Numerical methods for Mean Field Games: additional material Y. - PowerPoint PPT Presentation

Numerical methods for Mean Field Games: additional material Y. Achdou (LJLL, Universit´ e Paris-Diderot) July, 2017 — Luminy Y. Achdou Numerical methods for MFGs

Convergence results Outline 1 Convergence results 2 Variational MFGs 3 A numerical simulation at the deterministic limit 4 Applications to crowd motion 5 MFG with 2 populations Y. Achdou Numerical methods for MFGs

Convergence results A first kind of convergence result: convergence to classical solutions Assumptions ν > 0 u | t = T = u T , m | t =0 = m 0 , u T and m 0 are smooth 0 < m 0 ≤ m 0 ( x ) ≤ m 0 Y. Achdou Numerical methods for MFGs

Convergence results A first kind of convergence result: convergence to classical solutions Assumptions ν > 0 u | t = T = u T , m | t =0 = m 0 , u T and m 0 are smooth 0 < m 0 ≤ m 0 ( x ) ≤ m 0 The Hamiltonian is of the form H ( x, ∇ u ) = H ( x ) + |∇ u | β where β > 1 and H is a smooth function. For the discrete Hamiltonian: upwind scheme. Y. Achdou Numerical methods for MFGs

Convergence results A first kind of convergence result: convergence to classical solutions Assumptions ν > 0 u | t = T = u T , m | t =0 = m 0 , u T and m 0 are smooth 0 < m 0 ≤ m 0 ( x ) ≤ m 0 The Hamiltonian is of the form H ( x, ∇ u ) = H ( x ) + |∇ u | β where β > 1 and H is a smooth function. For the discrete Hamiltonian: upwind scheme. Local coupling: the cost term is C 1 F [ m ]( x ) = f ( m ( x )) f : R + → R , There exist constants c 1 > 0, γ > 1 and c 2 ≥ 0 s.t. mf ( m ) ≥ c 1 | f ( m ) | γ − c 2 ∀ m Strong monotonicity: there exist positive constants c 3 , η 1 and η 2 < 1 s.t. f ′ ( m ) ≥ c 3 min( m η 1 , m − η 2 ) ∀ m Y. Achdou Numerical methods for MFGs

Convergence results A first kind of convergence result: convergence to classical solutions Theorem Assume that the MFG system of pdes has a unique classical solution ( u, m ) . Let u h (resp. m h ) be the piecewise trilinear function in C ([0 , T ] × Ω) obtained by interpolating the values u n i (resp m n i ) of the solution of the discrete MFG system at the nodes of the space-time grid. � � lim � u − u h � L β (0 ,T ; W 1 ,β (Ω)) + � m − m h � L 2 − η 2 ((0 ,T ) × Ω) = 0 h, ∆ t → 0 Y. Achdou Numerical methods for MFGs

Convergence results A first kind of convergence result: convergence to classical solutions Theorem Assume that the MFG system of pdes has a unique classical solution ( u, m ) . Let u h (resp. m h ) be the piecewise trilinear function in C ([0 , T ] × Ω) obtained by interpolating the values u n i (resp m n i ) of the solution of the discrete MFG system at the nodes of the space-time grid. � � lim � u − u h � L β (0 ,T ; W 1 ,β (Ω)) + � m − m h � L 2 − η 2 ((0 ,T ) × Ω) = 0 h, ∆ t → 0 Main steps in the proof 1 Obtain energy estimates on the solution of the discrete problem, in particular on � f ( m h ) � L γ ((0 ,T ) × Ω) Y. Achdou Numerical methods for MFGs

