Mean Field Games: Numerical Methods Y. Achdou October 24, 2011 - - PowerPoint PPT Presentation

Mean Field Games: Numerical Methods

• Y. Achdou

October 24, 2011

Content

• A partial review on the theory of Lasry and Lions
• Numerical schemes for the mean field games system

– Description of the scheme – Existence, bounds, uniqueness

• Numerical tests
• The optimal planning problem

– Description of the scheme – Existence of the solution via convex programming – Numerical results

1

• I. Mean field games: some aspects of the theory of Lasry and Lions

Consider N identical players whose dynamics are dXi

t =

√ 2νdW i

t − γidt,

Xi

0 = xi ∈ Rd.

• (W 1

t , . . . , W N t ) independent Brownian motions in Rd,

• ν > 0,
• The control of the player i, i.e. γi is a bounded process assumed to be

adapted to (W 1

t , . . . , W N t ).

For simplicity, all the functions used below are periodic with period 1 in every direction. Let T be the unit torus of Rd.

2

The cost of the player i at time t is

Ji(t) = E @ Z T

t

@L(Xi

s, γi s) + V

2 4 1 N − 1 X

j=i

δXj

s

3 5 (Xi

s)

1 A ds + V0 2 4 1 N − 1 X

j=i

δXj

T

3 5 (Xi

T )

1 A

• V and V0 are operators which continuously map the set of probability

measures on T (endowed with the weak * topology) to a bounded subset of Lip(T).

• L is Lipschitz in x uniformly in γ bounded, and

lim

|γ|→∞ inf x

L(x, γ) |γ| = +∞. Introduce the Hamiltonian H(x, p) = sup

γ∈Rd (p · γ − L(x, γ)) ,

x ∈ T, p ∈ Rd. Assume that H is C1.

3

Nash equilibria: (¯ γ1, . . . , ¯ γN) is a Nash point, if ∀i = 1, . . . , N, Ji(t, ¯ γ1, . . . , ¯ γi−1, γi, ¯ γi+1, . . . , ¯ γN) ≥ Ji(t, ¯ γ1, . . . , ¯ γi−1, ¯ γi, ¯ γi+1, . . . , ¯ γN) Theorem There exist N functions (uj(t, x1, . . . , xN))j=1,...,N such that

8 > > > > > > < > > > > > > : ∂ui ∂t + ν∆Xui − H(xi, ∇xiui) − X

j=i

∂H ∂p (xj, ∇xjuj) · ∇xjui =−V 2 4 1 N − 1 X

j=i

δxj 3 5 (xi) ui(T, x1, . . . , xN) = V0 2 4 1 N − 1 X

j=i

δxj 3 5 (xi)

The feedbacks ¯ γi = ∂H ∂p (xi, ∇xiui) yield a Nash point. In general, no uniqueness

4

Assuming that the players have the same distribution m0 at t = 0, and passing to the limit as N → ∞, Lasry and Lions get the system of 2 PDEs      ∂u ∂t + ν∆u − H(x, ∇u) = −V [m], in (0, T) × T, ∂m ∂t − ν∆m − div

• m∂H

∂p (x, ∇u)

• =

0, in (0, T) × T, with the terminal and initial conditions u(t = T) = V0[m(t = T)], and m(0, x) = m0(x), in T, where m(t, ·) is the density of players at time t: m ≥ 0,

• T

m(t, x)dx = 1. Remark The full justification of the passage to the limit is done in special cases only.

5

Some results on the MFG system                      ∂u ∂t − ν∆u + H(x, ∇u) = V [m], in (0, T] × T, ∂m ∂t + ν∆m + div

• m∂H

∂p (x, ∇u)

• = 0,

in [0, T) × T,

• T

mdx = 1, m > 0 in T, u(t = 0) = V0[m(t = 0)], m(t = T) = m◦, (∗) Remark Note the special structure of the system:

• 1. forward/backward w.r.t. time.
• 2. the operator in the Fokker-Planck equation is the adjoint of the

linearized version of the operator in the HJB equation.

• 3. coupling: via V [m] in the HJB equation and ∂pH(t, x, ∇u) in the

Fokker Planck equation, and possibly via the initial condition on u.

