PDEs from Monge-Kantorovich Mass Transportation Theory Luca - PowerPoint PPT Presentation

PDEs from Monge-Kantorovich Mass Transportation Theory Luca Petrelli Math & Computer Science Dept. Mount St Mary’s University

Outline • Monge-Kantorovich mass transportation problem • Gradient Flow formalism • Time-step discretization of gradient flows • Application of theory to nonlinear diffusion problems • Signed measures

Monge’s original problem move a pile of soil from a deposit to an excavation with minimum amount of work + - from “Memoir sur la theorie des deblais et des remblais” - 1781

Mathematical Model of Monge’s Problem � d , nonnegative Radon measures on µ + µ − � d � � d � µ + � = µ − � < ∞ s : � d → � d one-to-one mapping rearranging into µ + µ − s # µ + = µ − ( s # ) or � � ∀ h ∈ C ( � d ; � d ) h ( s ( x )) dµ + ( x ) = h ( y ) dµ − ( y ) X Y for X = spt ( µ + ) , Y = spt ( µ − )

c(x,y) cost of moving a unit mass from x ∈ � d to y ∈ � d � total cost � d c ( x, s ( x )) dµ + ( x ) I [ s ] := Monge’s problem is then to find s ∗ ∈ A (admissable set) such that: I [ s ∗ ] = min s ∈ A I [ s ] ( M ) � � s # ( µ + ) = µ − � � with A = s

Problem is too hard! · Constraint is highly nonlinear! � � ∀ h ∈ C ( � d ; � d ) h ( s ( x )) dµ + ( x ) = h ( y ) dµ − ( y ) X Y Hard to identify minimum! · minimizing sequence such that { s k } ∞ k =1 ⊂ A I [ s k ] → inf s ∈ A I [ s ] Hard to find { s k j } subsequence such that s k j → s ∗ optimal. · Classical methods of Calculus of Variation fail! No terms create compactness for I [ · ] · does not involve gradients hence it can not I [ · ] · be shown coercive on any Sobolev space

Kantorovich’s relaxation -1940’s Kantorovich’s idea: transform (M) into linear problem Define : � � � prob. meas. µ on � d × � d � � proj x µ = µ + , proj y µ = µ − M := � � J [ µ ] := � d ×� d c ( x, y ) dµ ( x, y ) µ ∗ ∈ M Find such that J [ µ ∗ ] = min µ ∈ M J [ µ ] ( K )

Motivation as given we can define s ∈ A µ ∈ M E ⊂ � d × � d , E Borel � ( x, s ( x )) ∈ E x ∈ � d � µ ( E ) := µ + � � � � Problem need not be generated by any one-to-one mapping µ ∗ s ∈ A Solution only look for “weak” or generalized solutions

Linear programming analogy (Finite dimensional case) → µ + µ + ( x ) µ − ( y ) − → µ − − i j µ ( x, y ) − c ( x, y ) − → µ i,j → c i,j ( i = 1 , · · · , n, j = 1 , · · · , m ) n m � � Mass Balance Condition µ + µ − j < ∞ = i i =1 j =1 m n � � µ i,j = µ + Constraints µ i,j = µ − µ i,j > 0 i , j , j =1 i =1 n m Linear programming problem � � minimize c i,j µ i,j i =1 j =1 n m � � u i µ + maximize i + v j µ − j Then dual problem is i =1 j =1 subject to u i + v j ≤ c i,j

Kantorovich’s Dual Problem Define: � � x, y ∈ � d � � � u, v : � d → � + continuous , u ( x ) + v ( y ) ≤ c ( x, y ) � � L := ( u, v ) � � � � d v ( y ) dµ − ( y ) � d u ( x ) dµ + ( x ) + K ( u, v ) := Then dual problem to (K) is: Find such that K ( u ∗ , v ∗ ) = u ∗ , v ∗ ( u,v ) ∈ L K ( u, v ) max

Gradient Flows To define a gradient flow we need: a differentiable manifold M · a metric tensor on which makes a Riemannian manifold ( M , g ) M g · and a functional on M E · du Then is the gradient flow of on . ( M , g ) dt = − grad E ( u ) E where for all vector fields on . g (grad E, s ) = di ff E · s M s � du Then for all vector fields along . � g u dt , s + di ff E | u · s = 0 s u Main property of gradient flows: energy of system is decreasing along trajectories, i.e. · dt E ( u ) = di ff E | u · du d � du dt , du � dt = − g u dt

Partial Differential Equations as gradient flows � � � Let M := u ≥ 0 , measurable, with u dx = 1 define the tangent space to as M � � � T u M := s measurable, with s dx = 0 and identify it with � �� p measurable ∼ via the elliptic equation . −∇ · ( u ∇ p ) = s

Define � � � � g u ( s 1 , s 2 ) = u ∇ p 1 · ∇ p 2 dx ≡ s 1 p 2 dx � and E ( u ) = e ( u ) dx Then � � ∂ u � � du � ∂ t p − ∇ · ( u ∇ p ) e � ( u ) + di ff E | u · s = dx = g u dt , s � � ∂ u � � ∂ u � � ∂ t p + ∇ p · ( u ∇ e � ( u )) ∂ t − ∇ · ( u ∇ e � ( u )) = dx = dx = 0 p ∂ u ∂ t = ∇ · ( u ∇ e � ( u )) = ⇒

Examples of PDE that can be obtained as Gradient Flows ∂ u Heat Equation e ( u ) = u log u ∂ t = ∆ u ∂ u Fokker-Planck Equation e ( u ) = u log u + u V ∂ t = ∆ u + ∇ · ( u ∇ V ) 1 ∂ u Porous Medium Equation e ( u ) = m − 1 u m ∂ t = ∆ u m Note: equations are only solved in a weak or generalized way.

