An ergodic problem for Mean Field Games: qualitative properties and numerical simulations
Simone Cacace Fabio Camilli Annalisa Cesaroni Claudio Marchi
math.AP




-1.5cm

An ergodic problem for Mean Field Games:
qualitative properties and numerical simulations
    Simone Cacace[1],   Fabio Camilli[2],   Annalisa Cesaroni[3]   and Claudio Marchi[4] 
    
=========================================================================================



[1]

 Dip. di Matematica e Fisica, Università  degli Studi Roma Tre, Largo San Leonardo Murialdo 1, 00146 Roma, Italy, 
 (e-mail: cacace@mat.uniroma3.it)

[2]

 Dip. di Scienze di Base e Applicate per l'Ingegneria,  “Sapienza" Università  di Roma, 
 via Scarpa 16, 00161 Roma, Italy, (e-mail:camilli@dmmm.uniroma1.it)

[3]

 Dip. di Scienze Statistiche, Università di Padova, via Cesare Battisti 241-243, 35121 Padova, Italy 
 (e-mail: annalisa.cesaroni@unipd.it)

[4]

 Dip. di Ingegneria dell'Informazione, Università di Padova, via Gradenigo 6/B, 35131 Padova, Italy 
 (e-mail: claudio.marchi@unipd.it)



This paper is devoted to some qualitative descriptions and some numerical results for ergodic Mean Field Games systems which arise, e.g., in the homogenization with a small noise limit. We shall consider either power type potentials or logarithmic type ones. In both cases, we shall establish some qualitative properties of the effective Hamiltonian H̅ and of the effective drift b̅. In particular we shall provide two cases where the effective system keeps/looses the Mean Field Games structure, namely where ∇_P H̅(P,α) coincides or not with b̅(P, α).

On the other hand, we shall provide some numerical tests validating the aforementioned qualitative properties of H̅ and b̅. In particular, we provide a numerical estimate of the discrepancy ∇_P H̅(P,α)-b̅(P, α).


 

   AMS subject classification:   35B27, 35B30, 35B40, 35K40, 35K59, 65M06, 91A13 

  
  Keywords:  mean field games, periodic homogenization, small noise limit, ergodic problems, continuous dependence of solution on parameters, finite difference schemes

 



§ INTRODUCTION
 
This paper is devoted to some qualitative descriptions and some numerical results of ergodic problems arising in the homogenization of Mean Field Games (briefly, MFG) in the small noise limit. Our starting point is the homogenization result stated in <cit.>. In that paper, the authors tackle the asymptotic behaviour as ϵ→ 0 of the solution to
 
    -u_t^ϵ-ϵΔ u^ϵ +1/2|∇ u^ϵ|^2=V(x/ϵ, m^ϵ),     x∈ℝ^n, t∈ (0,T)
    
    m^ϵ_t-ϵΔ m^ϵ- div(m^ϵ∇ u^ϵ)=0,    x∈ℝ^n, t∈ (0,T)
with initial/terminal condition u^0(x,T)=u_0(x) and m^0(x,0)=m_0(x) where the potential V(y,m) is 1-periodic in y, increasing in m, C^1 and bounded.
It turns out that (u^ϵ,m^ϵ) converge (in a suitable sense) to the solution of the effective Cauchy problem 

    -u_t^0+H̅(∇ u^0, m^0)=0,     x∈ℝ^n, t∈ (0,T)
    
    m^0_t- div(m^0b̅(∇ u^0, m^0))=0,    x∈ℝ^n, t∈ (0,T)

 with the same initial/terminal condition. The effective operators H̅(P,α) and b̅(P,α) are obtained as follows: for every  P∈ℝ^n and α≥ 0, the value H̅(P,α) is the (unique) constant for which there exists a solution to the ergodic MFG system:  

    (i)   -Δ u     +1/2|∇ u  +P|^2-V(y, α m  ) =
     H̅(P,α),     y∈𝕋^n
    
    (ii)   -Δ m  - div(m (∇ u +P))=0,     y∈𝕋^n
     (iii)  ∫_𝕋^n u =0     ∫_𝕋^n m =1,
 
while b̅ is given by  
    b̅(P,α)= ∫_𝕋^n (∇ u +P) m dy,
 where (u,m) is the  solution to (<ref>).
Moreover the following relation holds true

