Optimal transport paths and their applications

Transcription

1 Optimal transport paths and their applications Qinglan Xia University of California at Davis March 4, 2009

2 Outline Monge-Kantorovich optimal transportation (Classical) Ramified optimal transportation Motivation and general set-up Properties of optimal transport paths (Existence, regularity, etc.) Numerical simulation Some Applications to applied mathematics Urban transport network; Formation of tree leaves; Transport system in placentas, lungs, etc. River channel networks Some Applications to Pure Mathematics In metric geometry: The geodesic problem in quasimetric spaces. Dimensional distance between probability measures Fractal Geometry: DLA driven by ramified optimal transportation First Prev Next Last Go Back Full Screen Close Quit

3 First Prev Next Last Go Back Full Screen Close Quit Monge s Transport Problem How do you best move a given pile of sand to fill a given hole of the same volume? Pile of Sand: a positive Radon measure µ + on a compact convex subset X R m. Hole: another positive Radon measure µ on X. Same Volume: 0 < µ + (X) = µ (X) < + move: a Borel, one-to-one map ψ : X X fill: ψ # µ + = µ (i.e. µ (A) = ψ # µ + (A) = µ + (ψ 1 (A))). best: minimum total work Work or cost of ψ: I (ψ) = X x ψ (x) dµ+ (x).

4 Monge s problem (1781) Find an optimal transport map in A = { ψ : X X Borel, one-to-one, ψ # ( µ + ) = µ } which minimizes the cost or in general case I [ψ] := I [ψ] := X X x ψ (x) dµ + (x) c (x, ψ (x)) dµ + (x) for some given cost density function c : X X [0, + ). Technical Difficulties: Highly nonlinear structure of I. No solution for X = [ 1, 1], µ + = δ 0, µ = 1 2 δ δ 1.

5 Kantorovich (1940 s) Transform it into a linear problem on a convex set. Transport map Transport Plan µ Relaxation µ µ + µ + Minimize J (γ) := in the class of transport plans X X c (x, y) dγ (x, y) M ={γ P (X X) π x# γ = µ +, π y# γ = µ } Existence: from a simple compactness argument of probability measures.

6 Wasserstein distances on P (X) Definition. Given p (0, + ) (usually [1, + )), for any µ +, µ P (X), define ( W p µ +, µ ) := ( min x y p dγ (x, y)) min(1,1/p). γ M X X distance between measures= minimal cost Proposition. W p is a distance on P (X) and metrizes the weak * topology of P (X). Many people has been working on this interesting problem. Applications: This problem has many applications in Economic; Fluid Mechanics; PDE; Optimization; meteorology and oceanography; surface reconstruction;.

7 Summary: For a given cost function c : X X [0, + ), we have considered Monge problem: Minimize I [ψ] := among all transport maps. X c (x, ψ (x)) dµ + (x) Monge-Kantorovich problem: Minimize J (γ) := c (x, y) dγ (x, y) among all transport plans. X X

8 Summary: For a given cost function c : X X [0, + ), we have considered Monge problem: Minimize I [ψ] := among all transport maps. X c (x, ψ (x)) dµ + (x) Monge-Kantorovich problem: Minimize J (γ) := c (x, y) dγ (x, y) among all transport plans. X X But, should we always define transportation cost as an integral of a cost function c(x, y)?

9 Summary: For a given cost function c : X X [0, + ), we have considered Monge problem: Minimize I [ψ] := among all transport maps. X c (x, ψ (x)) dµ + (x) Monge-Kantorovich problem: Minimize J (γ) := c (x, y) dγ (x, y) among all transport plans. X X But, should we always define transportation cost as an integral of a cost function c(x, y)? Answer: Not always.

