The Euler–Lagrange equation of the Jordan–Kinderlehrer–Otto step #
Let E be a separable Hilbert space, φ an energy on the quadratic Wasserstein space P₂(E) and
τ > 0 a time step. A step of the Jordan–Kinderlehrer–Otto scheme (JKO scheme) from a law μ
is a law ν minimizing
ρ ↦ φ ρ + W₂ (μ, ρ) ^ 2 / (2τ),
that is, a point of the resolvent TauCeti.moreauYosidaResolvent φ τ μ. This file derives the
first-order condition satisfied by such a minimizer, by perturbing ν along vector fields.
For a vector field ξ ∈ L²(ν; E) and h : ℝ, the map x ↦ x + h • ξ x lies in L²(ν; E), so
its law ν_h under ν lies in P₂(E) (TauCeti.exists_coe_eq_map_of_memLp). If γ is a
coupling of ν and μ, then (x, y) ↦ (x + h • ξ x, y) carries it to a coupling of ν_h and
μ, whence
W₂ (ν_h, μ) ^ 2 ≤ ∫ ‖x - y‖² dγ + 2h ∫ ⟪x - y, ξ x⟫ dγ + h² ∫ ‖ξ‖² dν
(TauCeti.wassersteinEDist_map_add_smul_sq_le). When γ is an optimal coupling, the first
integral is W₂ (ν, μ) ^ 2 (TauCeti.WassersteinSpace.dist_sq_eq_integral), and comparing the
minimizer ν with ν_h gives the variational inequality
φ ν ≤ φ ν_h + (h ∫ ⟪x - y, ξ x⟫ dγ + (h² / 2) ∫ ‖ξ‖² dν) / τ
for every h (TauCeti.apply_le_apply_add_of_mem_moreauYosidaResolvent). If the energy has a
derivative D along the perturbation, that is φ ν_h = f h near h = 0 for a function f
differentiable at 0 with f' 0 = D, then the right side minus the left has a local minimum at
h = 0, and its derivative there vanishes. This is the Euler–Lagrange equation of the JKO step:
∫ ⟪x - y, ξ x⟫ dγ(x, y) = -τ D
(TauCeti.integral_inner_eq_neg_mul_of_mem_moreauYosidaResolvent). Heuristically
D = ∫ ⟪∇ (δφ/δν), ξ⟫ dν for the first variation δφ/δν of the energy, and the equation is the
weak form of (x - y) / τ = -∇ (δφ/δν)(x), the displacement of the step being averaged over the
conditional law of y given x under γ.
For the potential energy ρ ↦ ∫ V dρ of a differentiable potential whose gradient grows at most
linearly, ‖∇V x‖ ≤ C (1 + ‖x‖), the potential is integrable against every law of finite second
moment (TauCeti.integrable_comp_of_norm_gradient_le), and differentiating under the integral sign
gives the derivative ∫ ⟪∇V x, ξ x⟫ dν along the perturbation
(TauCeti.hasDerivAt_integral_comp_add_smul). The Euler–Lagrange equation then reads
∫ ⟪x - y, ξ x⟫ dγ(x, y) = -τ ∫ ⟪∇V x, ξ x⟫ dν(x)
(TauCeti.integral_inner_eq_neg_mul_integral_inner_gradient_of_mem_moreauYosidaResolvent). This is
the drift part of the discrete Euler–Lagrange equation of Jordan, Kinderlehrer and Otto for the
Fokker–Planck equation.
Main results #
TauCeti.wassersteinEDist_map_add_smul_sq_le: the expansion of the squared Wasserstein distance along the perturbation of a law by a vector field ofL², against an arbitrary coupling.TauCeti.IsCoupling.integral_inner_eq_innerandTauCeti.IsOptimalCoupling.abs_integral_inner_le: the pairing∫ ⟪ξ x, y - x⟫ dγis anL²(γ)inner product, at most‖ξ‖ W₂(μ, ν)in absolute value for an optimal coupling.TauCeti.apply_le_apply_add_of_mem_moreauYosidaResolvent: the variational inequality of the JKO step.TauCeti.integral_inner_eq_neg_mul_of_mem_moreauYosidaResolvent: the Euler–Lagrange equation of the JKO step for an energy differentiable along the perturbation.TauCeti.integral_inner_eq_neg_mul_integral_inner_gradient_of_mem_moreauYosidaResolvent: the Euler–Lagrange equation of the JKO step for the potential energy.