    ∂ H/∂ P_i(P,α) = b̅_i(P,α)-α∫_^nV_m(y,α m) m̃_i m dy

where V_m=∂ V/∂ m and (ũ_i,m̃_i, ∂ H/∂ P_i(P,α) )∈ C^2,γ× W^1,p× is the solution of the ergodic problem 

    (i)      -Δũ_i + ∇ũ_i· (∇ u+P) + (∇ u+P)· e_i -V_m(y,α m)αm̃_i=∂ H/∂ P_i(P,α) 
     (ii)   
    -Δm̃_i- div((P+∇ u)m̃_i ) = div ( m(∇ũ_i+ e_i)) 
    (iii)    ∫_𝕋^nm̃_i=∫_𝕋^nũ_i=0.
 

It is worth to observe that, for the homogenization of a MFG system with finite noise, the cell problem is decoupled, see <cit.>; hence, from a mathematical point of view, the most interesting case is the small noise one as in (<ref>).

Let us recall that the ergodic system (<ref>) is not related only to the homogenization problem; actually, it has its own interpretation as the MFG system when agents pay an infinite horizon cost (see <cit.>) and it arises also in the study of the asymptotic behaviour as t→+∞ of evolutive MFG system (see <cit.>). For a general overview on MFG systems and on their applications, we refer the reader to the monographs <cit.> and references therein.

The aim of this paper is to study the ergodic systems (<ref>) and (<ref>) under more general assumptions, especially we shall consider unbounded potential of the form


  * Power type nonlinearities: V(y,m)=v(y)+ m^q for some q>0;

  * Logarithmic nonlinearities: V(y,m)= v(y)+log m  
where v:^n→ is Lipschitz continuous and 1-periodic. Without any loss of generality, we shall assume that v≥ 0. 
Let us stress that our methods apply to regular solutions to (<ref>),  that is (u,m)∈ C^2,γ(^n)× W^1,p(^n), for all γ∈ (0,1) and for all p>1. As a matter of fact, by the regularity of the Hamiltonian, under our assumptions, the solution to (<ref>) belongs to C^2,γ(^n)× C^2,γ(^n), for all γ∈ (0,1).

An interesting feature is that the limit system (<ref>) may loose the MFG structure, namely ∇_P H̅(P,α) may not coincide 
with b̅(P, α). This feature was already pointed out in <cit.> for the potential V(y,m)=v(y)+m; in Corollary <ref> we shall extend 
it to more general cases of power type nonlinearities. On the other hand, in Theorem <ref> we shall show ∇_P H̅(P,α)=b̅(P, α)  
for MFG systems with logarithmic potentials. We will finally explore these results numerically, in particular by computing the discrepancy ∇_P H̅- b̅. 



§ MEAN FIELD GAMES WITH POWER NONLINEARITIES

In this section we consider potential of the form V(y,m)=v(y)+ m^q with q>0. 
In this case, the ergodic problem (<ref>) reads 

    (i) -Δ u +|∇ u+P|^2/2 - v(y)- α^q m^q=H̅(P, α),     y∈𝕋^n
    
    (ii) -Δ m- div  ( m (∇ u+P))=0,     y∈𝕋^n
     (iii)  ∫_𝕋^n u =0     ∫_𝕋^n m =1.
  
Throughout this section we shall assume

    q>0  if n≤ 4,     0<q≤2/n-2  if  n>4.

We collect some qualitative properties of the effective operators. 
 There hold 


  (i) For every P∈^n, α≥ 0 there exists a unique constant H̅(P,α) such that (<ref>) admits a solution (u,m). Moreover this solution is unique, 
(u,m)∈ C^2,γ(^n)× W^1,p(^n) for every γ∈ (0,1) and p>1 and m>0.

  (ii) H̅(P,α)  is   decreasing in α.

  (iii)H̅(P,α) is coercive in P, with quadratic growth, in particular there exists a constant R_q≥ 1 depending only on q such that 

    |P|^2/2-(1+R_q)∫_^n v(y)dy-R_qα^q≤H̅(P,α)≤|P|^2/2 -α^q.