10 A simple example What is the best way to ship two items from nearby cities to the same destination far away. µ + µ

11 A simple example What is the best way to ship two items from nearby cities to the same destination far away. µ + First Attempt: Move them directly to their destination. µ

12 A simple example What is the best way to ship two items from nearby cities to the same destination far away. µ + Another way: put them on the same truck and transport together! µ

13 µ + µ + µ A V-shaped path A Y-shaped path Answer: Transporting two items together might be cheaper than the total cost of transporting them separately. As a result, A Y shaped path is preferable to a V shaped path. Here, the cost is naturally given by the actual transport path, while the transport maps for both types are trivially same. Knowing only maps is not enough here. In general, a ramified structure might be more efficient than a linear structure consisting of straight lines. µ

14 Examples of Ramified Structures Trees Circulatory systems Cardiovasular systems Railways, Airlines Electric power supply River channel networks Post office mailing system Urban transport network Marketing Ordinary life Communications Superconductor Conclusion: Ramified structures are very common in living and non-living systems. It deserves a more general theoretic treatment.

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17 Problem: Given two arbitrary probability measures µ + and µ P (X) on a convex compact subset X R m, find an optimal path transporting µ + to µ. Need: A class of transport paths. Broad enough to ensure the existence of optimal transport paths; A reasonable cost functional on the category. Optimal transport paths should allow some parts overlap in a cost efficient fashion. Should be Y-shaped rather than V shaped. Nice regularity of optimal transport paths. Idea: figuring out simple cases first!

18 Atomic measures An atomic measure is a (finite) sum of Dirac measures with positive multiplicities. a = i a i δ xi for some x i X and a i > 0. Let A(X) be the space of all atomic measures on X. Question: What is a transport path between two atomic probability measures a and b? a b

19 Transport atomic measures a A transport path from a to b is a weighted directed graph G = {V (G), E(G), w : E(G) (0, + )} satisfying Kirchhoff s laws (for eletrical circuits): V w (e) = w (e) v=e v=e + b for any interior vertex v. In other words, G is a polyhedral chain with boundary G = b a. Notation: For atomic measures a, b P (X), let P ath(a, b) be the family of all transport paths from a to b.

20 Cost Functionals Note that in general the space Path(a, b) might be very large. First Prev Next Last Go Back Full Screen Close Quit

21 Cost Functionals Note that in general the space Path(a, b) might be very large. Want: Find an optimal Y shaped or ramified transport path in Path(a,b).

22 Cost Functionals Note that in general the space Path(a, b) might be very large. Want: Find an optimal Y shaped or ramified transport path in Path(a,b). Thus, we need a suitable cost functional on transport paths.

23 Cost Functionals Note that in general the space Path(a, b) might be very large. Want: Find an optimal Y shaped or ramified transport path in Path(a,b). Thus, we need a suitable cost functional on transport paths. Answer: For each G = {V (G), E(G), w : E(G) (0, + )}, define the M α mass of G by for some α [0, 1). M α (G) := e w (e) α length (e) A Plateau type problem: Given the boundary b a, find a least M α mass minimizer in P ath(a, b).

24 Cost Functionals Note that in general the space Path(a, b) might be very large. Want: Find an optimal Y shaped or ramified transport path in Path(a,b). Thus, we need a suitable cost functional on transport paths. Answer: For each G = {V (G), E(G), w : E(G) (0, + )}, define the M α mass of G by for some α [0, 1). M α (G) := e w (e) α length (e) A Plateau type problem: Given the boundary b a, find a least M α mass minimizer in P ath(a, b). Result: an M α mass minimizer is indeed Y-shaped or ramified.

25 Example 1: Two points to one point It satisfies a balance equation: 3 m α i n i = 0. i=1 Using this equation, we have a formula to calculate the angles. In particular, if α = 0, then the angles are 120 o. Also, if α = 1/2, then the top angle must be 90 o.

26 Two points to two points First Prev Next Last Go Back Full Screen Close Quit

27 Some lemmas Lemma. For any G P ath(a, b), there exists a G P ath(a, b) such that G contains no cycles and ( ) M α G M α (G). L

28 Some lemmas Lemma. For any G P ath(a, b), there exists a G P ath(a, b) such that G contains no cycles and ( ) M α G M α (G). L Thus, we may consider only transport paths containing no cycles.

