A variational finite volume scheme for Wasserstein gradient flows es - PowerPoint PPT Presentation

A variational finite volume scheme for Wasserstein gradient flows es 1 , T. O. Gallou¨ et 2 , G. Todeschi 2 C. Canc` Inria Lille, Rapsodi 1 Inria Paris, MOKAPLAN 2 ICODE, January 8, 2010

Wasserstein gradient flows • domain Ω ∈ R d convex, bounded open • energy E : L 1 ( Ω ; R + ) → [0 , + ∞ ], convex • ρ 0 ∈ L 1 ( Ω ; R + ), E ( ρ 0 ) < + ∞ ∂ t � − ∇ · ( � ∇ δ E in Q T = Ω × (0 , T ) , δρ [ � ]) = 0 � ∇ δ E δρ [ � ] · n = 0 on Σ T = ∂ Ω × (0 , T ) , � ( · , 0) = ρ 0 in Ω .

JKO scheme • τ time discretization step � ρ 0 τ = ρ 0 , ρ n 2 τ W 2 1 2 ( ρ , ρ n − 1 τ ∈ argmin ρ ) + E ( ρ ) . τ dynamical formulation � t n � 1 ρ | v | 2 d x d t + E ( ρ ( t n )) , inf ρ , v 2 t n − 1 Ω with the constraints ( ρ ≥ 0) in Ω × ( t n − 1 , t n ) , ∂ t ρ + ∇ · ( ρ v ) = 0 ρ v · n = 0 on ∂ Ω × ( t n − 1 , t n ) , ρ ( t n − 1 ) = ρ n − 1 in Ω . τ

Inf-Sup problem • m = ρ v • φ is the Lagrange multiplier for the continuity equation � t n � � t n � | m | 2 2 ρ d x d t + ( ρ∂ t φ + m · ∇ φ ) d x d t ρ , m sup inf t n − 1 t n − 1 φ Ω Ω � [ φ ( t n − 1 ) ρ n − 1 − φ ( t n ) ρ ( t n )] d x + E ( ρ ( t n )) . + τ Ω minimize in m , m = − ρ ∇ φ . � t n � ( ∂ t φ − 1 2 | ∇ φ | 2 ) ρ d x d t sup inf ρ t n − 1 φ Ω � [ φ ( t n − 1 ) ρ n − 1 − φ ( t n ) ρ ( t n )] d x + E ( ρ ( t n )) . + τ Ω

Dual problem dual problem � � � � φ ( t n − 1 ) ρ n − 1 E ( ρ ( t n )) − φ ( t n ) ρ ( t n ) d x d x + inf sup , τ ρ ( t n ) φ ( t n − 1 ) Ω Ω subject to the constraints − ∂ t φ + 1 2 | ∇ φ | 2 ≤ 0 in Ω × ( t n − 1 , t n ) , φ ( t n ) ≤ δ E δρ [ ρ ( t n )] in Ω , φ ( t n ) = δ E δρ [ ρ ( t n )] ρ ( t n ) a.e .

Saddle point • Monotonicity of the initial value of HJ (second membre – final condition) • Saturation of the inequalities Optimality conditions : ∂ t φ − 1 2 | ∇ φ | 2 = 0 , in Ω × ( t n − 1 , t n ) in Ω × ( t n − 1 , t n ) ∂ t ρ − ∇ · ( ρ ∇ φ ) = 0 , with ρ ( t n − 1 ) = ρ n − 1 , in Ω τ φ ( t n ) = δ E δρ [ ρ ( t n )] , in Ω

weighted H − 1 distance dissipation �� � 1 / 2 ρ ) = 1 ρ | v | 2 d x D ( ρ ; ˙ 2 inf v Ω with the constraints ρ + ∇ · ( ρ v ) = 0 ˙ in Ω ρ v · n = 0 on ∂ Ω duality D ( ρ ; ξ ) = ( D ∗ ( ρ ; · )) ∗ D ( ρ ; ρ − µ ) = 1 2 � ρ − µ � ˙ = H − 1 ρ �� � 1 / 2 1 ρ = 1 ρ | ∇ ψ | 2 d x = D ∗ ( ρ ; ψ ) 2 � ψ � ˙ H 1 2 Ω with ψ solution to ρ − µ − ∇ · ( ρ ∇ ψ ) = 0 in Ω , ∇ ψ · n = 0 on ∂ Ω .