  (iv) There hold

    lim_|P|→ +∞H̅(P, α)/|P|^2 =1/2    and    lim_|P|→ +∞|b̅(P, α)-P |/|P|=0,
 
locally uniformly for α∈ [0,+∞).
 

(i) This existence result can be found in <cit.>, <cit.>, see also <cit.> and <cit.>. 

(ii) Following the same argument as in  <cit.>
we get that for α_1, α_2≥ 0, 

    (H̅(P,α_1)-H̅(P,α_2))(α_1-α_2)=-∫_^n (m_1^q-m_2^q)(m_1-m_2)dy

where m_1 and m_2 are respectively the solution to (<ref>) with α=α_1 and α=α_2. 
Since q>0, we get the statement.

(iii)  We multiply equation (i) in (<ref>) by m, we integrate and we get, recalling the periodicity assumptions  and that m has mean 1, 

    H̅(P,α)=∫_^n -mΔ u +1/2 |∇ u +P|^2 m-α^q m^q+1 -v(y)mdy.

We multiply equation (ii) in (<ref>)  by u, integrate and subtract by the previous one to get

    H̅(P,α)=∫_^n1/2 (|P|^2-|∇ u|^2)m -α^q m^q+1 -v(y)mdy.

Therefore, recalling that v≥ 0, and that by Jensen inequality ∫_^n m^q dy≥(∫_^n mdy)^q=1,  we get 
H̅(P,α)≤|P|^2/2 -α^q. 

 Integrating the first equation in (<ref>), we obtain 

    H̅(P,α)   =   ∫_^n|∇ u+P|^2/2 - v(y)- α^q m^q dy
    
       =   |P|^2/2+∫_^n|∇ u|^2/2- v(y)- α^q m^q dy.

From this, using  (<ref>), recalling that m has mean 1, we get 

    ∫_^n1/2  |∇ u|^2 (m+1)+(m-1) (α^q m^q +v(y))dy=0.

By monotonicity of m↦α^qm^q on m≥ 0, we observe that 
    α^q m^q(m -1) ≥α^q (m  -1)

and then we conclude from (<ref>) that 

    ∫_^n1/2  |∇ u|^2 (m+1)dy≤ - ∫_^n (m-1) v(y)dy≤∫_^n  v(y)dy.

Integrating (<ref>), we get 
∫_^n m^qdy≤∫_^n m^q+1dy. Hence, in (<ref>), we obtain 

    0≤∫_^nα^q m^q+1-α^q m^q dy≤∫_^n v(y)(1-m) dy≤∫_^n v(y)dy.

Therefore, by applying Hölder inequality, we obtain

    α^q ∫_^n m^q dy≤α^q ∫_^n m^q+1 dy ≤α^q  (∫_^n m^q+1 dy )^q/q+1+  
     ∫_^n  v(y) dy.

Let  us observe that this inequality can be written as A≤ B+ A^q/q+1. So, if we choose R_q>1   such that R_q≥ 1+ R_q^q/q+1, 
then the solutions to the inequality satisfies A≤ R_q(B+1). 
Hence, we conclude that 
    α^q  ∫_^n  m^q≤α^q∫_^n  m^q+1≤ R_q( ∫_^n v(y) dy +α^q).

Replacing this inequality in (<ref>), we obtain 

    H̅(P,α) 
    ≥|P|^2/2-(1+R_q)∫_^n v(y)dy-R_qα^q.
 

(iv)  The first limit is a direct consequence of item (iii). 
By  definition (<ref>), we get 
    b̅(P, α)-P =∫_^n∇ u(y) m dy.
 
 By Hölder's  inequality and (<ref>) we get

    ∫_^n  |∇ u(y)| m dy≤(∫_^n  |∇ u |^2m dy )^1/2(∫_^n   m dy )^1/2≤(∫_^n v(y) dy )^1/2 .

This permits to conclude. 


Using the properties stated in Proposition <ref>, we obtain regularity properties of the effective operators, and a relation between them. 
 