29 Some lemmas Lemma. For any G P ath(a, b), there exists a G P ath(a, b) such that G contains no cycles and ( ) M α G M α (G). L Thus, we may consider only transport paths containing no cycles. Lemma. If G contains no cycles, then 0 < w (e) 1 for any e E (G). Thus M (G) M α (G).

30 Some lemmas Lemma. For any G P ath(a, b), there exists a G P ath(a, b) such that G contains no cycles and ( ) M α G M α (G). L Thus, we may consider only transport paths containing no cycles. Lemma. If G contains no cycles, then 0 < w (e) 1 for any e E (G). Thus M (G) M α (G). Now, given any two probability measures µ + and µ, what is a transport path from µ + to µ? µ +?? µ

31 Transport general probability measures µ +?? µ?? a i G i b i Idea: Approximate µ +, µ by atomic measures a i, b i ; Transport a i to b i by a graph G i ; The limit T of G i (in a suitable sense) is a transportation of µ + to µ. The sequence of triples {a i, b i, G i } is called an approximating graph sequence of T.

32 First Prev Next Last Go Back Full Screen Close Quit Dyadic approximation of Radon measures Assume X Q, a cube in R m of the edge length d, with center c. Let Q i = {Q h i : h Zm [0, 2 i ) m } be a partition of Q into smaller cubes of edge length d 2 i. For any Radon measure µ on X, let Q 0 Q 1 Q 2 A i (µ) = h µ(q h i )δ c h i where c h i is the center of Qh i. Then, A i(h) converges to µ weakly as measures. This is called Dyadic approximation of µ.

33 How to take limits of G i s? Answer: View each G i as a 1 dimensional normal current with G i = b i a i. Let U R m be any open set. D n (U): C differential n forms in U with compact support. An n-current is an element of the dual space D n (U) of D n (U). i.e. an n current is a continuous linear functional on D n (U). Thus, 0 currents are just distributions. For any T D n (U), its boundary T D n 1 (U) is given by The mass of T D n (U) is given by T (ψ) = T (dψ), ψ D n 1 (U). M(T ) = sup{t (ω) : ω 1, ω D n (U)} T D n (U) is normal if M(T ) + M( T ) < +.

34 Examples of n-current Oriented n-dimensional submanifold M of U with H n (M) < +. [M](ω) = ω = < ω(x), ξ(x) > dh n (x) M M for any ω D n (U). Note that [M] = [ M] and M([M]) = H n (M). Differential m n forms φ D m n (U); φ(ω) = φ ω. Rectifiable currents τ(m, θ, ξ) τ(m, θ, ξ)(ω) = M U < ω(x), ξ(x) > θ(x)dh n (x) Here: M is a rectifiable n-set, θ is a locally H n integrable function and ξ(x) is the orientation of T x M.

35 Transport paths between Radon measures Definition. Given µ +, µ P (X), a normal 1-current T is called a transport path from µ + to µ if there exists a sequence of approximating graphs {a i, b i, G i } such that in the sense of distributions. a i µ +, b i µ, G i T Note that we automatically have T = µ + µ as distributions. For each transport path T, we define M α (T) := inf lim inf M α (G i ). {a i,b i,g i } i Let P ath(µ +, µ ) be the family of all transport paths from µ + to µ.

36 Example: How to transport a Lebesgue measure to a Dirac measure? 1 n 1 n First attempt: n ( 1 l i i=1 C n n i=1 ) α l i ( ) 1 α = Cn 1 α +. n

37 Example: How to transport a Lebesgue measure to a Dirac measure? 1 n 1 n l i First attempt: n ( ) 1 α l n i i=1 n ( ) 1 α C = Cn 1 α +. n i=1 1 2 Second attempt: n n=1 = C 2 n i=1 n=1 ( 1 2 n ) α l i C n=1 ( ) 1 α 2 n = C α 2 n i=1 ( ) 1 α 1 2 n 2 n

38 In higher dimension case, if α > 1 1 m, then n=1 C = C = C (2 n ) m i=1 n=1 n=1 n=1 ( ) 1 α (2 n ) m l i (2 n ) m i=1 ( 1 (2 n ) m ( 1 (2 n ) m ) α 1 2 n ) α 2 n(m 1) ( 2 m(1 α) 1) n < + Proposition. [Finite Cost] Suppose α > 1 1 m. For any µ P (X), there exists a T P ath(µ, δ c ) from µ to a Dirac measure δ c with M α (T) < +.