Linearized inf-sup problem LJKO scheme � � 1 � 2 ρ n � ρ − ρ n − 1 τ ∈ argmin ρ ( Ω ) + E ( ρ ) , n ≥ 1 . H − 1 ˙ τ 2 τ ρ ∈ P ( Ω ) Change of variable ( ρ , ψ ) �→ ( ρ , m = − ρ ∇ ψ ) � � ρ − ρ n − 1 + ∇ · m = 0 | m | 2 in Ω , τ 2 τρ d x + E ( ρ ) , inf subject to: m · n = 0 ρ , m on ∂ Ω . Ω saddle point � � � | m | 2 2 τρ d x − ( ρ − ρ n − 1 ) φ d x + m · ∇ φ d x + E ( ρ ) , ρ , m sup inf τ φ Ω Ω Ω

Linearized optimality conditions saddle point � � ( − φ − τ ρ n − 1 φ d x + inf 2 | ∇ φ | 2 ) ρ d x + E ( ρ ) . sup τ ρ φ Ω Ω optimality conditions τ + τ τ | 2 = δ E φ n 2 | ∇ φ n δρ [ ρ n τ ] , ρ n τ − ρ n − 1 τ − ∇ · ( ρ n τ ∇ φ n τ ) = 0 , τ monotonicity of discrete HJ equation = ⇒ saturation constraints

Space discretization Classicale finite volume mesh (ex: Cartesian grids, Delaunay triangulations or Vorono¨ ı tessellations.) � � T , Σ , ( x K ) K ∈ T • triplet • cell K ∈ T measure m K > 0. • face σ ∈ Σ measure m σ = H d − 1 ( σ ) > 0. • K ∈ T , Σ K of Σ such that ∂ K = � σ ∈ Σ K σ , � K ∈ T Σ K = Σ . • cell-centers ( x K ) K ∈ T orthogonal to K | L face of K , L ∈ T , same orientation as n KL outward w.r.t. K . • Σ ext = { σ ⊂ ∂ Ω } are not involved (no boundary fluxes) • N K the neighboring cells of K • d σ = | x K − x L | , diamond cell ∆ σ , • measure m ∆ σ = m σ d σ / d , transitivity a σ = m σ / d σ

Upstream weighted dissipation potentials L 2 ( R T ) scalar product � 〈 h , φ 〉 T = ∀ h = ( h K ) K ∈ T , φ = ( φ K ) K ∈ T , h K φ K m K , K ∈ T 1 2 � φ � 2 ρ dissipation, ˙ H 1 � T ( ρ ; φ ) = 1 a σ ρ σ ( φ K − φ L ) 2 ≥ 0 , D ∗ 2 σ ∈ Σ σ = K | L � ρ K if φ K > φ L , ∀ σ = K | L ∈ Σ . ρ σ = ρ L if φ K < φ L , not symmetric D ∗ T ( ρ ; φ ) ∕ = D ∗ T ( ρ ; − φ )

Upstream weighted dissipation potentials II � h = ( h K ) K ∈ T ∈ R T � � � 〈 h , 1 〉 T = 0 R T 0 = � F = ( F K σ , F L σ ) σ = K | L ∈ Σ ∈ R 2 Σ � � � F T = � F K σ + F L σ = 0 . discrete dissipation � ( F σ ) 2 ∀ h ∈ R T D T ( ρ ; h ) = inf d σ m σ ≥ 0 , 0 , 2 ρ σ F σ ∈ Σ subject to (continuity equation) � h K m K = m σ F K σ , ∀ K ∈ T . σ ∈ Σ K � ( F σ ) 2 0 if F σ = 0 and ρ σ = 0 , = 2 ρ σ + ∞ if F σ > 0 and ρ σ = 0 , upwind choice ρ σ = ρ K if F K σ > 0 , ρ L if F L σ > 0 ,

Discrete duality duality 〈 h , φ 〉 T − D ∗ ∀ h ∈ R T D T ( ρ ; h ) = sup T ( ρ ; φ ) , 0 . φ T ( ρ ; φ ) = 1 D T ( ρ ; h ) = D ∗ 2 〈 h , φ 〉 T . with (identification) � a σ ρ σ ( φ K − φ L ) , ∀ K ∈ T , h K m K = σ ∈ Σ K σ = K | L or φ K − φ L F K σ = ρ σ , ∀ σ = K | L ∈ Σ . d σ

Discrete JKO � � � = ( ρ 0 + R T � 〈 ρ , 1 〉 T = 〈 ρ 0 , 1 〉 T P T = ρ ∈ R T 0 ) ∩ R T + . + convexity of ρ �→ D T ( ρ ; µ − ρ ) 〈 µ − ρ , φ 〉 T − D ∗ D T ( ρ ; µ − ρ ) = sup T ( ρ ; φ ) . φ discrete JKO 1 ρ n ∈ argmin τ D T ( ρ ; ρ n − 1 − ρ ) + E T ( ρ ) , n ≥ 1 . ρ ∈ P T direct existence uniqueness (of ρ n ) and energy estimates

