I)
Expression of the Lagrangian:
L = ½ m v2
Expression of the Energy:
E = ½ m v2

48. 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 law I)
Expression of the Lagrangian:
L = ½ m v2
Expression of the Energy:
E = ½ m v2

49. 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 law II)
Free particle:
Theorem 3: v = cte (Newton law I)
Expression of the Lagrangian:
L = ½ m v2
Expression of the Energy:
E = ½ m v2

50. 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)
√

51. 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
Theorem 5:
E = ½ γ m v2 + mc2
γ = 1 / ( 1 – v2 / c2)√

52. The Least Action Principle
(quantum physics setting, just for fun)

53. Fluids

54. FluidsLagrange point of view

55. FluidsLagrange point of view
ρ nb particles per square
Euler point of view

56. FluidsLagrange point of view
ρ(x,y,t) nb particles per square
Euler point of view
v(x,y,t) speed of the particle under
grid point (x,y) at time t

57. Fluids
ρ(x,y,t) nb particles per square
Euler point of view
v(x,y,t) speed of the particle under
grid point (x,y) at time t
Q1: how to compute the
acceleration of the particles
from v(x,y,t) ?
dv
dt
=
∂v v. v
∂t
+
Q2: incompressible fluids ?
div(v) = 0
Q3: mass preservation ?
d ρ
dt = - div(ρv)
(Continuity equation)

58. Fluids
Start with Lagrange coordinates:
particle trajectories: X(t,x)
Minimize
Action:
∫t1
t2
1/2
∫Ω
∂X (t,x)
∂t
dxdt
(ρ = cte)
s.t. X satisfies mass preservation
(X is measure-preserving, more on
this later)
2

59. Fluids
Start with Lagrange coordinates:
particle trajectories: X(t,x)
Minimize
Action:
∫t1
t2
1/2
∫Ω
∂X (t,x)
∂t
dxdt
(ρ = cte)
s.t. X satisfies mass preservation
(X is measure-preserving, more on
this later)
Acceleration of the
particle under the grid
2

60. Fluids
Start with Lagrange coordinates:
particle trajectories: X(t,x)
Minimize
Action:
∫t1
t2
1/2
∫Ω
∂X (t,x)
∂t
dxdt
(ρ = cte)
s.t. X satisfies mass preservation
(X is measure-preserving, more on
this later)
The Lagrange multiplier
for the constraint = pressure
2

61. ?
T
Fluids – Benamou Brenier
ρ1 ρ2

62. ?
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

63. ?
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
Minimize C(T) =
∫Ω
|| x – T(x) ||2 dx
s.t. T is measure-preserving
ρ1(x)

64. Part. 3 Optimal Transport – Monge 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)

65. Part. 3 Optimal Transport – Monge problem
A map T is a transport map between μ and ν if
μ(T-1(B)) = ν(B) for any Borel subset B
B
(X;μ) (Y;ν)

66. Part. 3 Optimal Transport – Monge problem
A map T is a transport map between μ and ν if
μ(T-1(B)) = ν(B) for any Borel subset B
B
T-1(B)
(X;μ) (Y;ν)

67. Part. 3 Optimal Transport – Monge problem
A map T is a transport map between μ and ν if
μ(T-1(B)) = ν(B) for any Borel subset B
(or ν = T#μ the pushforward of μ)
(X;μ) (Y;ν)

68. Part. 3 Optimal Transport – Monge problem
Monge problem (1787):
Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x)
(X;μ) (Y;ν)

69. Part. 3 Optimal Transport – Kantorovich
Monge problem:
Find a transport map T that minimizes C(T) = ∫X || x – T(x) ||2 dμ(x)
Kantorovich 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)

70. Part. 3 Optimal Transport – Kantorovich
Transport plan – example 1/2 : translation of a segment

71. Part. 3 Optimal Transport – Kantorovich
Transport plan – example 1/2 : translation of a segment

72. Part. 3 Optimal Transport – Kantorovich

73. Part. 3 Optimal Transport – Dual of Kantorovich
Dual formulation of Kantorovich problem:
Find a c-concave function ψ
that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν
ψc(y) = inf x [ c(x,y) - ψ(x) ]where: (Legendre transform)

74. Part. 3 Optimal Transport – Dual of Kantorovich
Dual formulation of Kantorovich problem:
Find a c-concave function ψ
that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν
ψc(y) = inf x [ c(x,y) - ψ(x) ]where: (Legendre transform)
What about our initial problem ?
T(x) = x – grad ψ(x) = grad (½ x2- ψ(x) )
{{grad ψ(x) with ψ(x) := (½ x2- ψ(x))
[Brenier]

75. Part. 3 Optimal Transport – Dual of Kantorovich
Dual formulation of Kantorovich problem:
Find a c-concave function ψ
that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν
ψc(y) = inf x [ c(x,y) - ψ(x) ]where: (Legendre transform)
What about our initial problem ?
T(x) = x – grad ψ(x) = grad (½ x2- ψ(x) )
{{grad ψ(x) with ψ(x) := (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
[Brenier]

76. Part. 3 Optimal Transport – Dual of Kantorovich
Dual formulation of Kantorovich problem:
Find a c-concave function ψ
that maximizes ∫X ψ(x)dμ + ∫Y ψc(y)dν
ψc(y) = inf x [ c(x,y) - ψ(x) ]where: (Legendre transform)
What about our initial problem ?
T(x) = x – grad ψ(x) = grad (½ x2- ψ(x) )
{{grad ψ(x) with ψ(x) := (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
[Brenier]

