Dynamic formulations of Optimal Transportation and variational MFGs - - PowerPoint PPT Presentation

Dynamic formulations of Optimal Transportation and variational MFGs

Jean-David Benamou EPC MOKAPLAN CEMRACS-CIRM July 2017

2 / 36

Summary

• 1. Basic Introduction to Dynamic OT
• 2. Time Discretization and MultiMarginal OT
• 3. Entropic Regularization and IPFP/Sinkhorn
• 4. Scaling Algorithms
• 5. Schrödinger bridge and system
• 6. Application to Stochastic VMFGs

3 / 36

Modern setting of Monge Problem (1781)

■ Source/Target Data : d✚i✭x✮✭❂ ✚i✭x✮ dx✮❀ i ❂ 0❀ 1

✚i ✕ 0,

❩

D

✚0✭x✮dx ❂

❩

D

✚1✭x✮dx ❂ 1, D ✚ ❘n

■ Measure preserving Transport Maps :

▼ ❂ ❢T ✿ D ✦ D❀ T★✚0 ❂ ✚1❣ ✽B ✚ D T★✚0 ✭B✮ ❂ ✚0✭T 1✭B✮✮ det✭DT✮✚1✭T✭x✮✮ ❂ ✚0✭x✮

4 / 36

Modern setting of Monge Problem (1781)

■ Cost Function :

■✭T✮ ❂

❩

D

c✭x❀ T✭x✮✮ ✚0✭x✮dx

■ Monge Problem :

(MP) infT✷▼ ■✭T✮

■ Costs : typically c✭x❀ y✮ ❂ 1 p ❦y x❦p (Monge ✦ p ❂ 1). ■ Th. Brenier (1991) ✭p ❂ 2✮ : ✾✦ r✬, ✬ convex

such that ■✭r✬✭x✮✮ ❂ minT✷▼ ■✭T✮

■ Measure preserving property yields : ✭MABV 2✮ det✭D2✬✮✚1✭r✬✮ ❂ ✚0❀ r✬✭X0✮ ✚ X1 ■ Extensive Sobolev regularity theory develloped since by Cafarelli and Ambrosio schools ... O(N) Numerical methods : Monotone FD scheme B. Froese Oberman (2014)

• B. Collino Mirebeau (2016) and B. Duval (2017) and Semi-Discrete

approaches Mérigot (2011) Lévy (2015).

5 / 36

Adding the dynamics

■ Displacement Interpolation - McCann (1997). Def :

x ✼✦ ✂✭t❀ x✮ ❂ x ✰ t ✭r✬✭x✮ x✮❀ t ✷❪0❀ 1❬ ✚✄✭t❀ ✿✮ ❂ ✭✂✭t❀ ✿✮✮★✚0 (t ✼✦ ✚✄✭t❀ ✂✭t❀ x✮✮ ❂ ✚0✭x✮ det✭Dx✂✭t❀ x✮✮ )

■ for all t, ✂✭t❀ ✿✮ solves (MP) from ✚0 to ✚✄✭t❀ ✿✮. ■ W2✭✚0❀ ✚✄✭t❀ ✿✮✮ ❂

♣ ■✭✂✭t❀ ✿✮✮ is a geodesic distance on P✭D✮.

6 / 36

7 / 36

The CFD Formulation

■ Particles move in straigth line at constant speed

❴ ✂✭t❀ x0✮ ❂ ✭r✬✭x0✮ x0✮

def.

❂ v ✄✭t❀ ✂✭t❀ x0✮✮

■ B. Brenier (2000) : ✭✚✄❀ v ✄✮ is the unique minimum of

inf✭✚❀v✮satisfies✭CE✮

❩ 1 ❩

D

1 2✚✭t❀ x✮ ❦v✭t❀ x✮❦2 dx dt ✭CE✮ ❅t✚ ✰ div✭✚ v✮ ❂ 0❀ ❅✗v ❂ 0 on ❅D❀ ✚✭i❀ ✿✮ ❂ ✚i✭✿✮

■ This is a non-smooth convex relaxation (under ✭✚❀ v✮ ✦ ✭✚❀ ✛

def.

❂ ✚ v✮) proximal spliting methods achieve O✭N 3✮ heuristically. ■ A variational deterministic MFG - Lions Lasry (2007) : (CE) and v ❂ grad ✥ and ✭HJ✮ ❅t✥ ✰ 1

2 ❦r✥❦2 ❂ 0

see B. Carlier (2015).