The maps  (P,α)→H̅(P, α), b̅(P,α) are locally Lipschitz continuous and admit partial derivatives everywhere. 
Moreover, there hold

    ∇_P H̅(P,α)   =   b̅(P,α) -q/q+1α^q ∇_P  m_L^q+1^q+1
    ∂b̅/∂ P_i· e_i   ≥    0.

 

The proof of the local Lipschitz continuity of H̅ follows the same arguments of <cit.> by using the a priori bounds on m,∇ u obtained in the previous proof, in particular (<ref>) and (<ref>). 

We fix P∈^n, α>0; let (u,m) be  the solution to (<ref>) associated to (P,α).  
We introduce the following system for all i=1,…, n 

    (i)      -Δũ_i + ∇ũ_i· (∇ u+P) + (∇ u+P)· e_i -qα^q  m^q-1m̃_i=c_i(P,α) 
     (ii)   
    -Δm̃_i- div((P+∇ u)m̃_i ) = div (m(∇ũ_i+ e_i)) 
    
    (iii)   ∫_𝕋^nm̃_i=∫_𝕋^nũ_i=0.
 
Arguing as in <cit.>, we get that for every i there exists a unique c_i(P,α) such that this system admits a solution (ũ_i,m̃_i),
which is regular and unique. Moreover, the compatibility condition for existence of solutions reads

    c_i(P,α)= ∫_^n (∇ u+P)_i m- α^q q m^q m̃_i dy=b̅_i(P,α)-qα^q ∫_^n m^q m̃_i dy

where the latter equality is due to (<ref>).

We fix now δ∈ and define u_i^δ, m_i^δ the solution to (<ref>) associated to (P+δ e_i, α). Then it is possible to prove arguing as in <cit.> that u_i^δ-u/δ,m_i^δ-m/δ converges as δ→ 0 to a solution to (<ref>) in a suitable sense (weakly in W^1,2 for u_i^δ-u/δ and weakly in L^2 for m_i^δ-m/δ).
This implies by uniqueness that, up to subsequences,  c_i(P,α)=lim_δ→ 0H̅(P+δ e_i, α)-H̅(P,α)/δ. So relation (<ref>) is exactly the compatibility condition for existence of solutions to (<ref>). 

 
Following the same arguments as in <cit.>, we get that b̅ is locally Lipschitz continuous and moreover 

    ∂b̅/∂ P_i=e_i +∫_^nm̃_i ∇ u + m∇ũ_i dy.

 

We multiply (i) and (ii) in (<ref>)  respectively by m̃_i and by ũ_i, we subtract and we get

    ∫_^nm̃_i ∇ u · e_i   -qα^q  m^q-1m̃_i^2 - m(∇ũ_i +e_i)·∇ũ_i dy=0.

Therefore  

    ∂b̅/∂ P_i· e_i   =    1+∫_^nm̃_i ∇ u· e_i + m∇ũ_i · e_idy
       =   1+∫_^n qα^q  m^q-1m̃_i^2 +m(∇ũ_i +e_i)·∇ũ_i+ m∇ũ_i · e_idy
       =   1+ ∫_^n qα^q  m^q-1m̃_i^2 +m|∇ũ_i +e_i|^2- m dy
       =   ∫_^n qα^q  m^q-1m̃_i^2 +m|∇ũ_i +e_i|^2dy

and relation (<ref>) easily follows.
 
  
Under the assumption of Theorem <ref>, for n=1, there holds

    ∇_P H̅(P,α)≠b̅ (P,α).
 
 
  First of all we observe that by Jensen inequality m_L^q+1^q+1>1. 

Moreover, arguing as in <cit.> or as in <cit.> (see also <cit.>) and still using (<ref>), we have that √(m) is bounded in W^1,2(^n), uniformly in P. In other words, there exists a constant C>0 independent of P such that

    ∫_^n |∇√(m)|^2 dy
    ≤ C∫_^n v(y)dy.


So, since n=1, we conclude that m→m̅ strongly in L^q+1(^1). On the other side, reasoning as in <cit.> 
we can prove that m converges weakly to 1 in L^q+1. This implies that  m_L^q+1^q+1→ 1 as |P|→ +∞ and then in particular 
∇_P H̅(P,α)≠b̅ (P,α) at some (P,α). 
 