6

Theorem (Lasry-Lions) : Existence for (*) 1) If ν > 0 and

• V and V0 are operators which continuously map the set of probability

measures on T (endowed with the weak * topology) to a bounded subset of Lip(T), i.e. nonlocal smoothing operators,

• H is smooth on T × Rd and
• ∂H

∂x (x, p)

• ≤ C(1 + |p|),

∀x ∈ T, ∀p ∈ Rd,

• m0 is a smooth probabilty density,

then (*) has at least a classical solution. 2) Existence can also be proved if H is Lipschitz w.r.t. p uniformly in x and V [m](x) = F(m(x)) where F is a smooth function.

7

Uniqueness for (*) Theorem (Lasry-Lions) If the operators V and V0 are monotone, i.e.

• T

(V [m] − V [ ˜ m])(m − ˜ m) ≤ 0 ⇒ V [m] = V [ ˜ m],

• T

(V0[m] − V0[ ˜ m])(m − ˜ m) ≤ 0 ⇒ V0[m] = V0[ ˜ m], then (*) has a unique solution. Remark This assumption on V has an economical interpretation if V is local: crowd aversion.

8

Proof Consider two solutions of (*): (v1, m1) and (v2, m2):

• multiply HJB1 − HJB2 by m1 − m2

Z T Z

T

(−(v1 − v2)(∂tm1 − ∂tm2) + ν∇(v1 − v2) · ∇(m1 − m2)) + Z T Z

T

“ H(x, ∇v1) − H(x, ∇v2) ” (m1 − m2) = Z T Z

T

(V [m1] − V [m2])(m1 − m2) + Z

T

“ V0[m1(0)] − V0[m2(0)])(m1(0) − m2(0) ” .

• multiply FP1 − FP2 by v1 − v2

0 = Z T Z

T

−(v1 − v2)(∂tm1 − ∂tm2) + ν∇(v1 − v2) · ∇(m1 − m2) + Z T Z

T

„ m1 ∂H ∂p (x, ∇v1) − m2 ∂H ∂p (x, ∇v2) « · ∇(v1 − v2).

9

• subtract:

0 = 8 > > > > > > > > > > < > > > > > > > > > > : Z T Z

T

m1 „ H(x, ∇v1) − H(x, ∇v2) − ∂H ∂p (x, ∇v1) · ∇(v1 − v2) « + Z T Z

T

m2 „ H(x, ∇v2) − H(x, ∇v1) − ∂H ∂p (x, ∇v2) · ∇(v2 − v1) « + Z T Z

T

(V [m1] − V [m2])(m1 − m2) + Z

T

(V0[m1(t = 0)] − V0[m2(t = 0)])(m1(t = 0) − m2(t = 0))

Since H is convex, V and V0 are monotone, the 4 terms vanish. The strict monotonicity of V implies that V [m1] = V [m2] and v1(t = 0) = v2(t = 0). The identity v1 = v2 comes from the uniqueness for the HJB equation. The identity m1 = m2 comes from the uniqueness for the Fokker-Planck equation.

10

• II. Finite horizon: numerical methods

(Y.A, I. Capuzzo Dolcetta, SIAM J. Numerical Analysis, 2010)

11

Finite difference schemes Goal: use a (semi-)implicit finite difference scheme, robust when ν → 0, which guarantees existence, and possibly uniform bounds and uniqueness. Take d = 2:

• Let Th be a uniform grid on the torus with mesh step h, and xij be a

generic point in Th.

• Uniform time grid: ∆t = T/NT , tn = n∆t.
• The values of u and m at (xi,j, tn) are resp. approximated by U n

i,j and

M n

i,j. 12

Notation:

• The discrete Laplace operator:

(∆hW)i,j = − 1 h2 (4Wi,j − Wi+1,j − Wi−1,j − Wi,j+1 − Wi,j−1).

• Right-sided finite difference formulas for ∂w

∂x1 (xi,j) and ∂w ∂x2 (xi,j):

(D+

1 W)i,j = Wi+1,j − Wi,j

h , and (D+

2 W)i,j = Wi,j+1 − Wi,j

h .