39 Existence theorem First Prev Next Last Go Back Full Screen Close Quit Theorem. Given µ + and µ M Λ (X), α (1 m 1, 1], there exists an M α mass minimizer S in the family P ath(µ +, µ ). Moreover, M α (S) < Λ α 2 1 m(1 α) 1 md 2. Sketch of the proof: Pick {a i, b i, G i } with M α (G i ) inf{m α (T) : T Path(µ +, µ )} We may assume {G i } has no cycless M(G i ) M α (G i ) < C bounded. By the compactness of normal currents, lower semicontinuity of M α. G ik T P ath(µ +, µ )

40 A new distance on P (X) Definition. Given µ + and µ P (X), define ( d α µ +, µ ) := min{m α (T) : T Path(µ +, µ )}. Theorem. d α is a distance on P (X). Remark: d α is different from any of the Wassenstein distances. Theorem. d α metrizes the weak * topology of P (X).

41 Optimal transport paths Lemma. If G i P ath(a i, b i ) is an M α minimizer, then T P ath(µ +, µ ) is also an M α minimizer in P ath(µ +, µ ). T µ + µ a i G i b i Definition. A transport path T P ath(µ +, µ ) is called an optimal transport path if there exists a sequence of appximating graphs {a i, b i, G i } such that each G i P ath(a i, b i ) is an M α minimizer.

42 Error estimate By the lemma, we can pick our favorite approximating atomic measures {a i }, {b i }. We choose dyadic approximation {A n (µ)}. Proposition. For any µ P (X), d α (µ, A n (µ)) Cλ n with some constant C > 0 and λ = 2 m(1 α) 1 (0, 1). Corollary. If each G n is optimal, then M α (T) M α (G n ) + 2Cλ n T µ + µ A n (µ + ) A n (µ ) G n Optimal

43 Length Space Property Theorem. (P (X), d α ) is a length space. That is, for any µ +, µ P (X), there exists a continuous map ψ : [0, t] (P (M), d α ) with t = d α (µ +, µ ) such that ψ(0) = µ +, ψ(t ) = µ and for any 0 s 1 < s 2 t, d α (ψ(s 1 ), ψ(s 2 )) = s 2 s 1. In other words, an optimal transport path between Radon measures plays the role of a geodesic between two points. Later, we will see that in fact each ψ(s) is purely atomic for any 0 < s < t. 0 ψ s 1 s 2 d α (µ +, µ ) ψ(s 1 ) µ + ψ(s µ 2 ) (P (X), d α )

44 Atomic approximation (α = 0.1) First Prev Next Last Go Back Full Screen Close Quit

45 Atomic approximation (α = 0.5) First Prev Next Last Go Back Full Screen Close Quit

46 Atomic approximation (α = 0.95) First Prev Next Last Go Back Full Screen Close Quit

47 From Lebesgue to Dirac First Prev Next Last Go Back Full Screen Close Quit

48 Transporting general measures First Prev Next Last Go Back Full Screen Close Quit

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65 Transport Path & Transport Plan Let a and b be any two atomic measures. For example, 1 4 X 1 X X Y 1 Y 2 2 3

66 Transport Path & Transport Plan Let a and b be any two atomic measures. For example, 1 4 X 1 X X Y 1 Y Each transport plan γ P lan(a, b) is given by a real valued matrix e.g. U 1 = ( U = (u ij ). ) or U 2 = ( Each transport path G P ath(a, b) gives a 1-current valued matrix g(g) = (g ij ). (no cycles!) ) 1 4 X 1 X Y 1 Y X 3.

