Expectations over Fractal Sets

Transcription

1 Expectations over Fractal Sets Michael Rose Jon Borwein, David Bailey, Richard Crandall, Nathan Clisby 21st June 2015

2 Synapse spatial distributions R.E. Crandall, On the fractal distribution of brain synapses. In Computational and Analytical Mathematics (2013).

3 Classical box integrals B n (s) is the order-s moment of separation between a random point and a vertex of the n-cube: B n (s) := x s x [0,1] n = x s Dx x [0,1] n

4 Classical box integrals B n (s) is the order-s moment of separation between a random point and a vertex of the n-cube: B n (s) := x s x [0,1] n = x s Dx x [0,1] n n (s) is the order-s moment of separation between two random points in the n-cube: n (s) := x y s x,y [0,1] n = x y s DxDy x,y [0,1] n R.S. Anderssen, R.P. Brent, D.J. Daley and P.A.P. Moran, Concerning 1 1 (x xk 2 ) 1/2 dx 1 dx k and a Taylor Series Method. SIAM Journal on Applied Mathematics 30 (1976). D.H. Bailey, J.M. Borwein and R.E. Crandall, Box integrals. Journal of Computational and Applied Mathematics 206 (2007).

5 Outline: 1. Expectations over String-generated Cantor Sets (SCSs). 2. Expectations over Iterated Function System (IFS) attractors. 3. Ongoing Mathematics and Computation.

6 String-generated Cantor Set (SCS) expectations

7 String-generated Cantor Sets Ternary expansion for coordinates of x = (x 1,..., x n ) [0, 1] n (with x jk {0, 1, 2}): U(c) := #{1 s in ternary vector c} x 1 = 0. x 11 x 12 x x 2 = 0. x 21 x 22 x x n = 0. x n1 x n2 x n3... c 1 c 2 c 3...

8 String-generated Cantor Sets Ternary expansion for coordinates of x = (x 1,..., x n ) [0, 1] n (with x jk {0, 1, 2}): U(c) := #{1 s in ternary vector c} Definition (String-generated Cantor set) x 1 = 0. x 11 x 12 x x 2 = 0. x 21 x 22 x x n = 0. x n1 x n2 x n3... c 1 c 2 c 3... Given an embedding space [0, 1] n and an entirely-periodic string P = P 1 P 2... P p of non-negative integers with P i n for all i = 1, 2,..., p,

9 String-generated Cantor Sets Ternary expansion for coordinates of x = (x 1,..., x n ) [0, 1] n (with x jk {0, 1, 2}): U(c) := #{1 s in ternary vector c} Definition (String-generated Cantor set) x 1 = 0. x 11 x 12 x x 2 = 0. x 21 x 22 x x n = 0. x n1 x n2 x n3... c 1 c 2 c 3... Given an embedding space [0, 1] n and an entirely-periodic string P = P 1 P 2... P p of non-negative integers with P i n for all i = 1, 2,..., p, the String-Generated Cantor Set (SCS), denoted C n (P), is the set of all admissible x [0, 1] n, where x admissible U(c k ) P k k N

10 String-generated Cantor Sets D.H. Bailey, J.M. Borwein, R.E. Crandall and M.G. Rose, Expectations on fractal sets. Applied Mathematics and Computation 220 (2013).

11 Classical box integrals B n (s) is the order-s moment of separation between a random point and a vertex of the n-cube: B n (s) := x s x [0,1] n = x s Dx x [0,1] n n (s) is the order-s moment of separation between two random points in the n-cube: n (s) := x y s x,y [0,1] n = x y s DxDy x,y [0,1] n

12 Fractal Box Integrals

13 Definition of expectation Definition (Expectation over an SCS) The expectation of F : R n C on an SCS C n (P) is defined by: F (x) x Cn(P) := lim j 1 N 1 N j F (x y) x,y Cn(P) := lim j 1 N 2 1 N2 j when the respective limits exist. ( c1 F U(c i ) P i U(c i ) P i U(d i ) P i 3 + c c j 3 j ) ( c1 d 1 F c j d j 3 j )

14 Functional equations for expectations Proposition (Functional equations for expectations) For x, y in R n and appropriate F the expectations pertaining to the box integrals B and satisfy the functional equations: 1 F (x) x Cn(P) = p j=1 N F x p j 3 p + c j 3 j U(c k ) P k j=1 1 F (x y) x,y Cn(P) = p F x y p j=1 N2 j 3 p + U(b k ) P k U(a k ) P k j=1 (b j a j ) 3 j