References #
- R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker–Planck equation, SIAM J. Math. Anal. 29 (1998), 1–17, for the variation of the discrete scheme along the flow of a vector field.
- L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of
Probability Measures, 2nd ed., Birkhäuser 2008, Section 10.1, for the first-order condition of
the minimizing movement step in
P₂. - F. Santambrogio, Optimal Transport for Applied Mathematicians, Birkhäuser 2015, Section 8.3.
The pairing ⟪x - y, ξ x⟫ of the displacement of a coupling of finite quadratic cost with a
vector field of L²(ν) is integrable.
The squared Wasserstein distance along a perturbation. Let γ couple ν and μ with
finite quadratic cost and let ξ ∈ L²(ν; E). The law ν_h of x + h • ξ x under ν satisfies
W₂ (ν_h, μ) ^ 2 ≤ ∫ ‖x - y‖² dγ + 2h ∫ ⟪x - y, ξ x⟫ dγ + h² ∫ ‖ξ‖² dν, the right side being the
quadratic cost of the coupling (x, y) ↦ (x + h • ξ x, y) of ν_h and μ carried by γ.
The pairing ∫ ⟪ξ x, y - x⟫ dγ of a field ξ ∈ L²(μ; E) with the displacement of a coupling
γ out of μ is the L²(γ) inner product of ξ ∘ Prod.fst with the displacement.
The pairing with the square integrable displacement of a coupling is additive in the field.
The pairing with the square integrable displacement of a coupling is linear in the field.
Along the coupling (x, (1 - t) • x + t • y)₊ γ of μ with an interpolated law, the pairing
of a field ξ ∈ L²(μ; E) with the displacement is t times its pairing with the displacement
of γ.
Cauchy–Schwarz for the pairing. For an optimal coupling γ of two laws μ and ν of
P₂(E) for the quadratic cost and ξ ∈ L²(μ; E), |∫ ⟪ξ x, y - x⟫ dγ| ≤ ‖ξ‖ W₂(μ, ν).
The variational inequality of the JKO step. Let ν be a JKO step of time step τ > 0
from μ for the energy φ, let γ be an optimal coupling of ν and μ, and let
ξ ∈ L²(ν; E). For every h : ℝ, the law ν' of x + h • ξ x under ν satisfies
φ ν ≤ φ ν' + (h ∫ ⟪x - y, ξ x⟫ dγ + (h² / 2) ∫ ‖ξ‖² dν) / τ.
The Euler–Lagrange equation of the JKO step. Let ν be a JKO step of time step τ > 0
from μ for the energy φ, let γ be an optimal coupling of ν and μ, and let
ξ ∈ L²(ν; E). Suppose that the energy is differentiable along the perturbation of ν by ξ:
for h near 0, the law of x + h • ξ x under ν has energy f h, where f has derivative
D at 0. Then ∫ ⟪x - y, ξ x⟫ dγ(x, y) = -τ D.
The Euler–Lagrange equation of the JKO step for the potential energy. Let V be a
differentiable potential whose gradient grows at most linearly, ‖∇V x‖ ≤ C (1 + ‖x‖). Let ν
be a JKO step of time step τ > 0 from μ for the potential energy ρ ↦ ∫ V dρ, and γ an
optimal coupling of ν and μ. Then for every vector field ξ ∈ L²(ν; E),
∫ ⟪x - y, ξ x⟫ dγ(x, y) = -τ ∫ ⟪∇V x, ξ x⟫ dν(x).