8 / 36

Kantorovich Relaxation (1942)

■ Transport Plans :

✆✭✚0❀ ✚1✮ ❂ ❢✌ ✷ P✭D0 ✂ D1✮❀ PDi ★✌ ❂ ✚i❀ i ❂ 0❀ 1❣ ✽B0 ✚ D0 PD0

★ ✌ ✭B0✮ ❂ ✌✭B0❀ D1✮

✌✭B0❀ D1✮ ❂ ✚0✭B0✮

■ ✆✭✚0❀ ✚1✮ is non empty : ✚0 ✡ ✚1✭x0❀ x1✮ ❂ ✚0✭x0✮ ✚1✭x1✮

9 / 36

Kantorovich Relaxation (1942)

Deterministic Transport plan : ✌T

def.

❂ ✭Id❀ T✮★✚0 ✌T✭B0❀ B1✮ ❂ ✚0✭❢x ✷ B0❀ s✿t✿ T✭x✮ ✷ B1❣✮ T ✷ ▼ ✱ ✌T ✷ ✆✭✚0❀ ✚1✮

■ ✌r✬ solves ✭MK✮

inf

✌✷✆✭✚0❀✚1✮

❩

D0✂D1

c✭x0❀ x1✮d✌✭x0❀ x1✮

■ Linear program but N 2 unknowns Simplex or Interior point methods stuck to N ✬ 100.

10 / 36

Dynamic Kantorovich relaxation

■ Defs : ✡✭D✮ ❂ C✭❬0❀ 1❪❀ D✮ the set of abs. cont. path

✦ ✿ t ✷ ❬0❀ 1❪ ✼✦ ✦✭t✮ ✷ D. Q ✷ P✭✡✭D✮✮ a probability measure on ✡✭D✮. et ✿ ✡✭D✮ ✼✦ D the t-evaluation function - et✭✦✮ ❂ ✦✭t✮. ✽B ✚ D ✭et✮★Q ✭B✮ ❂ Q✭❢✦ ✷ ✡✭D✮❀ ✦✭t✮ ✷ B❣✮

11 / 36

Dynamic Kantorovich relaxation

■ Defs : ✡✭D✮ ❂ C✭❬0❀ 1❪❀ D✮ the set of abs. cont. path

✦ ✿ t ✷ ❬0❀ 1❪ ✼✦ ✦✭t✮ ✷ D. Q ✷ P✭✡✭D✮✮ a probability measure on ✡✭D✮. et ✿ ✡✭D✮ ✼✦ D the t-evaluation function - et✭✦✮ ❂ ✦✭t✮. ✽B ✚ D ✭e1✮★Q ✭B✮ ❂ ✚1✭B✮

12 / 36

Dynamic Kantorovich relaxation

✭DMK✮ inf

❢Q✷P✭✡✭D✮✮❀ ✭ei✮★Q❂✚i❀ i❂0❀1❣

❩

✡✭D✮

❩ 1

❦ ❴ ✦✭t✮❦2 dt dQ✭✦✮

■ ✟✂ ✿ D ✦ ✡✭D✮ , ✟✂✭x0✮ ❂ ✂✭✿❀ x0✮. ■ The solution Q✄ ❂ ✭✟✂✮★✚0 is deterministic.

✽O ✚ ✡✭D✮❀ Q✄✭O✮ ❂ ✚0✭❢x0 ✷ D0❀ s✿t✿ ✂✭x0❀ ✿✮ ✷ O❣✮

■ ✚✭t❀ ✿✮ ❂ ✭et✮★Q✄ is the CFD geodesic.

Analysis by Ambrosio school, see Santambrogio book (2015)

■ P✭✡✭D✮✮ is a BIG space : next section present an efficient

numerical method.

13 / 36

Summary

• 1. Basic Introduction to Dynamic OT
• 2. Time Discretization and MultiMarginal OT
• 3. Entropic Regularization and IPFP/Sinkhorn
• 4. Scaling Algorithms
• 5. Schrödinger bridge and system
• 6. Application to Stochastic VMFGs

14 / 36

Time Discretization

■ Discretize time : Set dt ❂ 1 M ti ❂ i dt❀ i ❂ 0✿✿M ■ Restrict to piecewise linear path ✦dt ❂ ❢x0❀ x1❀ ✿✿❀ xM ❣

(✦dt✭ti✮ ❂ xi).