67 Compatible Pair of Transport Path & Plan A transport path G and a transport plan γ are said to be compatible if G = u ij g ij. A compatible pair gives a decomposition of G.

68 Compatible Pair of Transport Path & Plan A transport path G and a transport plan γ are said to be compatible if G = u ij g ij. A compatible pair gives a decomposition of G. For instance, U 1 is compatible with G while U 2 is not. 1 4 X 1 X X 3 = 1 4 X 1 X 2 X X Y 1 Y Y 1 Y1 Y2 Y 2

69 Compatible Pair of Transport Path & Plan A transport path G and a transport plan γ are said to be compatible if G = u ij g ij. A compatible pair gives a decomposition of G. For instance, U 1 is compatible with G while U 2 is not. 1 4 X 1 X X 3 = 1 4 X 1 X 2 X X Y 1 Y Y 1 Y1 Y2 Y 2 A compatible pair of transport path and transport plan provides the necessary transporting information by its unique matrix representation ( (u ij ), (g ij ) ). u ij = amount of mass from x i to y j, while g ij = actual transport path.

70 Some Results There exists G P ath (a, b) compatible with all γ P lan (a, b). For any G P ath (a, b), there exists a γ P lan (a, b) compatible with G. Given a transport plan γ P lan ( µ +, µ ), there exists an optimal transport path T P ath ( µ +, µ ) with least finite M α cost among all compatible pairs (T, γ). (mailing problem) Given a transport path T P ath ( µ +, µ ), there exists an optimal transport plan γ P lan ( µ +, µ ) with least I (γ) cost among all compatible pairs (T, γ).

71 How nice is an optimal transport path? Let T P ath(µ +, µ ) be any transport path with M α (T) < +, not necessarily optimal. Theorem. (rectifiability)t is a real multiplicity 1-rectifiable current T = τ(m, θ, ξ) with T = µ + µ. Moreover, M α (T) = θ(x) α dh 1 (x) Idea of proof: Follows from the rectifiable slicing theorem. Now, assume that T is optimal. Let us see how nice T is. M

72 Interior regularity: a local finiteness property Suppose T P ath(µ +, µ ) is an M α minimizer. For any p spt(t ) \ spt( T ), there exists an open ball neighborhood B p of p such that T B p is a cone at p consisting of finite union of segments with suitable multiplicities. These segments are balanced by a simple balance equation. µ + µ B P P

73 How about the boundary? Observation: The support of T may not necessarily be 1-dimensional nearby its boundary, which is the difference of the given two measures. This is because the boundary itself may even be dense in the space, as demonstrated by letting the initial measure to be the Lebesgue measure. First Prev Next Last Go Back Full Screen Close Quit

74 How about the boundary? Observation: The support of T may not necessarily be 1-dimensional nearby its boundary, which is the difference of the given two measures. This is because the boundary itself may even be dense in the space, as demonstrated by letting the initial measure to be the Lebesgue measure. Solution: Relax yourself and enjoy the nature. The nature has provided a wonderful solution for us: the leaf vein.

75 How about the boundary? Observation: The support of T may not necessarily be 1-dimensional nearby its boundary, which is the difference of the given two measures. This is because the boundary itself may even be dense in the space, as demonstrated by letting the initial measure to be the Lebesgue measure. Solution: Relax yourself and enjoy the nature. The nature has provided a wonderful solution for us: the leaf vein. But, how to read this information?

76 Boundary Regularity To understand the boundary behavior, a suitable approach is to study the super-level sets of the rectifiable current T = τ(m, θ, ξ) instead. For each λ > 0, let M λ = {x M : θ(x) λ}. First Prev Next Last Go Back Full Screen Close Quit

77 Boundary Regularity To understand the boundary behavior, a suitable approach is to study the super-level sets of the rectifiable current T = τ(m, θ, ξ) instead. For each λ > 0, let M λ = {x M : θ(x) λ}. Theorem : Each super-level set of an optimal transport path is locally concentrated on a finite union of bilipschitz curves. These curves enjoy some nice properties similar to those satisfied by segments near an interior point.