§ MEAN FIELD GAME WITH LOGARITHMIC  NONLINEARITIES

In this section we consider potential of the form V(y,m)=v(y)+log m. 
In this case, defining H̅(P) as 
    H̅(P, α)=H̅(P)+logα    ∀α>0
 the system (<ref>) becomes

    (i) -Δ u + |∇ u+P|^2/2 - v(y)-log m=H̅(P),     y∈𝕋^n
    
    (ii) -Δ m- div ( m (∇ u+P))=0,     y∈𝕋^n
     (iii)  ∫_𝕋^n u =0     ∫_𝕋^n m=1.
 

We  collect some qualitative properties of the effective operators. 
     

  (i) For every P∈^n,  there exists a unique constant H̅(P) such that (<ref>) admits a solution (u,m). Moreover this solution is unique, 
(u,m)∈ C^2,γ(^n)× W^1,p(^n) for every γ∈ (0,1) and p>1 and m>0.

  (ii)H̅(P) is coercive in P, with quadratic growth, in particular  there exists C>0 such that

    |P|^2/2-C-2∫_^n  v(y)dy ≤H̅(P)≤|P|^2/2.


  (iii) H̅(P) is strictly convex. 

  (iv) There hold 
    lim_|P|→ +∞H̅(P)/|P|^2 =1/2    and    lim_|P|→ +∞|b̅(P)-P |/|P|=0.
 
 
 

 Following the arguments in <cit.>, we introduce the following minimization problem. For any ϕ smooth and periodic we define the energy 
 
    E_P(ϕ)= ∫_^n e^-Δϕ+  |∇ϕ+P|^2/2 - v(y) dy
 and consider the minimization problem
 
    inf{E_P(ϕ) | ϕ smooth and periodic}.
 
 Note that inf E_P(ϕ)≤  E_P(0)≤ e^|P|^2/2. 
According to <cit.>, there exists a unique (up to constants) smooth minimizer  ϕ   for this problem. We define 

    m_ϕ(y):= e^-Δϕ+ |∇ϕ+P|^2/2 - v(y)/E_P(ϕ).



(i) By computing the Euler Lagrange equation associated to the minimization problem we get that   u=ϕ, m=m_ϕ are solution to (<ref>) with H̅ given by  

    H̅(P)=loginf{ E_P(ϕ) | ϕ smooth and periodic}.
  So, <cit.> gives the existence  and uniqueness of a regular solution to (<ref>) (see also <cit.>). 

(ii) Using the representation formula (<ref>) for H̅(P), we get that  H̅(P)≤log  E_P(0)≤|P|^2/2. 

 We multiply the first equation in (<ref>) by m, the second by u, integrate and subtract, and we obtain 

    H̅(P)=∫_^n1/2 (|P|^2-|∇ u|^2)m  -mlog m -v(y)mdy.

 Integrating the first equation in (<ref>), 
we obtain 

    H̅(P)= ∫_^n|∇ u+P|^2/2- log m  - v(y)dy=
     |P|^2/2+∫_^n|∇ u|^2/2-  log m - v(y) dy.

The last two equalities give
 
    ∫_^n1/2  |∇ u|^2 (m+1)+(m-1) v(y)+ (mlog m-log m)dy=0.

Note that mlog m≥log m, so we conclude that also in this case  (<ref>) holds.  Moreover, we observe that there exists C>0 such that mlog m≥ 2log m-C and then

    ∫_^n (m-1) v(y)+ log m dy-C≤∫_^n (m-1) v(y)+ (mlog m-log m)dy≤ 0.

This in turns gives 

    ∫_^nlog m dy ≤ C+∫_^n  v(y)dy.

So, again by (<ref>), we conclude H̅(P)≥|P|^2/2-C-2∫_^n  v(y)dy.

 (iii) The strict convexity is  a consequence of the representation formula (<ref>), see <cit.>. 
 
 (iv) The proof is the same as in Proposition <ref>. 

As before, using the properties stated in Proposition <ref>, we obtain regularity properties of the effective operators, and a relation between them. 
 
The maps  P→H̅(P), b̅(P ) are locally Lipschitz continuous and admit partial derivatives everywhere. 
Moreover, there hold

    ∇_P H̅(P)=  b̅(P).