15 / 36

Time Discretization

■ Minimize w.r.t. Qdt✭x0❀ x1❀ ✿✿✿❀ xM ✮ ✷ P✭✡i❂0❀M Di✮ ■ ✭eti✮★Qdt ❂ ✚i becomes a margin condition :

❩

✡j ✻❂iDj

dQdt✭x0❀ x1❀ ✿✿❀ xM ✮ ❂ ✚i✭xi✮

■ Time integration of linear path in ✭DMK✮ :

inf

Q✷❊

❩

✡i❂0❀M Di

✵ ❅ ❳

i❂0❀M1

1 dt ❦xi✰1 xi❦2

✶ ❆ dQdt✭x0❀ x1❀ ✿✿❀ xM ✮

❊ ❂ ❢Qdt ✷ P✭✡i❂0❀M Di✮❀ ✭eti✮★Qdt ❂ ✚i❀ i ❂ 0❀ 1❣

16 / 36

Multi-Marginal OT

■ General Form of MMOT :

inf

Q✷❊

❩

✡i❂0❀M Di

c✭x0❀ x1❀ ✿✿❀ xM ✮ dQ✭x0❀ x1❀ ✿✿❀ xM ✮ ❊ ❂ ❢Q ✷ P✭✡i❂0❀M Di✮❀ ✭eti✮★Q ❂ ✚i❀ i ❂ 0❀ 1❀ ✿✿❀ M❣

■ Ex. : Density Functional Theory (Friesecke et al, Butazzo

et al (...) , Pass, ... ) c

def.

❂ P

i❁j 1 ❦xixj ❦ Margins : ✭ei✮★Q ❂ ✖

✚❀ i ❂ 0❀ ✿✿❀ M Existence of Maps open ...

■ Generalized Euler Geodesics (Brenier 89)

17 / 36

Multi-Marginal OT

■ Ex. : Wassertein Barycenters (Agueh/Carlier (2011))

c

def.

❂ P

i ✕i❦xi B✭x0❀ ✿✿❀ xM ✮❦2

B✭x0❀ ✿✿❀ xM ✮

def.

❂ P

i ✕ixi

Margins : ✭ei✮★Q ❂ ✚i❀ i ❂ 0❀ ✿✿❀ M Barycenter : B★Q ...

■ Solomon et al (2015)

18 / 36

Summary

• 1. Basic Introduction to Dynamic OT
• 2. Time Discretization and MultiMarginal OT
• 3. Entropic Regularization and IPFP/Sinkhorn
• 4. Scaling Algorithms
• 5. Schrödinger bridge and system
• 6. Application to Stochastic VMFGs

19 / 36

Entropic regularization of OT

See Christian Leonard surveys for the connection with the Schrödinger problem in the continuous setting.

Discretize in space D0 : ❢xi❣ and D1 : ❢xj ❣ ✚0 ❂ P

i ✖i ✍xi and ✚1 ❂ P j ✗j ✍yj

ci j ❂ c✭xi❀ xj ✮

■ Entropic regularisation of MK :

✭MK✧✮ min✌✷●

P

i j ✌✧ i j ci j ✰ ✧ ✌✧ i j ✭log ✌✧ i j 1✮

• ❂ ❢✌ ✷ ❘N✂N ❀ ✌✧

i j✟✟

✕ 0❀ P

j ✌✧ i j ❂ ✖i ❀ P i ✌✧ i j ❂ ✗j ❣ ■ Set ✌✧ i j ❂ e

ci j ✧

✭MK✧✮ min✌✧✷●

P

i j KL✭✌✧ i j ❥✌✧ i j ✮

KL✭f ❥g✮ ❂ f ✭log✭f g ✮ 1✮

20 / 36

Iterative Proportional Fitting Procedure

Sinkhorn (67) Ruschendorf (95) Galichon (09) Cuturi (13) ... min✌✧

i j max❢✬✧ i ❀✥✧ j ❣

P

ij ✥✧ j ✗j ✰✬✧ i ✖i✰✌✧ i j ✭ci j ✥✧ j ✬✧ i ✰✧ ✭log ✌✧ i j 1✮✮ ■ Optimal plan is a scaling :

✌❄❀✧

i j ❂ a✧ i b✧ j ✌✧ i j

where a✧

i ❂ e

✬✧ i ✧ and b✧

j ❂ e

✥✧ j ✧ .