Inf-Sup problem � ( F σ ) 2 1 ρ ∈ P T inf inf d σ m σ + E T ( ρ ) . F τ 2 ρ σ σ ∈ Σ φ Lagrange multiplier for � m K ( ρ n − 1 − ρ ) = m σ F K σ , ∀ K ∈ T . σ ∈ Σ K φ K − φ L minimize in F K σ , F K σ = ρ σ d σ � � � T − τ ρ n − 1 − ρ , φ a σ ρ σ ( φ K − φ L ) 2 + E T ( ρ ) . sup inf 2 ρ ≥ 0 φ σ ∈ Σ σ = K | L

Optimality conditions � � � T − τ ρ n − 1 − ρ , φ a σ ρ σ ( φ K − φ L ) 2 + E T ( ρ ) . sup inf 2 ρ ≥ 0 φ σ ∈ Σ σ = K | L Unique saddle point � � L ) + � 2 = ∂ E T K + τ m K φ n ( φ n K − φ n ( ρ n ) , a σ 2 ∂ρ K σ ∈ Σ K � ( ρ n K − ρ n − 1 a σ ρ n σ ( φ n K − φ n ) m K + τ L ) = 0 K σ ∈ Σ K up-winding leads saturation of the constraints

Monotonicity 2 | ∇ φ | 2 is monotone. the inverse of the operator φ �→ φ + τ � � ( φ K − φ L ) + � 2 , τ G K ( φ ) := φ K + ∀ K ∈ T . a σ 2 m K σ ∈ Σ K σ = K | L min φ implies | ∇ φ | 2 = 0 lemma f ∈ R T , there exists a unique solution to G ( φ ) = f , and it satisfies min f ≤ φ ≤ max f . let φ , � φ be the solutions corresponding to f and � f then f ≥ � φ ≥ � f = ⇒ φ .

Proof: f ≥ � f let K ∗ be the cell such that � � φ K ∗ − ˜ φ K − ˜ φ K ∗ = min φ K . K ∈ T φ K ∗ − ˜ φ K ∗ ≤ φ L − ˜ ⇒ φ K ∗ − φ L ≤ ˜ φ K ∗ − ˜ φ L = φ L � φ L ) + � 2 � � � ( φ K ∗ − φ L ) + � 2 ≤ τ τ (˜ φ K ∗ − ˜ a σ a σ . 2 m K 2 m K σ ∈ Σ K ∗ σ ∈ Σ K ∗ σ = K ∗ | L σ = K ∗ | L G K ∗ ( φ ) ≥ G K ∗ (˜ φ ) yields φ K ∗ ≥ ˜ φ K ∗ φ K ≥ ˜ φ K • uniqueness of the solution φ of G ( φ ) = f • maximum principle • Existence

Saturation of the constraints the inf-sup rewrites sup φ inf ρ ≥ 0 � � � T − τ ρ n − 1 − ρ , φ a σ ρ σ ( φ K − φ L ) 2 + E T ( ρ ) 2 σ ∈ Σ σ = K | L � � � � � ( φ K − φ L ) + � 2 T − τ ρ n − 1 − ρ , φ = E T ( ρ ) + a σ ρ K 2 σ ∈ Σ K K σ = K | L � � ρ n − 1 , φ = E T ( ρ ) + T − 〈 ρ , G ( φ ) 〉 T . at ρ n , φ n is optimal in � � E T ( ρ n ) + ρ n − 1 , φ T − 〈 ρ n , G ( φ ) 〉 T . sup φ

Energy estimates direct estimate E T ( ρ n ) + 1 τ D T ( ρ n ; ρ n − 1 − ρ n ) ≤ E T ( ρ n − 1 ) improved estimate � n � E T ( ρ n ) + τ D ∗ T ( ρ n ; φ n ) + τ D ∗ ρ n ; ˇ ≤ E T ( ρ n − 1 ) , ˇ φ T ρ n solution of classical backward Euler ˇ � 1 ∂ E T ρ n K − ρ n − 1 ρ n σ (ˇ φ n K − ˇ φ n φ n ˇ ρ n ) (ˇ ) m K + τ a σ ˇ L ) = 0 , K = (ˇ K m K ∂ρ K σ ∈ Σ K

Convergence � � L ) + � 2 = ∂ E T K + τ m K φ n ( φ n K − φ n ( ρ n ) , a σ 2 ∂ρ K σ ∈ Σ K � ( ρ n K − ρ n − 1 a σ ρ n σ ( φ n K − φ n ) m K + τ L ) = 0 K σ ∈ Σ K • weak solution of ∂ t � − ∇ · ( � ∇ δ E δρ [ � ]) = 0 • Fokker-Planck, non linear di ff usion without drift � � � ρ K log ρ K e − V K − ρ K + e − V K E T ( ρ ) = m K K ∈ T 2 | ∇ φ | 2 → 0 everywhere • di ffi culty : τ