15 Special case - second moments The functional expectation relations lead directly to: Theorem (Closed forms for B(2, C n (P)) and (2, C n (P))) For any embedding dimension n and SCS C n (P) the box integral B(2, C n (P)) is rational, given by the closed form: B(2, C n (P)) = n p p k=1 ( ) Pk 1 n 2 j=0 j n j (n j) 9 k Pk ( n ) j=0 j 2 n j

16 Special case - second moments The functional expectation relations lead directly to: Theorem (Closed forms for B(2, C n (P)) and (2, C n (P))) For any embedding dimension n and SCS C n (P) the box integral B(2, C n (P)) is rational, given by the closed form: B(2, C n (P)) = n p p k=1 ( ) Pk 1 n 2 j=0 j n j (n j) 9 k Pk ( n ) j=0 j 2 n j and the corresponding box integral (2, C n (P)) is also rational, given by: (2, C n (P)) = 2B(2, C n (P)) n 2

17 Special case - second moments The classical box integrals over the unit n-cube are: B n (2) = n 3 and n (2) = n 6 which matches the output of our closed forms when P = n.

18 Iterated Function System (IFS) attractor expectations

19 IFS Attractors Let (X, d) be a metric space and let (H(X ), h(d)) be the associated space of non-empty compact subsets of X equipped with the Hausdorff metric h(d).

20 IFS Attractors Let (X, d) be a metric space and let (H(X ), h(d)) be the associated space of non-empty compact subsets of X equipped with the Hausdorff metric h(d). Definition For each i {1, 2,..., m} (where m 2), let f i : X X be a contraction mapping with contractivity factor 0 < c i < 1 (so d (f i (x), f i (y)) c i d (x, y)) and associated probability 0 < p i < 1 (where m i=1 p i = 1). A hyperbolic iterated function system (IFS) is the collection. {X ; f 1,..., f m }

21 IFS Attractors Theorem Let {X ; f 1,..., f m } be a hyperbolic IFS. Then the transformation F : H(X ) H(X ) defined by F(S) = m n=1 f n(s) for all S H(X ) is a contraction mapping on H(X ) with contractivity factor C = max {c 1,..., c m }.

22 IFS Attractors Theorem Let {X ; f 1,..., f m } be a hyperbolic IFS. Then the transformation F : H(X ) H(X ) defined by F(S) = m n=1 f n(s) for all S H(X ) is a contraction mapping on H(X ) with contractivity factor C = max {c 1,..., c m }. Theorem (The Contraction Mapping Theorem) The mapping F possesses a unique fixed point A H(X ), which satisfies: m A = F(A) = f n (A) n=1 and which is referred to as the attractor of the IFS.

23 IFS Attractors Theorem Let {X ; f 1,..., f m } be a hyperbolic IFS. Then the transformation F : H(X ) H(X ) defined by F(S) = m n=1 f n(s) for all S H(X ) is a contraction mapping on H(X ) with contractivity factor C = max {c 1,..., c m }. Theorem (The Contraction Mapping Theorem) The mapping F possesses a unique fixed point A H(X ), which satisfies: m A = F(A) = f n (A) n=1 and which is referred to as the attractor of the IFS. We will take as our deterministic fractals those sets that can be so expressed as an IFS attractor.

24 IFS Attractors ( x f 1 (x, y) = 2, y ) 2 ( x + 1 f 2 (x, y) = 2, y + 3 ) 2 ( x + 2 f 3 (x, y) = 2, y ) 2

25 SCS in IFS framework Any given SCS can be expressed as the attractor of an IFS in the following manner: Proposition The IFS corresponding to the SCS C n (P) is: {[0, 1] n R n ; f 1, f 2,..., f i,..., f m } where f i (x) = ( 1 p ( 3) x + 1 ) 3 c1i + ( ) c2i ( 1 p 3) cpi for i {1, 2,..., m} ranging over all admissible columns c k, where m = p k=1 N k and N k = P k ( n ) j=0 j 2 n j.

26 Code Space Definition Given an IFS {X ; f 1,..., f m }, the associated code space Σ m is defined as: Σ m := {σ = σ 1 σ 2... σ i {1, 2,..., m} i N}

27 Code Space Definition Given an IFS {X ; f 1,..., f m }, the associated code space Σ m is defined as: Σ m := {σ = σ 1 σ 2... σ i {1, 2,..., m} i N} The address function φ : Σ m X is defined by: for any x A. φ(σ) := lim k f σ 1 f σ2... f σk (x)