■ Margin constraints give :

a✧

i ❂

✖i

P

j ✌✧ i j b✧ j

and b✧

j ❂

✗j

P

i ✌✧ i j a✧ i

.

■ IPFP is the relaxation :

a

✧❀k✰ 1

2

i

❂ ✖i

P

j ✌✧ i j b✧❀k j

b✧❀k✰1

j

❂ ✗j

P

i ✌✧ i j a ✧❀k✰ 1

2

i

✿

21 / 36

1-D IPFP/Sinkhorn

22 / 36

Remarks on IPFP

■ It. are contractions in the Hilbert metric

dH ✭p❀ q✮ ❂ log✭

maxi✭ pi

qi ✮

mini✭ pi

qi ✮ ✮.

■ Convergence with ✧ (Cominetti San Martin (94) , Carlier et al

(15) ) .

■ On a cartesian grid and d ✕ 2 ( xi ❂ ❢x 1 i1❀ x 2 i2❣ )

✌✧

i j ❂ e

❦xi xj ❦2 ✧

❂ e

❦x1 i1 x1 j1 ❦2 ✧

e

❦x2 i2 x2 j2 ❦2 ✧

is separable Store ✭ ♣ N ✂ ♣ N✮ matrices. One Iteration costs O✭N 1✿5✮.

■ ★ iterations increase with 1 ✧. Stability problems can be

fixed.

■ Many Generalizations including MMOT check B. et al

(2015) Chizat et al (2017) .

23 / 36

IPFP for the MMOT

■ M scalings :

Q❄❀✧

i1❀i1❀✿✿❀iM ❂ a1 i1 a2 i2✿✿✿aM iM e c✭x1❀✿✿❀xM ✮

✧

c✭x1❀ ✿✿❀ xM ✮ ❂ ❥ ❥xi2 xi1❥ ❥2 ✰ ❥ ❥xi3 xi2❥ ❥2 ✰ ✿✿ ✰ ❥ ❥xiM xiM1❥ ❥2 dt

■ IPFP algebra amounts to

am❀✭k✮

im

❂ ✖im

P

i1❀✿✿❀im1❀✚

✚

im ❀im✰1❀✿✿❀iM ❢✿❣

❢✿❣

def.

❂ a1❀✭k✮

i1

✿✿ am1❀✭k✮

im1

✚ ✚

am

im am✰1❀✭k1✮ im✰1

✿✿ aM❀✭k1✮

iM

Qi1❀✿✿❀iM

■ Cost again separable (also along dimensions)

Qi1❀✿✿✿❀iM ❂ ◗M1

m❂1 ✘im im✰1

✘i j ❂ e

❥ ❥xi xj ❥ ❥2 dt ✧

■ Store (M

♣ N ✂ ♣ N) matrices - one iteration costs O✭M N 1✿5✮

24 / 36

W2 geodesic between characteristic functions

25 / 36

Summary

• 1. Basic Introduction to Dynamic OT
• 2. Time Discretization and MultiMarginal OT
• 3. Entropic Regularization and IPFP/Sinkhorn
• 4. Scaling Algorithms
• 5. Schrödinger bridge and system
• 6. Application to Stochastic VMFGs

26 / 36

A more general formulation Chizat et al (2017)

■ Primal :

min✌ F0✭PD0★✌✮ ✰ F1✭PD1★✌✮ ✰ ✧ KLD0✂D1✭✌❥✌✮ Fi l.s.c. convex and proper ...

■ Dual :

max✭u❀v✮ F ✄

0 ✭u✮ F ✄ 1 ✭v✮ ✧❤e

1 ✧ ✭u✟v✮ 1❀ ✌✐D0✂D1

■ Pbm is well posed and the solution is again a scaling :

✌❄✭x0❀ x1✮ ❂ e

u❄✭x0✮ ✧

✌✭x0❀ x1✮ e

v❄✭x1✮ ✧

✌ ❂ e ❦x1x0❦2

✧

■ Scaling Algorithm:

uk✰ 1

2 ❂ arg maxu✭F ✄

0 ✭u✮ ✧❤e

1 ✧ u 1❀ e 1 ✧ v k ✐D0✮

v k✰1 ❂ arg maxv✭F ✄

1 ✭v✮ ✧❤e

1 ✧ v 1❀ T e 1 ✧ uk✰ 1 2 ✐D1✮

❤e

1 ✧ ✭u✟v✮ 1❀ ✌✐D0✂D1 ❂ ❤e 1 ✧ u 1❀ e 1 ✧ v✐D0 ❂ ❤e 1 ✧ v 1❀ T e 1 ✧ u✐D1

27 / 36

A more general formulation Chizat et al (2017)

■ Ex 1 - Sinkhorn/IPFP : F✭✚✮ ❂ 0 if ✚ ❂ ✚0 and ✰✶ else . ■ Ex 2 - Unbalanced OT : F✭✚✮ ❂ ✕KLD✭✚❥✚0✮ .