78 Key Idea of Proof: Decomposition! For any optimal weighted directed graph G P ath(a, b), if M α (a) + M α (b) is bounded above, then we can decompose a, b, G a = a P + a R, b = b P + b R, G = P + R so that P P ath(a P, b P ), R P ath(a R, b R ), the total number of vertices and edges of P are uniformly bounded. The level set G λ is contained in P. Edges of P are nice. Taking the limits to get the decomposition of optimal transport paths. Advantage: Graphs are much easier to deal with. Just using combinatory.

79 Key Idea of Proof: Decomposition! For any optimal weighted directed graph G P ath(a, b), if M α (a) + M α (b) is bounded above, then we can decompose a, b, G a = a P + a R, b = b P + b R, G = P + R so that P P ath(a P, b P ), R P ath(a R, b R ), the total number of vertices and edges of P are uniformly bounded. The level set G λ is contained in P. Edges of P are nice. Taking the limits to get the decomposition of optimal transport paths. Advantage: Graphs are much easier to deal with. Just using combinatory. Feedback? A natural question: Can we use this idea to understand the dynamic formation of a tree leaf?

80 Key Idea of Proof: Decomposition! For any optimal weighted directed graph G P ath(a, b), if M α (a) + M α (b) is bounded above, then we can decompose a, b, G a = a P + a R, b = b P + b R, G = P + R so that P P ath(a P, b P ), R P ath(a R, b R ), the total number of vertices and edges of P are uniformly bounded. The level set G λ is contained in P. Edges of P are nice. Taking the limits to get the decomposition of optimal transport paths. Advantage: Graphs are much easier to deal with. Just using combinatory. Feedback? A natural question: Can we use this idea to understand the dynamic formation of a tree leaf? YES!!

81 α=0.6, β=0.5, ε= α=0.6, β=0.5, ε= α=0.6, β=0.5, ε= α=0.6, β=0.5, ε=

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83 15 α=0.66, β=0.7, totalcost=

84 α=0.68, β=0.38, totalcost=

85 Applications in metric geometry For any two atomic measures on X: a = a i δ xi, b = b j δ yj and any transport plan γ = γ ij δ (xi,y j ) P lan(a, b) we may define H α (γ) = (γ ij ) α x i y j for any α < 1 (α is allowed to be negative here!) Then, we define J α (a, b) = min{h α (γ) : γ P lan(a, b)}. Theorem. J α is a quasi-metric on the space A(X) of atomic probability measures on X. A quasi-metric d is a metric except that it satisfies a relaxed triangle ineq.: d(x, y) C(d(x, z) + d(z, y)).

86 One may calculate the length of a curve f in the quasi-metric space (A(X), J α ) as in geometry: L Jα (f) := where the quasi-metric derivative 1 0 f Jα dt f J α (f(t), f(t + s)) Jα := lim. s 0 s Key observation: each transport path G in P ath(a, b) defines a curve f G in (A(X), J α with M α (G) = L Jα (f G ) Theorem. The quasi-metric J α induces a metric on A(X). The induced metric is the d α metric that we have studied earlier. An α-optimal transport path is exactly a geodesic in this induced metric space in the usual sense of metric geometry. (A(X), d α ) is a length space. But it may not be a complete metric space.

87 Completion of (A(X), d α ) Let P α (X) be the completion of A(X) with respect to d α for any α < 1. (P α (X), d α ) is still a length space. Clearly, any µ, ν P α (X), we have d α (µ, ν) < + In particular, d α (µ, δ O ) < +. Now, we want to study the dimension of measures. For any probability measure µ, we define Theorem. dim(µ) := inf α<1 { 1 1 α : µ P α(x)} dim Hausdorff (µ) dim(µ) dim Minkowski (µ) When µ is concentrated on a nice set, the equalities hold. Which dimension is better if the concentrated set is not nice?