28 Code Space Definition Given an IFS {X ; f 1,..., f m }, the associated code space Σ m is defined as: Σ m := {σ = σ 1 σ 2... σ i {1, 2,..., m} i N} The address function φ : Σ m X is defined by: φ(σ) := lim k f σ 1 f σ2... f σk (x) for any x A. The attractor of the IFS can also be represented by: A = {φ(σ) σ Σ m }

29 Expectations on IFS Attractors Definition (Fundamental definition of expectation) Let {X ; f 1,..., f m } be an IFS with attractor A H(X ). Let F : X C be a complex-valued function over X. The expectation of F over A, F (x) x A, is defined as: when the limit exists. 1 F (x) x A := lim j m j σ j Σ m F ( φ(σ j ) )

30 Expectations on IFS Attractors Corollary (Fundamental definition of separation (using code-space)) Let {X ; f 1,..., f m } be an IFS with attractor A H(X ). Let F : X C be a complex-valued function over X. The separation expectation of F over A, F (x y) x,y A, is defined as: 1 F (x y) x,y A := lim j m 2j when the limit exists. F σ j Σ m τ j Σ m ( φ(σ j ) φ(τ j ) )

31 The invariant IFS measure Definition Let B be a Borel subset of a metric space (X, d). The residence measure is defined as: 1 µ(b) := lim n n ( { }) # k : f k (x) B, 1 k n The residence measure is a normalised, invariant measure over the attractor of any IFS.

32 The invariant IFS measure Theorem (Elton s Theorem - special case) Let (X, d) be a compact metric space and let {X ; f 1,..., f m } be a hyperbolic IFS.

33 The invariant IFS measure Theorem (Elton s Theorem - special case) Let (X, d) be a compact metric space and let {X ; f 1,..., f m } be a hyperbolic IFS. Let {x n } n=0 denote a chaos game orbit of the IFS starting at x 0 X,

34 The invariant IFS measure Theorem (Elton s Theorem - special case) Let (X, d) be a compact metric space and let {X ; f 1,..., f m } be a hyperbolic IFS. Let {x n } n=0 denote a chaos game orbit of the IFS starting at x 0 X, that is, x n = f σn... f σ1 (x 0 ) where the maps are chosen independently according to the probabilities p 1,..., p m for n N.

35 The invariant IFS measure Theorem (Elton s Theorem - special case) Let (X, d) be a compact metric space and let {X ; f 1,..., f m } be a hyperbolic IFS. Let {x n } n=0 denote a chaos game orbit of the IFS starting at x 0 X, that is, x n = f σn... f σ1 (x 0 ) where the maps are chosen independently according to the probabilities p 1,..., p m for n N. Let µ be the unique invariant measure for the IFS. Then, with probability 1, lim n 1 n + 1 n F (x k ) = k=0 X F (x)dµ(x)

36 Expectations over IFS attractors Corollary Let {X ; f 1, f 2,..., f m } be a contractive IFS with attractor A H(X ). Given a complex-valued function F : X C, the expectation of F over A is given by the integral: F (x) x A = F (x)dµ(x) X

37 Functional equations Proposition (Functional equations for expectations) For points x, y in the attractor A of a non-overlapping IFS, the expectations for a complex-valued function F satisfy the functional equations: F (x) = 1 m m F (f j (x)) j=1 F (x y) = 1 m 2 m j=1 k=1 m F (f j (x) f k (y))

38 Functional equations Proposition (Functional equations for expectations) For points x, y in the attractor A of a non-overlapping IFS, the expectations for a complex-valued function F satisfy the functional equations: and more generally F (x) = 1 m m F (f j (x)) j=1 F (x y) = 1 m 2 m F (x 1, x 2,..., x n ) = 1 m n m j=1 k=1 m j 1 =1 j 2 =1 m F (f j (x) f k (y)) m F (f j1 (x 1 ), f j2 (x 2 ),..., f jn (x n )) j n=1

39 Ongoing mathematics and computation

40 Exact evaluation of even moments Substitute a given IFS and function F into the functional equation: F (x y) = 1 m 2 m j=1 k=1 m F (f j (x) f k (y)) Simplify the resulting expression into a (linear) combination of n simpler expectations. Feed these expectations back into the functional equation to generate a system of n (linear) equations in the n unknown expectations. Solve the system of equations and hence determine the expectation.