• The set of 4 finite difference formulas at xi,j:

[DhW]i,j =

• (D+

1 W)i,j, (D+ 1 W)i−1,j, (D+ 2 W)i,j, (D+ 2 W)i,j−1

• .

13

Discrete HJB equation ∂u ∂t − ν∆u + H(x, ∇u) = V [m] ↓ U n+1

i,j

− U n

i,j

∆t − ν(∆hU n+1)i,j + g(xi,j, [DhU n+1]i,j) = (Vh[M n])i,j

• g(xi,j, [DhU n+1]i,j)

=g “ xi,j, (D+

1 U n+1)i,j, (D+ 1 U n+1)i−1,j, (D+ 2 U n+1)i,j, (D+ 2 U n+1)i,j−1

” ,

• for instance,

(Vh[M])i,j = V [mh](xi,j), calling mh the piecewise constant function on T taking the value Mi,j in the square |x − xi,j|∞ ≤ h/2.

14

Classical assumptions on the discrete Hamiltonian g (q1, q2, q3, q4) → g (x, q1, q2, q3, q4) .

• Monotonicity: g is nonincreasing with respect to q1 and q3 and

nondecreasing with respect to to q2 and q4.

• Consistency:

g (x, q1, q1, q3, q3) = H(x, q), ∀x ∈ T, ∀q = (q1, q3) ∈ R2.

• Differentiability: g is of class C1, and
• ∂g

∂x

• x, (q1, q2, q3, q4)
• ≤ C(1 + |q1| + |q2| + |q3| + |q4|).
• Convexity: (q1, q2, q3, q4) → g (x, q1, q2, q3, q4) is convex.

15

The discrete version of ∂m ∂t + ν∆m + div

• m∂H

∂p (x, ∇v)

• = 0.

(†) It is chosen so that

• each time step leads to a linear system with a matrix

– whose diagonal coefficients are negative, – whose off-diagonal coefficients are nonnegative, in order to hopefully use some discrete maximum principle.

• The argument for uniqueness should hold in the discrete case, so the

discrete Hamiltonian g should be used for (†) as well.

16

Principle Discretize −

• T

div

• m∂H

∂p (x, ∇u)

• w =
• T

m∂H ∂p (x, ∇u) · ∇w by −h2

i,j

Bi,j(U, M)Wi,j := h2

i,j

Mi,j∇qg(xi,j, [DhU]i,j) · [DhW]i,j, which leads to

Bi,j(U, M) = 1 h B B B B B B B B @ B @ Mi,j ∂g ∂q1 (xi,j, [DhU]i,j) − Mi−1,j ∂g ∂q1 (xi−1,j, [DhU]i−1,j) +Mi+1,j ∂g ∂q2 (xi+1,j, [DhU]i+1,j) − Mi,j ∂g ∂q2 (xi,j, [DhU]i,j) 1 C A + B @ Mi,j ∂g ∂q3 (xi,j, [DhU]i,j) − Mi,j−1 ∂g ∂q3 (xi,j−1, [DhU]i,j−1) +Mi,j+1 ∂g ∂q4 (xi,j+1, [DhU]i,j+1) − Mi,j ∂g ∂q4 (xi,j, [DhU]i,j) 1 C A 1 C C C C C C C C A

17

This yields the semi-implicit scheme:        U n+1

i,j

− U n

i,j

∆t − ν(∆hU n+1)i,j + g(xi,j, [DhU n+1]i,j) = (Vh[M n])i,j M n+1

i,j

− M n

i,j

∆t + ν(∆hM n)i,j + Bi,j(U n+1, M n) = 0 = 0 It is important to see that the operator M → ν(∆hM)i,j + Bi,j(U, M) is the adjoint of the linearized version of U → ν(∆hU)i,j − g(xi,j, [DhU]i,j). The discrete system has the same structure as the continuous MFG system

18

Classical discrete Hamiltonians g can be chosen. For example, if the Hamiltonian is of the form H(x, ∇u) = ψ(x, |∇u|), a possible choice is the Godunov scheme g(x, q1, q2, q3, q4) = ψ

• x,
• min(q1, 0)2 + max(q2, 0)2 + min(q3, 0)2 + max(q4, 0)2
• .