 
 The regularity comes from the properties of H̅ proved in Proposition <ref>.

 We fix P∈^n; let (u,m) be the solution to (<ref>) associated to P.  
We introduce the following system for all i=1,…, n 

    (i)      -Δũ_i + ∇ũ_i· (∇ u+P) + (∇ u+P)· e_i -m̃_i/m=c_i(P) 
     (ii)   
    -Δm̃_i- div((P+∇ u)m̃_i ) = div (m(∇ũ_i+ e_i)) 
    
    (iii)   ∫_𝕋^nm̃_i=∫_𝕋^nũ_i=0.
 Note that by <cit.> 1/m∈ L^∞(^n).
Arguing as in <cit.>, we get that for every i there exists a unique c_i(P) for which this system admits a solution (ũ_i,m̃_i),
which is regular and unique. Moreover, the compatibility condition for existence of solutions reads

    c_i(P)= ∫_^n (∇ u+P)_i m- m̃_i dy =∫_^n (∇ u+P)_i m dy= b̅_i(P,α).

Moreover as in Theorem <ref>, we get that, up to subsequences,  c_i(P)=lim_δ→ 0H̅(P+δ e_i)-H̅(P)/δ. So relation (<ref>) is exactly the compatibility condition for existence of solutions to (<ref>). 
 



§ NUMERICAL TESTS

This section is devoted to the numerical solution of the cell problems presented in the previous sections, 
in particular we provide a numerical validation of the properties involving the effective Hamiltonian H̅ and the effective drift b̅. 
We recall that the computation of effective operators is an expensive task, since it requires the solution of a nonlinear problem for each pair (P,α).
To this end we apply the direct method we recently introduced in  <cit.>, 
an efficient solver for ergodic problems associated to the homogenization of Hamilton-Jacobi equations. 
The new method avoids completely classical approximations such as the small-delta or the large-time methods (see <cit.>). 
The ergodic constant H̅ appearing in the cell problem (<ref>) is handled as an additional unknown, so that the corresponding nonlinear system is 
formally overdetermined, having more equations than unknowns. 
The solution is then meant in a least-squares sense and computed by a generalized Newton method for inconsistent systems. 
We refer the reader to <cit.> for technical details and for the actual implementation of the method. Here we just remark that, 
after introducing a discretization of the torus ^n with N nodes, collecting the approximations U_i, M_i, H̅ (i=1,...,N) of the unknowns 
in a single vector X=(U,M,H̅) of length 2N+1, and choosing a scheme 
which is consistent with the notion of viscosity solutions, we end up with a nonlinear system of 2N+2 equations. 
The first 2N equations correspond to the partial differential equations in (<ref>), whereas the last 2 equations correspond to the normalization conditions 
∫_^n u=0 and ∫_^nm=1. Defining a suitable  non linear map F:^2N+1→^2N+2, the resolution of the cell problem is then reduced to 

    .

Note that, once a solution X is computed, we obtain the effective drift b̅ in (<ref>) by a simple trapezoidal quadrature rule. 
Moreover,  the system (<ref>) can be discretized  in a similar  way and its solution easily computed in a single iteration, being an overdetermined but linear system. 
Hence we get the solution  (Ũ,M̃,∂H̅/∂ P), from which we can readily compute the integral in (<ref>), 
again by a trapezoidal quadrature rule. This gives, in particular, a practical strategy to validate numerically the relationship described by  (<ref>).

Here we consider for simplicity the one dimensional case n=1, but the method can be easily implemented in any dimension. 
In all the tests we choose 

a nonnegative potential of the form v(x)=A(1+1/2(sin(2π x)+cos(4π x))) with A>0. 
We observe that, for small values of the amplitude A, the corresponding correctors u and m, i.e. solutions to (<ref>), are quite close to 0 and 1 
respectively, as shown in Figure <ref> in the case V(y,m)=v(y)+m with P=10 and α=2. 
This considerably affects the shape of the effective Hamiltonian and effective drift, 
making difficult to explore numerically the features we are interested in. That's why we enforce non trivial solutions by choosing A=100 in what follows.  