Convergence results A first kind of convergence result: convergence to classical solutions Theorem Assume that the MFG system of pdes has a unique classical solution ( u, m ) . Let u h (resp. m h ) be the piecewise trilinear function in C ([0 , T ] × Ω) obtained by interpolating the values u n i (resp m n i ) of the solution of the discrete MFG system at the nodes of the space-time grid. � � lim � u − u h � L β (0 ,T ; W 1 ,β (Ω)) + � m − m h � L 2 − η 2 ((0 ,T ) × Ω) = 0 h, ∆ t → 0 Main steps in the proof 1 Obtain energy estimates on the solution of the discrete problem, in particular on � f ( m h ) � L γ ((0 ,T ) × Ω) Plug the solution of the system of pdes into the numerical scheme, take 2 advantage of the consistency and stability of the scheme and prove that �∇ u − ∇ u h � L β ((0 ,T ) × Ω) and � m − m h � L 2 − η 2 ((0 ,T ) × Ω) converge to 0 Y. Achdou Numerical methods for MFGs

Convergence results A first kind of convergence result: convergence to classical solutions Theorem Assume that the MFG system of pdes has a unique classical solution ( u, m ) . Let u h (resp. m h ) be the piecewise trilinear function in C ([0 , T ] × Ω) obtained by interpolating the values u n i (resp m n i ) of the solution of the discrete MFG system at the nodes of the space-time grid. � � lim � u − u h � L β (0 ,T ; W 1 ,β (Ω)) + � m − m h � L 2 − η 2 ((0 ,T ) × Ω) = 0 h, ∆ t → 0 Main steps in the proof 1 Obtain energy estimates on the solution of the discrete problem, in particular on � f ( m h ) � L γ ((0 ,T ) × Ω) Plug the solution of the system of pdes into the numerical scheme, take 2 advantage of the consistency and stability of the scheme and prove that �∇ u − ∇ u h � L β ((0 ,T ) × Ω) and � m − m h � L 2 − η 2 ((0 ,T ) × Ω) converge to 0 To get the full convergence for u , one has to pass to the limit in the Bellman 3 equation. To pass to the limit in the term f ( m h ), use the equiintegrability of f ( m h ). Y. Achdou Numerical methods for MFGs

Convergence results A second kind of convergence result: convergence to weak solutions  − ∂u   ∂t − ν ∆ u + H ( x, ∇ u ) = f ( m ) in [0 , T ) × Ω    � �  ∂m m ∂H ∂t − ν ∆ m − div ∂p ( x, Du ) = 0 in (0 , T ] × Ω (MFG)     u | t = T ( x ) = u T ( x )   m | t =0 ( x ) = m 0 ( x ) Assumptions discrete Hamiltonian g : consistency, monotonicity, convexity growth conditions : there exist positive constants c 1 , c 2 , c 3 , c 4 such that c 1 | g q ( x, q ) | 2 − c 2 , g q ( x, q ) · q − g ( x, q ) ≥ | g q ( x, q ) | ≤ c 3 | q | + c 4 . f is continuous and bounded from below u T continuous, m 0 bounded Y. Achdou Numerical methods for MFGs

Convergence results A second kind of convergence result: convergence to weak solutions Theorem (stated in the case d = 2 ) Let u h, ∆ t , m h, ∆ t be the piecewise constant functions which take the values u n +1 i and m n i , respectively, in ( t n , t n +1 ) × ( ih − h/ 2 , ih + h/ 2) . There exist functions ˜ u , ˜ m such that m in L β ( Q ) 1 after the extraction of a subsequence, u h, ∆ t → ˜ u and m h, ∆ t → ˜ for all β ∈ [1 , 2) m belong to L α (0 , T ; W 1 ,α (Ω)) for any α ∈ [1 , 4 2 ˜ u and ˜ 3 ) 3 (˜ u, ˜ m ) is a weak solution to the system (MFG) in the following sense: 1 u ) ∈ L 1 ( Q ) , m ) ∈ L 1 ( Q ) , H ( · , D ˜ mf ( ˜ ˜ � � ∈ L 1 ( Q ) m ˜ H p ( · , D ˜ u ) · D ˜ u − H ( · , D ˜ u ) 2 (˜ u, ˜ m ) satisfies the system (MFG) in the sense of distributions m ∈ C 0 ([0 , T ]; L 1 (Ω)) and ˜ 3 u, ˜ ˜ u | t = T = u T , ˜ m | t =0 = m 0 . Y. Achdou Numerical methods for MFGs