If ψ(x, w) is convex and nondecreasing w.r.t. w, then g is a convex function

• f (q1, q2, q3, q4); g is nonincreasing w.r.t. q1 and q3 and nondecreasing

w.r.t. q2 and q4. Finally, it can be proven that the global scheme is consistent if H is smooth enough.

19

Existence for the discrete problem Theorem Assume that M NT ≥ 0 and that h2

i,j M NT i,j = 1. Under the

assumptions above on V , V0 and g, the discrete problem has a solution and there is a Lipschitz estimate on U n

h uniform in n, h and ∆t.

Strategy of proof K =   (Mi,j)0≤i,j<N : h2

i,j

Mi,j = 1, Mi,j ≥ 0    . Apply Brouwer fixed point theorem to a well chosen mapping χ : KNT − → KNT , (M n)n → (U n)n → (M n)n.

20

Proof: a fixed point method in KNT , Step 1: a map Φ : (M n)n → (U n)n. Given (M NT

i,j ), define the map Φ: (M n)0≤n<NT ∈ KNT → (U n)0≤n≤NT :

     U n+1

i,j

− U n

i,j

∆t − ν(∆hU n+1)i,j + g(xi,j, [DhU n+1]i,j) = (Vh[M n])i,j , U 0

i,j = V0[m0 h](xi,j).

• Existence is classical: (Leray-Schauder fixed point theorem at each

time step, making use of the monotonicity of g, the uniform boundedness assumption on V and of H(·, 0)).

• Uniqueness stems from the monotonicity of g.

21

Step 2: estimates

• There exists a constant C independent of (M n)n and h s.t.

U n∞ ≤ C(1 + T).

• The map Φ is continuous, from the continuity of V and well known

results on continuous dependence on the data for monotone schemes.

• There exists a constant L independent of (M n)n and h s.t.

DhU n∞ ≤ LT, ∀n, proved by using the assumption

• ∂g

∂x(x, q1, q2, q3, q4)

• ≤ C(1 + |q1| + |q2| + |q3| + |q4|).

22

Step 3: A map χ : (M n)n → ( M n)n

• Choose a positive constant µ > 0 large enough.
• For (U n)n = Φ((M n)n), backward linear parabolic problem for

M n:

8 > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > : f M NT = M NT , −µM n

i,j =

f M n+1

i,j

− f M n

i,j

∆t − ν(∆h f M n)i,j − µf M n

i,j

+ 1 h B @ f M n

i,j

∂g ∂q1 (xi,j, [DhU n+1]i,j) − f M n

i−1,j

∂g ∂q1 (xi−1,j, [DhU n+1]i−1,j) +f M n

i+1,j

∂g ∂q2 (xi+1,j, [DhU n+1]i+1,j) − f M n

i,j

∂g ∂q2 (xi,j, [DhU n+1]i,j) 1 C A + 1 h B @ f M n

i,j

∂g ∂q3 (xi,j, [DhU n+1]i,j) − f M n

i,j−1

∂g ∂q3 (xi,j−1, [DhU n+1]i,j−1) +f M n

i,j+1

∂g ∂q4 (xi,j+1, [DhU n+1]i,j+1) − f M n

i,j

∂g ∂q4 (xi,j, [DhU n+1]i,j) 1 C A

23

From the previous estimates on (U n)n, one can find µ large enough and independent of (M n)n such that the iteration matrix is the opposite of a M-matrix, thus there is a discrete maximum principle. Therefore, there exists a unique solution ( M n)n. Moreover, M n ≥ 0 ⇒

• M n ≥ 0,

∀n, h2

i,j M n = 1

⇒ h2

i,j

M n = 1, ∀n. Thus M n ∈ K for all n. Define the map χ : KNT → KNT , (M n)0≤n<NT → ( M n)0≤n<NT

24

Step 4: existence of a fixed point of χ From the boundedness and continuity of the mapping Φ, and from the fact that g is C1, we obtain that χ : KNT → KNT is continuous. From Brouwer fixed point theorem, χ has a fixed point, which yields a solution of the full system.

