Computational Homology in Topological Dynamics

Transcription

1 1 Computational Homology in Topological Dynamics ACAT School, Bologna, Italy MAY 26, 2012 Marian Mrozek Jagiellonian University, Kraków

2 Outline 2 Dynamical systems Rigorous numerics of dynamical systems Homological invariants of dynamical systems Computing homological invariants Homology algorithms for subsets of R d Homology algorithms for maps of subsets of R d Applications

3 Goal 3 Input: representation of a subset X R d Output: Betti numbers, torsion coefficients, homology generators

4 Goal 3 Input: representation of a subset X R d Output: Betti numbers, torsion coefficients, homology generators Input: representation of a continuous map f : X Y of subsets of R d Output: matrix of map induced in homology

5 Prototype homology algorithm 4 In principle, the homology algorithm may look as follows:

6 Prototype homology algorithm 4 In principle, the homology algorithm may look as follows: (1) Triangulate the space

7 Prototype homology algorithm 4 In principle, the homology algorithm may look as follows: (1) Triangulate the space (2) Construct the matrices of boundary maps

8 Prototype homology algorithm 4 In principle, the homology algorithm may look as follows: (1) Triangulate the space (2) Construct the matrices of boundary maps (3) Compute the kernel and the image

9 Prototype homology algorithm 4 In principle, the homology algorithm may look as follows: (1) Triangulate the space (2) Construct the matrices of boundary maps (3) Compute the kernel and the image (4) Compute the quotient space

10 Set representation 5 Simplicial complex classical

11 Set representation 6 Simplicial complex classical Cubical set typical in imaging and rigorous numerics very efficient and fast representation (bitmaps)

12 Set representation 7 Simplicial complex classical Cubical set typical in imaging and rigorous numerics very efficient and fast representation (bitmaps) General polyhedrons most general obtaining the chain complex is not straightforward

13 Cube triangulation 8 How many simplices do we need to triangulate a d-cube?

14 Cube triangulation 8 How many simplices do we need to triangulate a d-cube? Not more than d! but can we do better?

15 Cube triangulation 8 How many simplices do we need to triangulate a d-cube? Not more than d! but can we do better?

16 Cube triangulation 9 Theorem. Hughes, Anderson (1995), Bliss, Su (2005) d T v (d) T (d) ??? Theorem. Smith (2000) C(d) 6 d/2 d! 2(d + 1) (d+1)/2

17 Input 10 Algebraists expect matrices of boundary maps as input of homology computations.

18 Input 10 Algebraists expect matrices of boundary maps as input of homology computations. On input: a set represented as a list of top dimensional cells (cubes, simplices,...) Generation of faces, incidence coefficients and boundary maps, whenever necessary, must be considered a part of the job!

19 Elementary intervals and cubes 11 An elementary interval is an interval [k, l] R such that l = k (degenerate) or l = k + 1 (nondegenerate).

20 Elementary intervals and cubes 11 An elementary interval is an interval [k, l] R such that l = k (degenerate) or l = k + 1 (nondegenerate). An elementary cube Q in R d is I 1 I 2 I d R d.

21 Elementary intervals and cubes 11 An elementary interval is an interval [k, l] R such that l = k (degenerate) or l = k + 1 (nondegenerate). An elementary cube Q in R d is I 1 I 2 I d R d. The dimension of Q is the number of nondegenerate I i.

22 Elementary intervals and cubes 11 An elementary interval is an interval [k, l] R such that l = k (degenerate) or l = k + 1 (nondegenerate). An elementary cube Q in R d is I 1 I 2 I d R d. The dimension of Q is the number of nondegenerate I i. K the set of all elementary cubes in R d

23 Elementary intervals and cubes 11 An elementary interval is an interval [k, l] R such that l = k (degenerate) or l = k + 1 (nondegenerate). An elementary cube Q in R d is I 1 I 2 I d R d. The dimension of Q is the number of nondegenerate I i. K the set of all elementary cubes in R d K k the set of all elementary cubes in R d of dimension k

24 Elementary intervals and cubes 11 An elementary interval is an interval [k, l] R such that l = k (degenerate) or l = k + 1 (nondegenerate). An elementary cube Q in R d is I 1 I 2 I d R d. The dimension of Q is the number of nondegenerate I i. K the set of all elementary cubes in R d K k the set of all elementary cubes in R d of dimension k An elementary cube is full if its dimension is d.