88 A geometric criterion of dimensions For any two probability measures µ, ν P (X), we define D(µ, ν) := inf α<1 { 1 1 α : µ, ν P α(x)} Theorem. D is a pseudometric on P (X). i.e. D is a metric except that D(µ, ν) = 0 does not imply µ = ν. By introducing an equivalent relation µ ν if D(µ, ν) = 0 Then define D([µ], [ν]) = D(µ, ν). Then, D is a metric on P (X)/. Theorem. D(µ, δ O ) = dim(µ) That is, the dimension of a measure is the distance to any atomic measure! For this reason, I call D the dimensional distance.

89 Example: µ =Lebesgue measure, ν =Dirac mass If α > 1 1 m, i.e., m > 1 1 α then n=1 C = C = C (2 n ) m i=1 n=1 n=1 n=1 ( ) 1 α (2 n ) m l i (2 n ) m i=1 ( 1 (2 n ) m ( 1 (2 n ) m ) α 1 2 n ) α 2 n(m 1) ( 2 m(1 α) 1) n < + So, the dimension of µ = m = inf { 1 α<1 1 α : d α(µ, δ 0 ) < + } = D(µ, ν), the distance from µ to δ 0

90 Example: µ =Cantor measure, ν =Dirac mass Here again, n=1 ( ) 2 n 1 α ( ) 1 n 2 n = 3 ( n=1 21 α 2 1 α 3 < α < α > ln 2 ln 3 ) n < the dimension of µ = ln 2 ln 3 = inf { 1 α<1 1 α : d α(µ, δ 0 ) < + } = D(µ, ν), the distance from µ to δ 0 Note, here α is allowed to be negative.(α )

91 Example: µ =Fat Cantor measure, ν =Dirac mass Examples: µ = Fat λ Cantor set (i.e. remove an interval of length λ from the middle of [0, 1]). n=1 ( ) 2 n 1 α 1 + λ 2 n 4 ( ) 1 λ n 1 = λ 2 (1 λ) 2 1 α p < 1 n=1 ( 2 1 α p) n < 2 1 α < 1 p 1 1 α > ln 2 ln p = ln 2 ln 2 ln (1 λ) where p = 1 λ 2. Again, we have dimension of µ = inf α<1 { 1 1 α : d α(µ, δ 0 ) < + }

92 Example: µ =self-similar measure, ν =Dirac mass Example: A =finite union of A i for i = 1, k. Each A i is a σ rescale of A. n=1 ( ) k n 1 α k n σ n 1 L = L σ n=1 k 1 α σ < α > ln k ln σ ( k 1 α σ) n < + Therefore, D (µ) = ln k ln σ. Here again, self-similar dimension of µ = inf α<1 { 1 1 α : d α(µ, δ 0 ) < + }

93 DLA driven by optimal transportation Diffusion limited aggregation model (Witten and Sander): A particle is released far away and undergoes a random walk (Brownian motion). Once it meets the existing aggregate it sticks and the process starts over. Off-lattice DLA in the plane evolves a cluster with fractal dimension Can we use DLA to generate clusters of other fractal dimensions? Now, we want to use the idea of optimal transportation to propose a modification of the standard diffusion-limited aggregation (DLA).

94 Our key idea: The probability of sticking is inversely proportional to the additional cost of transporting the new particle to the root via the existing transport system in the current aggregate. We used a family of cost functions parameterized by a parameter α as studied in ramified optimal transportation. α < 0 promotes growth near the root; α > 0 promotes growth at the tips of the cluster. α = 0 is a phase transition point and corresponds to standard DLA.

95 Modified DLA with negative α. When α is negative enough the final cluster is an one dimensional curve.

96 Modified DLA with positive α. When α is positive enough we get a nearly two dimensional disk. Thus our model encompasses the full range of fractal dimension from 1 to 2.

97 Thank You and Enjoy the Nature First Prev Next Last Go Back Full Screen Close Quit