25

Uniqueness Theorem Same assumptions as above on V , V0, H and g. Assume also that the operators Vh and V0,h are strictly monotone, i.e.

• Vh[M] − Vh[

M], M − M

• 2 ≤ 0

⇒ Vh[M] = Vh[ M],

• V0,h[M] − V0,h[

M], M − M

• 2 ≤ 0

⇒ V0,h[M] = V0,h[ M]. The discrete problem has a unique solution. Proof The choice of the scheme makes it possible to mimic the proof used in the continuous case: uses the convexity and monotonicity assumptions on g.

26

An example ν = 1, T = 1, m(T) = 1 H(x, p) = sin(2πx2) + sin(2πx1) + cos(4πx1) + |p|2, F(x, m) = m2, V0[m](x) = m2 + cos(πx1) cos(πx2).

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 3.89e-16

• 0.2
• 0.4
• 0.6
• 0.8
• 1
• 1.2
• 1.4
• 1.6
• 1.8
• 2
• 2.2
• 2.4
• 2.6
• 2.8

The potential H(x, 0) = sin(2πx2) + sin(2πx1) + cos(4πx1).

27

Evolution of m(top) and u(bottom) Snapshots at t = (0, 4, 8, 100, 180, 190, 196, 200)/200

28

Comparison with the solution of the infinite horizon MFG system The solution around t = T/2 is very close to the solution of the infinite horizon MFG system

0.05 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 1.73e-18

• 0.005
• 0.01
• 0.015
• 0.02
• 0.025
• 0.03
• 0.035

1.08 1.07 1.06 1.05 1.04 1.03 1.02 1.01 1 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.9

29

The infinite horizon MFG system Find (u, m, λ ∈ R) such that              − ν∆u + H(x, ∇u) + λ = V [m], − ν∆m − div

• m∂H

∂p (x, ∇u)

• = 0,
• T

udx = 0,

• T

mdx = 1, and m > 0 in T.

30

A remark about the infinite horizon MFG system The Hamiltonian is of the form H(x, p) = |p|2 + g(x). In such cases, the infinite horizon MFG system is equivalent to a generalized Hartree equation: −ν2∆φ − gφ + φV [φ2] = λφ, in T2, and

• T2 φ2 = 1,

where φ(x) = φ0 exp (−u(x)/ν) and taking m = φ2, and the constant φ0 is fixed by the equation

• T2 log(φ/φ0) = 0. As a consequence, m can be

written as a function of u. This gives a way to test the accuracy of the scheme.

31

Order of the scheme

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 relative error 100 * h Err vs. h

32

Same test except ν = 0.01, ∆t = 1/200.

33

Evolution of m(top) and u(bottom) Snapshots at t = (0, 4, 8, 100, 180, 190, 196, 200)/200

34

The solution of the infinite horizon problem

0.15 0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01

• 0.01

1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

ν = 0.01, left: u, right m. Note that the supports of ∇u and of m tend to be disjoint as ν → 0.

35

A case when V has the “bad” monotonicity ν = 0.2, V (m) = − log(m) Note that near t ∼ T/2, the population concentrates (fashion effect).

36

Infinite horizon and a nonlocal operator V

0.3 0.25 0.2 0.15 0.1 0.05

• 1.39e-17
• 0.05
• 0.1
• 0.15
• 0.2

7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5

ν = 0.1, H(x, p) = sin(2πx2) + sin(2πx1) + cos(4πx1) + |p|3/2, V [m] = 200(1 − ∆)−1(1 − ∆)−1m left: u, right m.

37

"u.gp" "m.gp"

ν = 0.001, H(x, p) = sin(2πx2) + sin(2πx1) + cos(4πx1) + |p|2, V [m] = (1 − ∆)−1(1 − ∆)−1m left: u, right m.

38

About the solvers (Y. A, V. Perez, to appear in Network and Heterogeneous Media)

• Newton method

– linearized discrete MFG systems : well-posed if M > 0, which is not sure. Hence, breakdowns of the Newton method may occur. – Careful initial guess avoids breakdown. – Initial guesses: continuation method, by decreasing ν progressively.