25 Elementary intervals and cubes 11 An elementary interval is an interval [k, l] R such that l = k (degenerate) or l = k + 1 (nondegenerate). An elementary cube Q in R d is I 1 I 2 I d R d. The dimension of Q is the number of nondegenerate I i. K the set of all elementary cubes in R d K k the set of all elementary cubes in R d of dimension k An elementary cube is full if its dimension is d. For A K we use notation A := A.

26 Elementary intervals and cubes 11 An elementary interval is an interval [k, l] R such that l = k (degenerate) or l = k + 1 (nondegenerate). An elementary cube Q in R d is I 1 I 2 I d R d. The dimension of Q is the number of nondegenerate I i. K the set of all elementary cubes in R d K k the set of all elementary cubes in R d of dimension k An elementary cube is full if its dimension is d. For A K we use notation A := A. For A R d we use notation K(A) := { Q K Q A }.

27 Cubical Chains 12 Given an elementary cube Q we define the associated elementary chain by { 1 if P = Q Q(P ) = 0 otherwise.

28 Cubical Chains 12 Given an elementary cube Q we define the associated elementary chain by { 1 if P = Q Q(P ) = 0 otherwise. A cubical chain is a finite linear combination of elementary chains of the same dimension, called the dimension of the chain.

29 Cubical Chains 12 Given an elementary cube Q we define the associated elementary chain by { 1 if P = Q Q(P ) = 0 otherwise. A cubical chain is a finite linear combination of elementary chains of the same dimension, called the dimension of the chain. All cubical chains of dimension q form an Abelian group, denoted C q and called the group of q-chains.

30 Cubical Product 13 Given two elementary chains P, Q, we define their cubical product by P Q := P Q. and we extend this definition linearly to arbitrary chains.

31 Cubical Product 14

32 Boundary Operator 15 Boundary operator is a homomorphism : C q C q 1 given on generators by 0 if Q = [l], Q := [l + 1] [l] if Q = [l, l + 1]. Î P + ( 1) dim I Î P if Q = I P.

33 Boundary Operator 15 Boundary operator is a homomorphism : C q C q 1 given on generators by 0 if Q = [l], Q := [l + 1] [l] if Q = [l, l + 1]. Î P + ( 1) dim I Î P if Q = I P. Theorem. = 0

34 Chain groups of a cubical set 16 For an elementary chain c = n i=1 α i Q i we define its support by c := { Q i α i 0 }

35 Chain groups of a cubical set 16 For an elementary chain c = n i=1 α i Q i we define its support by c := { Q i α i 0 } Given a cubical set X we define the group of q-chains of X by C q (X) := { c C q c X }.

36 Chain groups of a cubical set 16 For an elementary chain c = n i=1 α i Q i we define its support by c := { Q i α i 0 } Given a cubical set X we define the group of q-chains of X by C q (X) := { c C q c X }. Is is easy to verify that we have the induced boundary operator q X : C q (X) C q 1 (X).

37 Cubical Homology 17 The kernel of q X Z q (X). is called the group of q-cycles of X and denoted by

38 Cubical Homology 17 The kernel of q X is called the group of q-cycles of X and denoted by Z q (X). The image of q+1 X is called the group of q-boundaries of X and denoted by B q (X).

39 Cubical Homology 17 The kernel of q X is called the group of q-cycles of X and denoted by Z q (X). The image of q+1 X is called the group of q-boundaries of X and denoted by B q (X). One can verify that B q (X) Z q (X), which allows us to define the qth homology group of X by H q (X) := Z q (X)/B q (X)

40 Cubical Homology 17 The kernel of q X is called the group of q-cycles of X and denoted by Z q (X). The image of q+1 X is called the group of q-boundaries of X and denoted by B q (X). One can verify that B q (X) Z q (X), which allows us to define the qth homology group of X by H q (X) := Z q (X)/B q (X) By homology of X we mean the collection of all homology groups H(X) := {H q (X)}.