Important fact! Can implement gradient flow without making explicit use of gradient operator through time-discretization and then passing to the limit as the time step goes to 0. Jordan, Kinderlehrer and Otto (1998) ∂ u ( x, t ) � � u ∇ ψ ( x ) − ∆ u = 0 − div ∂ t Otto (1998) ∂ u ( x, t ) − ∆ u 2 = 0 ∂ t Kinderlehrer and Walkington (1999) ∂ u ( x, t ) − ∂ � � u ∇ ψ ( x ) + K ( u ) x = g ( x, t ) ∂ t ∂ x Agueh (2002) ∂ u ( x, t ) � � u ∇ c ∗ [ ∇ ( F � ( u ) + V ( x )] = 0 − div ∂ t Petrelli and Tudorascu (2004) ∂ u ( x, t ) − ∇ · ( u ∇ Ψ ( x, t )) − ∆ f ( t, u ) = g ( x, t, u ) ∂ t

Time-discretized gradient flows 1. Set up variational principle u h Let h > 0 be the time step. Define the sequence recursively � � k k ≥ 0 as follows: is the intial datum ; given , define as the u h u h u h u 0 k − 1 k 0 solution of the minimization problem � 1 � � 2 + E ( u ) u h � min ( P ) 2 h d k − 1 , u u ∈ M where d, the Wasserstein metric, is defined as �� � d ( µ + , µ − ) 2 := | x − y | 2 dµ ( x, y ) inf µ ∈ M � d ×� d µ + i.e. d is the least cost of Monge-Kantorovich mass reallocation of to µ − for . c ( x, y ) = | x − y | 2

2 . Euler-Lagrange Equations Use Variation of Domain method to recover E-L eqns. � � � � d φ ( u h � d ×� d ( y − x ) · ξ ( y ) dµ ( x, y ) − h k ) ∇ · ξ dx = 0 where φ ( s ) =: e � ( s ) s − e ( s ) or in Gradient Flow terms: u h k − u h k − 1 = − grad E ( u h k ) h Then recover approximate E-L eqns., i.e. � 1 � � � 1 � h ( u h k − u h k − 1 ) ζ − φ ( u h 2 h �∇ 2 ζ � ∞ d ( u h k , u h � � k − 1 ) 2 k ) ∆ ζ � ≤ dx � � � � d

3 . Linear time interpolation Define u h ( x, t ) := u h k ( x ) if kh ≤ t < ( k + 1) h After integration in each interval over time we obtain � 1 � � n � � � � � k − 1 ) 2 u h ( x, t + τ ) − u h ( x, t ) ζ − φ ( u h ) ∆ ζ d ( u h k , u h � � dxdt � ≤ C � � h � � [0 ,T ] ×� d � k =1 n � 2 ≤ C h � Necessary inequality: u h k , u h � d k − 1 k =1

4. Convergence result as time step h goes to 0 Linear case Through a Dunford-Pettis like criteria show existence of function u u h � u such that, up to a subsequence, in some space. L p Nonlinear case Stronger convergence is needed, through precompactness result in . Also needed discrete maximum principle: L 1 u 0 bounded ⇒ u h bounded Then, passing to the limit in the general Euler-Lagrange equation shows that u is a “weak” solution of � � � ∂ u � u ∇ e � ( u ) = ∇ · ≡ ∆ φ ( u ) ∂ t

Nonlinear Diffusion Problems  u t − ∇ · ( u ∇ Ψ ( x, t )) − ∆ f ( t, u ) = g ( x, t, u ) in Ω × (0 , T ) ,  � � u ∇ Ψ + ∇ f ( t, u ) · ν x = 0 on ∂ Ω × (0 , T ) , ( NP ) u ( · , 0) = u 0 ≥ 0 in Ω .  Theorem 4. Assume (f1)-(f3), (g1)-(g4) and ( Ψ ), then the problem (NP) admits a nonnegative essentially bounded weak solution provided that Ω is bounded and convex and the initial u 0 data is nonnegative and essentially bounded.

Hypothesis ( u − v )( f ( t, u ) − f ( t, v )) ≥ c | u − v | ω for all u, v ≥ 0 , ( f 1) f ( · , s ) are Lipschitz continuous for s in bounded sets ( f 2) f ( t, · ) di ff erentiable, ∂ f ∂ s positive and monotone in time ( f 3) for all x ∈ � d g ( x, · , · ) nonnegative in [0 , ∞ ) × [0 , ∞ ) ( g 1) g ( x, t, u ) ≤ C (1 + u ) locally uniformly w.r.t. ( x, t ) , t ≥ 0 ( g 2) g ( x, t, · ) is continuous on [0 , ∞ ) ( g 3) { g ( x, · , u ) } ( x,u ) is equicontinuous on [0 , ∞ ) w.r.t. ( x, u ) ( g 4) Ψ : � d × [0 , ∞ ) → � di ff .ble and locally Lipschitz in x ∈ � d ( Ψ )

Novelties Time-dependent potential and diffusion coefficient Ψ ( · , t ) f ( t, · ) Non homogeneous forcing term g ( x, t, u ) � ( k +1) h Ψ k := 1 Averaging in time for , and , e.g. Ψ f Ψ ( · , t ) dt g h kh � kh New variational principle for v k − 1 := u k − 1 + g ( · , t, u k − 1 ) dt ( k − 1) h � 1 � � 2 + E ( u ) v h � ( P � ) min 2 h d k − 1 , u u ∈ M