The scaling Alg. : u

✧❀k✰ 1

2

i

❂

✥

✖i

P

j ✌✧ i j b✧❀k j

✦

✕ ✕✰✧

■ Ex 3 - JKO Gradient Flows :

F0✭✚✮ ❂ 0 if ✚ ❂ ✚0 and ✰✶ else and F1✭✚✮ ❂ dt E✭✚✮ an internal energy (simplest is E ❂ V ✚).

Check Carlier et al (15) for the analysis of the impact of the entropic regularization on the JKO GF. ■ Congestion :

F✭✚✮ ❂ 0 if ✚ ✔ ☛ and ✰✶ else . The scaling Alg. : a

✧❀k✰ 1

2

i

❂

min❢☛i❀P

j ✌✧ i j b✧❀k j

❣

P

j ✌✧ i j b✧❀k j

28 / 36

Summary

• 1. Basic Introduction to Dynamic OT
• 2. Time Discretization and MultiMarginal OT
• 3. Entropic Regularization and IPFP/Sinkhorn
• 4. Scaling Algorithms
• 5. Schrödinger bridge and system
• 6. Application to Stochastic VMFGs

29 / 36

Schrödinger bridge

Long list of refs, check Léonard survey, see also Leger (17) ■ Schrödinger problem

inf✭✚❀v✮satisfies✭FP✮

❩ 1 ❩

D

1 2✚✭t❀ x✮ ❦v✭t❀ x✮❦2 dx dt ✭FP✮ ❅t✚ ✰ div✭✚ v✮ ❂ ✧✁✚❀ ✚✭i❀ ✿✮ ❂ ✚i✭✿✮

■ A variational stochastic "planning" MFG :

✽ ❁ ✿

❅t✚ ✰ div✭✚ r✥✮ ❂ ✧✁✚❀ ✚✭i❀ ✿✮ ❂ ✚i✭✿✮ ❅t✥ ✰ 1

2❦r✥❦2 ❂ ✧✁✥

30 / 36

Schrödinger system

■ Hopf-Cole Change of variable :

✑ ❂ e

1 2 ✧✥

✑✄ ❂ ✚ e 1

2 ✧✥

gives

✽ ❁ ✿

❅t✑ ✰ ✧✁✑ ❂ 0 ❅t✑✄ ✧✁✑✄ ❂ 0

Guéant (11?) ■ Solve Schrödinger System : Find ✭a❀ b✮ s.t.

✽ ❁ ✿

✚0✭x0✮ ❂ a✭x0✮

❘ ✭x0❀ x1✮ b✭x1✮ dx1

✭t ❂ 0✮ ✚1✭x1✮ ❂ b✭x1✮

❘ ✭x0❀ x1✮ a✭x0✮ dx0

✭t ❂ 1✮ ✭x0❀ x1✮ ❂ 1 ✭4✙✧✮

d 2

e 1

4✧❦x1x0❦2

■ Back to IPFP/Sinkhorn ...

31 / 36

Increasing ✧

32 / 36

Summary

• 1. Basic Introduction to Dynamic OT
• 2. Time Discretization and MultiMarginal OT
• 3. Entropic Regularization and IPFP/Sinkhorn
• 4. Scaling Algorithms
• 5. Schrödinger bridge and system
• 6. Application to Stochastic VMFGs

33 / 36

P✭✡✭D✮✮ Lagrangian relaxation of VMFGs

See B. Carlier Santambrogio survey (16)

■ PVMFG quadratic Hamiltonian

inf✭✚❀v✮satisfies✭CE✮

❩ 1 ❩

D

1 2✚✭t❀ x✮ ❦v✭t❀ x✮❦2 ✰ G✭t❀ x❀ ✚✭t❀ x✮✮ dx dt ✭CE✮ ❅t✚ ✰ div✭✚ v✮ ❂ 0❀ ❅✗v ❂ 0 on ❅D❀ ✚✭i❀ ✿✮ ❂ ✚i✭✿✮

