Numerical Approximation of L 1 Monge-Kantorovich Problems

Transcription

1 Rome March 30, 2007 p. 1 Numerical Approximation of L 1 Monge-Kantorovich Problems Leonid Prigozhin Ben-Gurion University, Israel joint work with J.W. Barrett Imperial College, UK

2 Rome March 30, 2007 p. 2 Monge, 1781 Given two distributions of mass with the same finite total, embankment f + and excavation f, find a transport map T which carries f + into f, f + = f B R n Borel, T 1 (B) and minimizes the total transportation cost c(x,t(x))f + (x) B where c(x,y) is the cost of transporting one unit of mass from x to y and f ± M + (R n ), the set of nonnegative measures.

3 Rome March 30, 2007 p. 3 Kantorovich, 1942 Find an optimal transport plan, a measure γ M + (R n R n ) satisfying γ = f +, γ = B R n B R n B and minimizing the transportation cost R n R n c(x,y)γ(x,y) B f The difference is that splitting of mass is allowed: the mass from a point x is distributed as γ(x, ). This is a linear programming problem. The infimum is known as Kantorovich-Wasserstein distance.

4 Rome March 30, 2007 p. 4 Recent much progress is due to Ambrosio, Brenier, Caffarelli, Evans, Feldman, Gangbo, McCann,... and was obtained for very general cost functions. The most studied: c(x,y) = x y p, with p = 1, 2. p = 2: The optimal plan is unique, it is an optimal map, and this map is a gradient of a convex function. Numerical algorithms: Benamou and Brenier 00, Angenent, Haker, Tannenbaum 03. p = 1: The cost is not strictly convex, not every optimal plan is a map, an optimal map exists but is not unique. Numerical algorithms: the issue of this talk.

5 Rome March 30, 2007 p. 5 transport density and potential Dual formulation of L 1 MK problem, Kantorovich 42, { } max uf : u Lip 1 (R n ), R n where f = f + f. This problem is equivalent to the following PDE (MK equation, Evans 97:) (a u) = f u 1, a 0, u < 1 a = 0 Here u is Kantorovich potential, a - transport density. A weak solution {u,a} Lip 1 (R n ) M + (R n ) exists for any f with finite total variation and zero average, Bouchitté, Buttazzo, and Seppecher 97.

6 Rome March 30, 2007 p. 6 transport flux Although optimal transport plan is not unique, the density a is the same for all such plans (under mild regularity conditions on f, Feldman and McCann 02). The transport flux q = a u is also the same for all optimal plans and contains all information on the local transport. The Kantorovich-Wasserstein distance between f + and f (the minimal cost of transportation) can be found as C = a = q Our goal: computing the optimal transport flux.

7 Rome March 30, 2007 p. 7 dual formulation for flux Let q = a u, where {a,u} solve the MK equation, (a u) = f u 1, a 0, u < 1 a = 0. For any test vector function v, u (v q) u v + u q v q. Hence we arrive at q = f and v q u (v q) v, a mixed formulation of the equivalent dual to MK problem (Bouchitté, Buttazzo, and Seppecher 97): min { q : q [M(R n )] n and q = f }

8 Rome March 30, 2007 p. 8 generalization the transport should be confined to the closure of a Lipschitz domain Ω R n with spt(f) Ω. the cost of transporting one unit of mass to the distance dx near a point x is k(x)dx, where k : Ω R >0 is a given function. In this case the equivalent flux formulation of MK problem, Pratelli 05, reads: { } min k(x) q q = f in Ω, q n = 0 on Ω Ω where the constraint is understood in the sense of distributions, (q, φ) = (f,φ), φ C 1.