We start with the case of power nonlinearities, setting V(y,m)=v(y)+m^q, with q=1,2. We discretize the torus ^1 with N=400 
nodes and we consider a uniform grid of 51× 51 nodes for discretizing the space of parameters [-10,10]×[0,20]∋ (P,α). 

Figure <ref> 
shows the surfaces and the level sets of the computed effective Hamiltonian H̅(P,α) and the effective drift b̅(P,α) as functions of (P,α) for q=1, 
whereas Figure <ref> shows the case q=2. 


In both cases q=1,2 we observe that H̅(P,α) is “almost” quadratic in P and nonincreasing in α, while b̅(P,α) exhibits a cubic-like 
behavior for P close to zero. Let us take a closer look to some slices of these effective surfaces, in order to analyze their behavior for P large. 
We choose α=10 and we plot in Figure <ref> the graphs of H̅(P,α)/P^2 and |b̅(P,α)-P|/P as functions of P for q=1,2.
We clearly observe convergence to 1/2 and 0 respectively, as expected by Proposition <ref>-(iv). 




We proceed by examining the relationship between the effective operators described in the previous section.
First of all we want to show that, in the case of power nonlinearities, the structure of mean field game is lost during the homogenization process, namely we have 
∇_PH̅(P,α)≠b̅(P,α). To this end, we consider the residual in (<ref>) as a measure of this discrepancy:

    ℛ(P,α)=|α∫_^nV_m(y,α m) m̃ m dy| ,

where we recall that, for each (P,α), the auxiliary m̃ solves the linearized cell problem (<ref>), in which the previously computed 
correctors (u,m) appear as given data. 

Figure <ref> shows the residuals respectively for q=1 and q=2 on the space of parameters 
[-10,10]×[0,20], while Figure <ref> shows the behavior of the corresponding L^∞ norms under grid refinement. 
We clearly observe that, as N increases, the residual converges but not vanishes. 
Note that, in the case of power nonlinearities, the residual can be computed using also expression (<ref>), in which appears 
the derivative in P of the q+1 norm of the corrector m. Indeed, denoting by m_(P,α) the solution of (<ref>), 
we can approximate ∂/∂ P m_(P,α)_L^q+1^q+1 by finite differences of the form  
m_(P+δ,α)_L^q+1^q+1- m_(P,α)_L^q+1^q+1/δ for some small δ>0. This approach leads to similar results, 
up to an additional error of order 𝒪(δ). 







We evaluate the numerical error in the relationship (<ref>) by defining 


    ℰ_∞=
    ∂ H/∂ P(P,α) - b̅(P,α)+α∫_^nV_m(y,α m) m̃ m dy_∞ ,

Figure <ref> shows that, for both q=1 and q=2, the error ℰ_∞ has order of convergence 1, under grid refinement for a space step h=1/N. 



We now consider the logarithmic case, setting V(y,m)=v(y)+log m. We already know that in this case the dependency of H̅ on P and α is separated, 
however we solve the full cell problem (<ref>) and not the reduced one in (<ref>), in order to recover this feature numerically.    
Again, we discretize the torus ^1 with N=400 nodes and we consider a uniform grid of 
51× 51 nodes for discretizing the space of parameters [-10,10]×[1,20]∋ (P,α) (here we avoid the singularity of the log function just starting with α>0). 

Figure <ref> shows the surfaces and the level sets of the computed effective Hamiltonian H̅(P,α) and the effective drift b̅(P,α) as functions of (P,α). 
We observe in particular the convexity of H̅ in P and the independence of b̅ on α. 
We finally recall that in this case the residual ℛ(P,α)=|α∫_^nV_m(y,α m) m̃ m dy|=0, being V_m=1/m and m̃ with zero mean. 
Then, in Figure <ref> we show the error 
ℰ(P,α)=|∇_PH̅(P,α)-b̅(P,α)| and the behavior of its L^∞ norm ℰ_∞ under grid refinement for a space step h=1/N. 
We see that ℰ_∞ does not depend 
on α and it exihibits again an order of convergence 1 as the space step goes to zero. This confirm the relation ∇_PH̅(P,α)=b̅(P,α) in the logarithmic case and also the separated dependency 
of H̅ on P and α.





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