Variational MFGs Outline 1 Convergence results 2 Variational MFGs 3 A numerical simulation at the deterministic limit 4 Applications to crowd motion 5 MFG with 2 populations Y. Achdou Numerical methods for MFGs

Variational MFGs An optimal control problem driven by a PDE Assumptions Φ : L 2 ( Q ) → R , C 1 , strictly convex. Set f [ m ] = ∇ Φ[ m ]. Ψ : L 2 (Ω) → R , C 1 , strictly convex. Set g [ m ] = ∇ Ψ[ m ]. Running cost: L : Ω × R d → R , C 1 convex and coercive. γ ∈ R d {− p · γ − L ( x, γ ) } , C 1 , coercive. Hamiltonian H ( x, p ) = sup L ( x, γ ) = sup p ∈ R d {− p · γ − H ( x, p ) } . Optimization problem: Minimize on ( m, γ ), m ∈ L 2 ( Q ), γ ∈ { L 2 ( Q ) } d : � T � � � J ( m, γ ) = Φ( m ) + m ( t, x ) L ( x, γ ( t, x ))) dxdt + Ψ( m ( T, · )) 0 Ω � ∂m ∂t − ν ∆ m + div( m γ ) = 0 , in (0 , T ) × Ω , subject to m (0 , x ) = m 0 ( x ) in Ω . Y. Achdou Numerical methods for MFGs

Variational MFGs An optimal control problem driven by a PDE The optimization problem is actually the minimization of a convex functional with linear constraints: Set  mL ( x, z m ) if m > 0 ,  � L ( x, m, z ) = 0 if m = 0 and z = 0 ,  + ∞ if m = 0 and z � = 0 . ( m, z ) �→ � L ( x, m, z ) is convex and LSC. The optimization problem can be written: � T � � � � m ∈ L 2 ( Q ) ,z ∈{ L 1 ( Q ) } d Φ[ m ] + inf L ( x, m ( t, x ) , z ( t, x )) dxdt + Ψ( m ( T, · )) 0 Ω subject to  ∂m   ∂t − ν ∆ m + div( z ) = 0 , in (0 , T ) × Ω , m ( T, x ) = m 0 ( x ) in Ω ,   m ≥ 0 . Y. Achdou Numerical methods for MFGs

Variational MFGs Optimality conditions (1/2) � ∂ t δm − ν ∆ δm + div( δm γ ) = − div( m δγ ) , in (0 , T ] × Ω , δγ �→ δm : δm (0 , x ) = 0 in Ω , � T � � � δJ ( m, γ ) = δm ( t, x ) L ( x, γ ( t, x )) + f [ m ]( t, x )) 0 Ω � T � � δγ ( t, x ) m ( t, x ) ∂L + ∂γ ( x, γ ( t, x )) + δm ( T, x ) g [ m ( T, · )]( x ) dx. 0 Ω Ω � − ∂u ∂t − ν ∆ u − γ · ∇ u = L ( x, γ ) + f [ m ]( t, x ) in [0 , T ) × Ω Adjoint problem u ( t = T ) = g [ m | t = T ] � T � � � δJ ( m, γ ) = u ( t, x ) ∂ t δm − ν ∆ δm + div( δm γ ) 0 Ω � T � m ( t, x ) δγ ( t, x ) ∂L + ∂γ ( x, γ ( t, x )) 0 Ω � ∂L � � T � = m ( t, x ) ∂γ ( x, γ ( t, x )) − ∇ u ( t, x ) δγ ( t, x ) 0 Ω Y. Achdou Numerical methods for MFGs