9 Rome March 30, 2007 p. 9 mixed method Let V M 0 = v [M(Ω)]n v M(Ω), v,φ = v, φ φ C 1 (Ω) Problem (Q): Find q V M 0 and u C(Ω) such that q,φ = f,φ φ C(Ω), v,k q,k (v q),u v V M 0

10 Rome March 30, 2007 p. 10 regularization Replace by 1 r r with r > 1 and consider the regularized Problem (Q r ): Find q r V0 r and u r L p I (Ω) such that ( q r,φ) = (f r,φ) φ L p (Ω), (k q r r 2 q r,v) = ( q r,u r ) v V0 r Here V0 r = {v [Lr (Ω)] n : v L r (Ω), v n Ω = 0} L p I (Ω) = {ψ Lp (Ω) : Ω ψ = 0}, is a regularization of f. 1 r + 1 p = 1, and f r L r I

11 Rome March 30, 2007 p. 11 Theorem 1. The problem (Q r ), r > 1, has a unique solution. As r 1 there exists a subsequence of {q r,u r } such that q r q -weakly in [M(Ω)] n ; q r q -weakly in M(Ω); u r u strongly in C(Ω). Moreover, {q,u} V M 0 C I (Ω) and solves (Q).

12 Rome March 30, 2007 p. 12 approximation of (Q r ) q r approximated using divergence-conforming Raviart-Thomas finite elements of the lowest order, RT h 0, and vertex sampling on nonlinear term; u r by piecewise constant functions, their set is denoted S h, and S h I = {φ Sh : Ω φ = 0}. Here h is the maximal element diameter. Problem (Q h r): Find q h r V h 0 and uh r Sh I such that ( q h r,φh ) = (f r,φ h ) φ h S h, (k q h r r 2 q h r,vh ) h = ( q h r,uh r ) vh RT h 0.

13 Theorem 2. The unique solution {q h r,u h r } of (Q h r) converges, for r > 1 fixed and h 0, to {q r,u r }, the unique solution of (Q r ). Implementation: iterations, ( Q j+1 2,φ h ) =(f r,φ h ) φ h S h, (k{ Q j r 2 Q j + Q j r 2 ε (Q j+1 2 Q j )},v h ) h = (U j+1, v h ) v h RT h 0 where the regularization Q ε = ( Q 2 + ε 2 ) 1 2 makes all terms well defined, successive over-relaxation Q j+1 = γq j (1 γ)q j, and adaptive mesh generation. Rome March 30, 2007 p. 13

14 Rome March 30, 2007 p. 14 inhomogeneous domain f + k 1 f k = 1 Ω is a square, f + and f are uniform in their supports, left and right ellipses, respectively. The cost parameter: k = 1 outside the inclined ellipse, inside the ellipse k = k 1.

15 Rome March 30, 2007 p. 15 k 1 = 0.01 and k 1 = 1 mesh flux q q -levels

16 Rome March 30, 2007 p. 16 k 1 = 3 and k 1 = mesh flux q q -levels

17 f+ = Pl 1 i=1 l δ(x yi ), f = Pn 1 i=1 n δ(x xi ); yi, xi, Upper row: exact optimal transport rays (linear programming); lower row: approximate solution (mixed scheme). Transport cost errors: 0.05% left (l = 9, n = 6); 0.3% right (l = 11, n = 7). Rome March 30, 2007 p. 17

18 Rome March 30, 2007 p. 18 opt. couplings: f + 1/ Ω, f = n i=1 α iδ(x x i ), αi = 1 Solution characterization, Rüschendorf and Uckelmann 00: Let F(x) = max i { x x i + a i } and A i = {x Ω : F(x) = x x i + a i }. The coupling {A i } is optimal iff a i are such that A i = α i, i. Left: all a i are zero, points are coupled to the closest sink, exact boundaries can be found. Right: all sinks are equal, the boundaries are found as small-level contours of q.

19 Rome March 30, 2007 p. 19 partial MK problem Let only a given amount, m min( f +, f ), should be optimally transported from f + to f ; the distributions can be unbalanced. It is needed to find both the optimal transportation domains and the optimal transport plan for them. Free boundary problem in optimal transportation: Caffarelli and McCann 07, cost = (distance) 2. We now derive a variational formulation for the partial L 1 MK problems, cost = distance, and solve them numerically.