• Solvers for linearized discrete MFG systems:

– Due to the forward-backward structure, marching in time is not possible. – Preconditioned iterative method for the whole system in (U, M) – A good understanding of the PDE system and multigrid lead to solvers with optimal linear complexity.

39

• III. Optimal planning with MFG

(Y.A, F. Camilli, I. Capuzzo Dolcetta, to appear in SIAM J. Control and Optimization)      ∂u ∂t − ν∆u + H(x, ∇u) = V (m(x)), in (0, T) × T, ∂m ∂t + ν∆m + div

• m∂H

∂p (x, ∇u)

• =

0, in (0, T) × T, with the initial and terminal conditions m(0, x) = m0(x), m(T, x) = mT (x), in T, and m ≥ 0,

• T

m(t, x)dx = 1.

40

Existence results (P-L. Lions)

• Ok if ν = 0, if H coercive, if V is a strictly increasing function and if

m0 and mT are smooth positive functions. Principle of the (difficult) proof: eliminate m (thanks to the HJB equation) and get a boundary value problem for u with a strictly elliptic quasilinear second order PDE, and nonlinear Fourier conditions.

• OK if ν = 0, if V = 0 (optimal transport) and if m0 and mT are

smooth positive functions.

• Ok if ν > 0 and if H(p) = c|p|2, if V is a smooth and bounded

function and if m0 and mT are smooth positive functions. Principle of the proof: use a clever change of unknowns: φ = exp(−u) and χ = m/φ.

• If ν > 0 and H(p) = c|p|2 ?
• Non-existence if H is sublinear, m0 = mT and T small enough.

41

Optimal control (on PDEs) approach Assumption: V = W ′ where W : R → R is a strictly convex function. H is a smooth Hamiltonian (convex): H(x, p) = sup

γ∈Rd (p · γ − L(x, γ)),

where L is a strictly convex function and lim

|γ|→∞ inf x L(x, γ)/|γ| = +∞

A weak form of the MFG system can be found by considering the problem

• f optimal control on PDEs:

minimize (m, γ) → T

• T
• m(t, x) L(x, γ(t, x))+W(m(t, x))
• dt dx,

subject to the constraints        ∂tm + ν∆m + div(m γ) = 0, in (0, T) × T, m(T, x) = mT (x) in T, m(0, x) = m0(x) in T.

42

Making the contraints linear by the change of variable z = mγ Introduce Φ(x, m, z) = sup

p∈Rd

• p · z − mH(x, p)
• for

x ∈ T, m ≥ 0, z ∈ Rd. Φ is convex and LSC w.r.t. (m, z) and        m > 0 ⇒ Φ(x, m, z) = mL(x, z

m),

z = 0 ⇒ Φ(x, m, z) ≤ mL(x, 0), m = 0, z = 0 ⇒ Φ(x, m, z) = +∞.

43

The new problem is to minimize over vector fields z = z(t, x) the convex functional T

• T
• Φ(x, m(t, x), z(t, x)) + W(m(t, x))
• dt dx

subject to the linear constraints              ∂tm + ∆m + div z = 0, in (0, T) × T, m ≥ 0, in (0, T) × T, m(T, x) = mT (x), in T, m(0, x) = m0(x), in T. The set defined by the constraints is convex and nonempty if m0 > 0 and if m0 and mT are smooth enough.

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A more rigorous setting

• Call χ the indicator function of the set R+.
• Call the convex and continuous functional Θ:

Θ : C([0, T] × T) ×

• C([0, T] × T)

d → R ∪ {+∞} (α, β) → T

• T
• W + χ

∗(α + H(β)) (

• W + χ

∗ is convex, continuous and non decreasing).

• Let M = M
• [0, T] × T
• denote the set of Radon measures. The

Legendre-Fenchel transform of Θ is Θ∗ : M ×

• M

d → R ∪ {+∞} (m, z) → sup

α,β

• m, α + z, β −

T

• T
• W + χ

∗(α + H(β))

• .