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

78. 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)

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

80. 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

81. 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

82. 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

83. Part. 3 Power Diagrams

84. 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)

85. Part. 3 Optimal Transport
Optimal transport Centroidal Voronoi Tesselation
[L - A numerical Algorithm for L2 semi-discrete OT (ESAIM M2AN 2015)]

86. Part. 3 Power Diagrams & Transport

87. Part. 3 Power Diagrams & Transport

88. Part. 3 Power Diagrams & Transport

89. Part. 3 Power Diagrams & Transport

90. Part. 3 Power Diagrams & Transport

91. Part. 3 Power Diagrams & Transport

92. Part. 3 Power Diagrams & Transport
hi

93. Part. 3 Power Diagrams & Transport

94. Part. 3 Power Diagrams & Transport

95. Part. 3 Power Diagrams & Transport

96. Part. 3 Power Diagrams & Transport

97. Part. 3 Power Diagrams and Transport

98. Part. 3 Optimal Transport – 2D examples
Numerical Experiment: A disk becomes two disks

99. Part. 3 Optimal Transport – 3D examples
Numerical Experiment: A sphere becomes two spheres

100. Part. 3 Optimal Transport – 3D examples
Numerical Experiment: Armadillo to sphere

101. Part. 3 Optimal Transport – 3D examples
Numerical Experiment: Other examples

102. Part. 3 Optimal Transport – 3D examples
Numerical Experiment: Varying density

103. Part. 1 Optimal Transport
[Schwartzburg et.al, SIGGRAPH 2014]

104. Part. 1 Optimal Transport
Optimal transport E.U.R.
let there be light !

105. Part. 1 Optimal Transport
Optimal transport E.U.R.
let there be light !
The millenium simulation project,
Max Planck Institute fur Astrophysik
pc/h : parsec (= 3.2 light years)

106. Part. 4 Optimal Transport – 3D examples
Numerical Experiment: Performances
2002: several hours of supercomputer time were needed
for computing OT with a few thousand Dirac masses, with a combinatorial
algorithm in O(n2log(n))

107. Part. 4 Optimal Transport – 3D examples
Numerical Experiment: Performances
2002: several hours of supercomputer time were needed
for computing OT with a few thousand Dirac masses, with a combinatorial
algorithm in O(n2log(n))
2015:
3D version of [Mérigot] (multilevel + BFGS) + several tricks [L 2015]

108. Part. 4 Optimal Transport – 3D examples
Numerical Experiment: Performances
2002: several hours of supercomputer time were needed
for computing OT with a few thousand Dirac masses, with a combinatorial
algorithm in O(n2log(n))
2015:
3D version of [Mérigot] (multilevel + BFGS) + several tricks [L 2015]

109. Part. 4 Optimal Transport – 3D examples
Numerical Experiment: Performances
2002: several hours of supercomputer time were needed
for computing OT with a few thousand Dirac masses, with a combinatorial
algorithm in O(n2log(n))
2015:
3D version of [Mérigot] (multilevel + BFGS) + several tricks [L 2015]
2016: Damped Newton [Mérigot, Thibert] + several tricks [L] for 3D:
1 million Dirac masses in 240 seconds

110. Part. 4 Optimal Transport – 3D examples
Numerical Experiment: Performances
2002: several hours of supercomputer time were needed
for computing OT with a few thousand Dirac masses, with a combinatorial
algorithm in O(n2log(n))
2015:
3D version of [Mérigot] (multilevel + BFGS) + several tricks [L 2015]
2016: Damped Newton [Mérigot, Thibert] + several tricks [L] for 3D:
1 million Dirac masses in 240 seconds
10 million Dirac masses in 90 minutes

111. Part. 4 Optimal Transport – 3D examples
Numerical Experiment: Performances
2002: several hours of supercomputer time were needed
for computing OT with a few thousand Dirac masses, with a combinatorial
algorithm in O(n2log(n))
2015:
3D version of [Mérigot] (multilevel + BFGS) + several tricks [L 2015]
2016: Damped Newton [Mérigot, Thibert] + several tricks [L] for 3D:
1 million Dirac masses in 240 seconds
10 million Dirac masses in 90 minutes
Semi-discrete OT is now scalable ! (new tool in Num. Ana. Toolbox)

112. Other topics
•Euler equation in more complicated setting:
[Merigot & Mirebeau, Merigot & Galouet]
•Using semi-discrete OT to solve other PDEs
[Benamou, Carlier, Merigot , Oudet]
•Faster solvers

113. Online resources
Source code: alice.loria.fr/software/geogram
Demos: www.loria.fr/~levy/GEOGRAM/
www.loria.fr/~levy/GLUP/
L., A numerical algorithm for semi-discrete L2 OT in 3D,
ESAIM Math. Modeling and Analysis, 2015
Video (presentation with the proofs)
www.loria.fr/~levy (Ecole d été calcul des variations 2015)
New projects:
MAGA Q. Mérigot - ANR
EXPLORAGRAM (Inria) with E. Schwindt, Q. Mérigot and J.-D. Benamou

114. merci
http://alice.loria.fr