20 Rome March 30, 2007 p. 20 partial L 1 MK problem We assume again transport is allowed only in Ω, the local cost of transporting one unit of mass is k(x)dx. Let us define Γ m M+ (Ω Ω) as the set of nonnegative measures satisfying Ω Ω γ = m and dominated by f ±, i.e. for every Borel set B Ω B Ω γ B f+, Ω B γ B f. The optimal transport plan minimizes the total cost of transportation in Γ m. The plan can be presented as a part of an optimal plan for a balanced problem.

21 Rome March 30, 2007 p. 21 auxiliary balanced problem imbed R n into R n+1 as a subspace x n+1 = 0; introduce two auxiliary sources, w and w +, supported in hyperplanes x n+1 = l and x n+1 = l, respectively; postulate free transportation, k = 0, in auxiliary planes; everywhere else we assume k = 1. The MK problem obtained is balanced. It contains Dirichlet regions, Bouchitté and Buttazzo 01. Flux formulation holds true, Pratelli 05.

22 Rome March 30, 2007 p. 22 auxiliary balanced problem Transport in auxiliary planes is free; If l > l 0 = 1 2 max{d Ω(x,y) : x spt(f + ), y spt(f )} transport from f + to f via an auxiliary plane is not paying; The cost of interplane transport is the same for all reasonable plans. If q is an optimal flux, Q is a solution to the partial { problem. Flux formulation } min k q : q = f Ω [ l,l] can be simplified.

23 Rome March 30, 2007 p. 23 new variational formulation Problem (PMK): min K P Ω { Q + l( q + + q ) } where K P V M 0 [M(Ω)] 2 is the set of triples {Q,q +,q } satisfying three mass balance conditions: Q + q + q = f + f, q + = f + m, Ω Ω q = f m. Ω Ω

24 Rome March 30, 2007 p. 24 special case Let m = f + < f Then w = 0 and q + = 0, so q = Q f + + f and q = f m, Q V M 0 Problem (PMK 0 ): { Q +l Q f + +f } min Q V M 0 Same if m = f < f +.

25 Rome March 30, 2007 p. 25 back to balance In the balanced case, m = f + = f, the fluxes q +, q are zero. Hence Q = f + f and we return to the known flux formulation, { min Q V M 0 Q } Q = f+ f Alternatively, for any l > l 0 we have also { Q + l Q f + + f } min Q V M 0 as an equivalent flux formulation of L 1 MK problem.

26 Rome March 30, 2007 p. 26 existence for partial MK Barrett and P., in preparation: Regularization: 1 r r, where 0 < r 1 1; For any r > 1 the Euler-Lagrange equations for regularized problem have a unique solution: the fluxes Q r V r 0, q ± r L r (Ω ± ) and Lagrange multipliers u r L p I (Ω), λ± r R. Here spt(f ± ) Ω ± Ω, p = r r 1. As r 1 the solutions converge (for a subsequence) to a solution of Euler-Lagrange equations for the flux formulation of the partial L 1 MK problem.

27 Rome March 30, 2007 p. 27 solution of partial problems: regularization: 1 r r where 0 < r 1 1; constraint optimization: augmented Lagrangian method; approximation: Raviart-Thomas elements of the least order and vertex sampling on nonlinear terms for Q r, piecewise constant elements for q r ± and the Lagrange multiplier u r of the pointwise divergence constraint; nonlinearity: iterations, successive overrelaxation; singularities: adaptive mesh generation.

28 Rome March 30, 2007 p. 28 example Left: f + = 1 in the left rectangle, f = 1 in the right one; m = 1 3 f + = 1 2 An optimal plan is easy to find, transportation cost error is less than 0.05%; red line " " is the free boundary. Right: elliptic obstacle is placed between the rectangles. f.

29 Rome March 30, 2007 p. 29 example: partial coupling f + = 1 in the whole unit square Ω f is a sum of five equal point sinks of total mass m < 1. " " is the free boundary. Left: m = 0.5; right: m = 0.75;

30 Rome March 30, 2007 p. 30 partial coupling: mesh Results of three mesh adaptions, elements.