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The primal problem is to minimize Θ∗(m, z) on M ×

• M

d subject to the linear constraints              ∂tm + ∆m + div z = 0, in (0, T) × T, m ≥ 0, in (0, T) × T, m(T, x) = mT (x), in T, m(0, x) = m0(x), in T.

46

Convex programming The primal problem can be written: minimize Θ∗(m, z) + Σ∗(−m, −z),

• Θ and Σ are convex and resp. continuous, LSC and there exists (˜

α, ˜ β) such that Σ(˜ α, ˜ β) < +∞.

• Θ∗ and Σ∗ are convex and LSC and there exists (

m, z) such that    Θ∗( m, z) < +∞, Σ∗(− m, − z) < +∞, Θ∗ is continuous near m, z. (comes from the monotonicity and coercivity of the Hamiltonian). Fenchel-Rockafeller duality theorem There exists a saddle point: min (Θ + Σ) = − min

• Θ∗(m, z) + Σ∗(−m, −z)
• .

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Optimality conditions for the saddle point

• If m > 0, we recover the MFG system of PDEs.
• In the continuous setting, we were not able to prove that m > 0.
• Discrete problem: same programm, but it is possible to prove that

m > 0.

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The semi-implicit scheme                  U n+1

i,j

− U n

i,j

∆t − ν(∆hU n+1)i,j + g(xi,j, [DhU n+1]i,j) = V (M n

i,j),

M n+1

i,j

− M n

i,j

∆t + ν(∆hM n)i,j + Bi,j(U n+1, M n) = 0, M n ∈ K, M NT

i,j = (mT )i,j,

M 0

i,j

= (m0)i,j. Convex programming yields the existence of (M, U) under rather general assumptions.

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Existence of (M, U) via convex programming Theorem: If

• V = W ′ where W is a strictly convex and coercive C2 function.
• The discrete Hamiltonian is convex and coercive:

lim

q1→−∞

g(x, q1, q2, q3, q4) |q1| = lim

q2→+∞

g(x, q1, q2, q3, q4) q2 = +∞, lim

q3→−∞

g(x, q1, q2, q3, q4) |q3| = lim

q4→+∞

g(x, q1, q2, q3, q4) q4 = +∞.

• m0, mT ∈ K with (m0)i,j > 0,

then a solution of the discrete MFG system can be found by solving a saddle-point problem. The primal problem is the discrete analogue of the

• ptimal control of pdes problem above.

Moreover M is unique (same usual proof).

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Open question For non quadratic Hamiltonian, find bounds on M and U independent of h, ∆t.

51

A penalized scheme              U ǫ,n+1

i,j

− U ǫ,n

i,j

∆t − ν(∆hU ǫ,n+1)i,j + g(xi,j, [DhU ǫ,n+1]i,j) = V

• M ǫ,n

i,j

• ,

M ǫ,n+1

i,j

− M ǫ,n

i,j

∆t + ν(∆hM ǫ,n)i,j + Bi,j(U ǫ,n+1, M ǫ,n) = 0, M ǫ,n ∈ K, with the final time and initial time conditions U ǫ,0

i,j = 1

ǫ (M ǫ,0

i,j − (m0)i,j),

M ǫ,NT

i,j

= (mT )i,j, ∀ 0 ≤ i, j < Nh. Theorem As ǫ → 0, M ǫ → M, given by the discrete MFG system.

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T = 1, ν = 1, V (m) = m2, H(p) = sin(2πx2) + sin(2πx1) + cos(4πx1) + |p|2

Snapshots at t = (0, 4, 8, 100, 180, 190, 196, 200)/200

53

T = 0.01

Snapshots at t = (0, 4, 8, 100, 180, 190, 196, 200)/200

54

T = 0.01, ν = 0.1, H(p) = sin(2πx2) + sin(2πx1) + cos(4πx1) + |p|3

Snapshots at t = (0, 4, 8, 100, 180, 190, 196, 200)/200

55

T = 0.1, ν = 0.125, V (m) = − log(m)

Snapshots at t = (0, 4, 8, 100, 180, 190, 196, 200)/200

56