41 Orthogonal Sierpinski Triangle B(2, A) = 10 27

42 Orthogonal Sierpinski Triangle B(2, A) = (2, A) = 8 27

43 Equilateral Sierpinski Triangle B(2, A) = 4 9

44 Equilateral Sierpinski Triangle B(2, A) = 4 9 (2, A) = 2 9

45 Koch Curve B(2, A) = 1 3

46 Koch Curve B(2, A) = 1 3 (2, A) = 4 27

47 Barnsley Fern

48 Barnsley Fern

49 Barnsley Fern

50 Barnsley Fern B(2, A) =

51 Barnsley Fern

52 Barnsley Fern

53 Barnsley Fern

54 Barnsley Fern (2, A) =

55 Current research: Poles for box integrals For classical box integrals over unit hypercubes, we have the following: Theorem (Absolutely-convergent analytic series for B n (s)) For all s C, B n (s) = n1+s/2 s + n ( 2 γ n 1,k n k=0 where the γ m,k are fixed real coefficients defined by the two-variable recursion: ) k (1 + 2k/m)γ m,k = (k 1 s/2)γ m,k 1 + γ m 1,k for m, k 1, with initial conditions γ 0,k := δ 0,k, γ m,0 := 1.

56 Current research: Poles for box integrals For classical box integrals over unit hypercubes, we have the following: Theorem (Absolutely-convergent analytic series for B n (s)) For all s C, B n (s) = n1+s/2 s + n ( 2 γ n 1,k n k=0 where the γ m,k are fixed real coefficients defined by the two-variable recursion: ) k (1 + 2k/m)γ m,k = (k 1 s/2)γ m,k 1 + γ m 1,k for m, k 1, with initial conditions γ 0,k := δ 0,k, γ m,0 := 1. Note the single pole at s = n, the negated dimension of the embedding space.

57 Current research: Poles for box integrals Theorem (Fractal Dimension with the Open Set Condition) Suppose that the open set condition holds for the IFS F = {R n ; f 1, f 2,..., f m } (with associated contraction factors {c 1, c 2,..., c m }). That is, the attractor contains a non-empty set O A which is open in the metric space A such that 1. f i (O) f j (O) = for all i, j {1, 2,..., m} with i j 2. m i=1 f i(o) O Then the Hausdorff dimension and Minkowski box-counting dimension of the attractor of the IFS are equal and take the value δ, where: m (c i ) δ = 1. i=1

58 Current research: Poles for box integrals Using the functional expectation relations, we can prove: Proposition (SCS: Pole of B(s, C n (P))) For any SCS C n (P), the box integral B(s, C n (P)) has a pole at s = δ(c n (P)).

59 Current research: Poles for box integrals Using the functional expectation relations, we can prove: Proposition (SCS: Pole of B(s, C n (P))) For any SCS C n (P), the box integral B(s, C n (P)) has a pole at s = δ(c n (P)). Proposition (IFS: Pole of B(s) over Uniform Affine IFSs) Let F = {X ; f 1, f 2,..., f m } be a contractive affine IFS satisfying the open set condition with uniform contraction factors; that is, c 1 = c 2 =... = c m. Then the box integral B(s, A) over the attractor A H(X ) has a pole at s = δ(a)

60 Current research: Poles for box integrals Proposition (IFS: Bounds on pole of (s) over Similarity IFSs) Let F = {X ; f 1, f 2,..., f m } be a contractive similarity IFS satisfying the open set condition; that is, f i (x) f i (y) = c i x y for all i. Then, if the box integral A (s) over the attractor A H(X ) has a pole on the real axis, the pole is bounded by: log(m) log(c max ) s log(m) log(c min ) where c max = c = max{c 1,..., c m } and c min = min{c 1,..., c m }.

61 Current research: Odd-order moments n s B(s, C 2 (2)) π log(1 + 2) log(1 + 2) log(1 + 2) Table : Closed-form results for B box integrals of various order s over the unit square and unit cube. For the unit n-cube all integer values for 1 n 5 have closed forms. Ti 2 is a generalized tangent (polylog) value and G is Catalan s constant.

62 Current research: Odd-order moments n s B(s, C 3 (3)) π arctan G π log(1 + 2) + 3 Ti 2 (3 2 2) π log ( ) π log ( ) π log ( )

63 Current research: Odd-order moments n s (s, C 2 (2)) ,-3, log(1 + 2) log(1 + 2) Table : Closed-form results for box integrals of various order s over the unit square and unit cube. For the unit n-cube all integer values for 1 n 5 have closed forms. Ti 2 is a generalized tangent (polylog) value and G is Catalan s constant.

64 Current research: Odd-order moments n s (s, C 3 (3)) 3 2 ( ) ( ) 2π 12 G + 12 Ti π log ( +2 log log arctan π + 2 ( ) log ( ) +12 log log ( ) π log ( ) ( + 8 log π ) 2 ) log ( ) log ( )

65 Current research: What are the odd moments? Insight into the evaluation of odd moments can be gleaned from computer-assisted mathematics. In particular, we are considering a modified Richardson Extrapolation Technique combined with the PSLQ Integer Relation Algorithm to hunt for closed forms (joint work with Nathan Clisby).