■ Lagrangian Relaxation

inf

❢Q✷P✭✡✭D✮✮❀ ✭ei✮★Q❂✚i❀ i❂0❀1❣

❩

✡✭D✮

K✭✦✮ dQ✭✦✮ ✰

❩ 1

• ✭t❀ ✭et✮★Q✮ dt

K✭✦✮ ❂

❘ 1

0 ❦ ❴

✦✭t✮❦2 dt

• ✭t❀ ✚✮ ❂

❘

D G✭t❀ x❀ ✚✭t❀ x✮✮ dx if ✚ ✜ ▲d and ✰✶ else. ■ The minimizer is an equilibrium : Jh❄✭Q✮ ✕ Jh❄✭Q❄✮ forall adm. Q Jh❄✭Q✮ ❂

❩

✡✭D✮

K✭✦✮ dQ✭✦✮ ✰

❩ 1 ❩

D

h❄✭t❀ x✮ d✭et✮★Q h❄✭t❀ x✮ ❂ G✵✭t❀ x❀ ✚❄✭t❀ x✮✮ ✚❄✭t❀ ✿✮ ❂ ✭et✮★Q❄

34 / 36

Entropic Regularization ❄ ❂❄ Stochastic MFG

Carlier et al (WIP) inf

❢Q✷✿✿✿❣

❩

✡✭D✮

K✭✦✮ dQ✭✦✮ ✰

❩ 1

• ✭t❀ ✭et✮★Q✮ dt ✰ ✧❍✭Q❥▲d✮

❍✭Q❥R✮ ❂

❘

✡✭D✮ log✭ dQ dR ✮dQ✭✦✮ if P ✜ R and ✰✶ else. ■ Same as :

inf✭✚❀v✮satisfies✭FP✮

❩ 1 ❩

D

1 2✚✭t❀ x✮ ❦v✭t❀ x✮❦2 ✰ t●✭t❀ x❀ ✚✭t❀ x✮✮ dx dt ✭FP✮ ❅t✚ ✰ div✭✚ v✮ ❂ ✧✵✁✚❀ ✚✭i❀ ✿✮ ❂ ✚i✭✿✮

■ Same as :

inf❢Q✷✿✿✿❣ ✧❍✭Q❥Q✮ ✰

❩ 1

• ✭t❀ ✭et✮★Q✮ dt

Q✭✿✮ ❂ ❘

D ❲x ✭✿✮ dx the reference measure

❲x : the Wiener measure induced by a Brownian motion starting at x.

35 / 36

Discretization and Scaling Alg.

■ Piecewise linear time discr. ✭M✮ + grid in space ✭N✮ :

Q✧❀dt

i1❀i1❀✿✿❀iM ✷ ▼M✂N ■ Density Margins :

✚im ❂ P

i1❀✿✿❀im1❀✚

✚

im ❀im✰1❀✿✿❀iM Q✧❀dt i1❀i1❀✿✿❀iM ■ Discrete MFG :

infQdt ✧KL✭Q✧❀dt❥Q✧❀dt✮ ✰ P

j ✷i1❀i1❀✿✿❀iM Gdt j ✭✚j ✮

Q✧❀dt

i1❀✿✿✿❀iM ❂ ◗M1 m❂1 ✘im im✰1

✘i j ❂ e

❥ ❥xi xj ❥ ❥2 dt ✧

■ M scalings :

Q❄

i1❀i1❀✿✿❀iM ❂ a1 i1 a2 i2✿✿✿aM iM Q✧❀dt i1❀✿✿✿❀iM ■ Scaling Alg :

Iterate (marginwise) am❀k

im

❂ arg max

b ✭✭Gdt im✮✄✭b✮ ✧

❳

im

bim

❳

i1❀✿✿❀im1❀✚

✚

im ❀im✰1❀✿✿❀iM

❢✿❣✮ ❢✿❣

def.

❂ a1❀✭k✮

i1

✿✿ am1❀✭k✮

im1

✚ ✚

am

im am✰1❀✭k1✮ im✰1

✿✿ aM❀✭k1✮

iM

Q✧❀dt

i1❀✿✿❀iM

36 / 36

Tests with moving obstacles (strong congestion)

G✭t❀ x❀ ✚✭t❀ x✮✮ ❂ 0 if ✚✭t❀ x✮ ❂ 0 for x ✷ OBS✭t✮ and ✰✶ else .