66 Current research: High-precision numerics A vector x = (x 1, x 2 x n ) of real numbers has an integer relation if there exists integers a i, not all zero, such that: n a i x i = 0. i=1

67 Current research: High-precision numerics A vector x = (x 1, x 2 x n ) of real numbers has an integer relation if there exists integers a i, not all zero, such that: n a i x i = 0. i=1 Given a vector x = (x 1, x 2 x n ), PSLQ iteratively constructs a sequence of integer-matrices B n that reduce the vector xb n. The process continues until either: 1. The smallest entry of the latest B n abruptly decreases to within drops to within ɛ of 0. This signals the detection of an integer relation, which PSLQ will produce as one of the columns of the last B n. 2. The available precision is exhausted. In this case, PSLQ will establish a bound on the size of any possible integer relation.

68 Current research: High-precision numerics A vector x = (x 1, x 2 x n ) of real numbers has an integer relation if there exists integers a i, not all zero, such that: n a i x i = 0. i=1 Given a vector x = (x 1, x 2 x n ), PSLQ iteratively constructs a sequence of integer-matrices B n that reduce the vector xb n. The process continues until either: 1. The smallest entry of the latest B n abruptly decreases to within drops to within ɛ of 0. This signals the detection of an integer relation, which PSLQ will produce as one of the columns of the last B n. 2. The available precision is exhausted. In this case, PSLQ will establish a bound on the size of any possible integer relation. In order to find an integer relation among n terms, PSLQ requires at least nd accurate digits in all terms, where d is the number of digits of the largest integer a i.

69 Current research: High-precision numerics The Richardson Extrapolation Technique combines multiple lower-accuracy evaluations to eliminate the highest-order error terms and thereby obtain a higher-accuracy evaluation. Start with an approximation formula A 1 (h) for quantity of interest x, accurate to O(h 1 ). That is, x = A 1 (h) + O(h 1 ) = A 1 (h) + c 1 h + c 2 h

70 Current research: High-precision numerics The Richardson Extrapolation Technique combines multiple lower-accuracy evaluations to eliminate the highest-order error terms and thereby obtain a higher-accuracy evaluation. Start with an approximation formula A 1 (h) for quantity of interest x, accurate to O(h 1 ). That is, x = A 1 (h) + O(h 1 ) = A 1 (h) + c 1 h + c 2 h Thus, halving the step-size h in the power series, ( ) h h x = A 1 + c c h

71 Current research: High-precision numerics The Richardson Extrapolation Technique combines multiple lower-accuracy evaluations to eliminate the highest-order error terms and thereby obtain a higher-accuracy evaluation. Start with an approximation formula A 1 (h) for quantity of interest x, accurate to O(h 1 ). That is, x = A 1 (h) + O(h 1 ) = A 1 (h) + c 1 h + c 2 h Thus, halving the step-size h in the power series, ( ) h h x = A 1 + c c h Combine these equations to eliminate c 1 h, yielding: x = A 2 (h) 1 2 c 2h c 3h = A 2 (h) + O(h 2 ) ( where A 2 (h) = A h ( ( 1 2) + 1 h ) 2 1 A1 1 2 A1 (h) ).

72 Current research: High-precision numerics Successive iterations over smaller step-sizes yield x = A k (h) + O(h k ), where ) + A k (h) = A k 1 ( h k 1 1 ( ( ) ) h A k 1 A k 1 (h). 2 Adapting this process to IFS attractors, Nathan Clisby has computed 112 digits of B(1, Sierpiński Triangle):

73 Current research: High-precision numerics Successive iterations over smaller step-sizes yield x = A k (h) + O(h k ), where ) + A k (h) = A k 1 ( h k 1 1 ( ( ) ) h A k 1 A k 1 (h). 2 Adapting this process to IFS attractors, Nathan Clisby has computed 112 digits of B(1, Sierpiński Triangle):

74 Current research: High-precision numerics Successive iterations over smaller step-sizes yield x = A k (h) + O(h k ), where ) + A k (h) = A k 1 ( h k 1 1 ( ( ) ) h A k 1 A k 1 (h). 2 Adapting this process to IFS attractors, Nathan Clisby has computed 112 digits of B(1, Sierpiński Triangle): We are aiming to improve this method by constructing an analogue of Bulirsch-Stöer extrapolation modified for IFS attractors, wherein the sequence of estimates is fitted to a rational function of h and evaluated at h = 0.

75 Thank you!