41 Standard approach 18 Immediate algebraization:

42 Standard approach 19 Immediate algebraization: generate the faces

43 Standard approach 20 Immediate algebraization: generate the faces construct the boundary maps D k =

44 Standard approach 21 D k = Immediate algebraization: generate the faces construct the boundary maps find Smith diagonalization and read Betti numbers B k = Q 1 D k R B k =

45 Standard approach 21 D k = Immediate algebraization: generate the faces construct the boundary maps find Smith diagonalization and read Betti numbers Advantages: standard linear algebra may be easily adapted to homology generators B k = Q 1 D k R B k =

46 Standard approach 21 D k = B k = Q 1 D k R B k = Immediate algebraization: generate the faces construct the boundary maps find Smith diagonalization and read Betti numbers Advantages: standard linear algebra may be easily adapted to homology generators Problems: constructing faces immediately may increase data size complexity: Cn 3 sparseness of matrices may not help (fill-in process) C large for sparse matrices (dynamic storage allocation)

47 Geometric reduction algorithms 22 Geometric Reductions Reduce the set so that the representation used is preserved the homology is not changed build chain complex compute homology

48 Shaving 23 If X = X is cubical and Q X is an elementary cube such that Q X is acyclic and X = (X \ {Q}) then H(X) = H(X ) full cubes representation is used! acyclicity tests via lookup tables: 2 3d 1 entries extremely fast in dimension 2 and 3

49 Shaving 23 If X = X is cubical and Q X is an elementary cube such that Q X is acyclic and X = (X \ {Q}) then H(X) = H(X ) full cubes representation is used! acyclicity tests via lookup tables: 2 3d 1 entries extremely fast in dimension 2 and 3 not enough memory for dimension above 3 partial acyclicity tests in higher dimensions

50 Acyclic subspace 24 If X is cubical and A X is acyclic then { H n (X) H n (X, A), for n 1 = Z H n (X, A), for n = 0 breadth-first search construction

51 Acyclic subspace 24 If X is cubical and A X is acyclic then { H n (X) H n (X, A), for n 1 = Z H n (X, A), for n = 0 breadth-first search construction

52 Free face reductions 25 foreach σ do if cbd(σ) = {τ} then remove(σ); remove(τ); endif; endfor; free face - a generator with exactly one generator in coboundary a combinatorial counterpart of deformation retraction on algebraic level:

53 Dual reductions?

54 Dual reductions? free coface - a generator with exactly one generator in boundary

55 Dual reductions? free coface - a generator with exactly one generator in boundary one space homology theory with compact supports for locally compact sets (Steenrod 1940, Massey 1978)

56 Dual reductions? free coface - a generator with exactly one generator in boundary one space homology theory with compact supports for locally compact sets (Steenrod 1940, Massey 1978) combinatorial version (MM, B. Batko, 2006)

57 Q := empty queue; enqueue(q,s); while Q do s:=dequeue(q); if bd S s = {t} then remove(s); remove(t); enqueue(q, cbd K t); else if bd S s = then enqueue(q, cbd K s); endif; endwhile ; Coreduction algorithm 27

58 Q := empty queue; enqueue(q,s); while Q do s:=dequeue(q); if bd S s = {t} then remove(s); remove(t); enqueue(q, cbd K t); else if bd S s = then enqueue(q, cbd K s); endif; endwhile ; Coreduction algorithm 27

59 Coreduction algorithm 28

60 Coreductions for S-complexes 29 S-complex - a free chain complex with a fixed basis S which allows computation of incidence coefficients κ(s, t) directly from the coding of the basis

61 Coreductions for S-complexes 29 S-complex - a free chain complex with a fixed basis S which allows computation of incidence coefficients κ(s, t) directly from the coding of the basis Examples: cubical complexes, simplicial complexes

62 Coreductions for S-complexes 29 S-complex - a free chain complex with a fixed basis S which allows computation of incidence coefficients κ(s, t) directly from the coding of the basis Examples: cubical complexes, simplicial complexes Rectangular CW-complexes (P. D lotko, T. Kaczynski, MM, T. Wanner, 2010)

63 Augmentible S-complexes 30 Definition. An S-complex is augmentible iff there exists ɛ : S 0 R (augmentation) such that ɛ(t) 0 for t S 0 t κ(s, t)ɛ(t) = 0 for s S 1 Coreductions may be applied to any augmentible S-complexes.

64 Coreduction algorithm 31 Unlike torus, coreductions of Bing s House result in a non-augmentible S- complex.

65 CAPD::RedHom 32

66 CAPD::RedHom 32 A subproject of CAPD (

67 CAPD::RedHom 32 A subproject of CAPD ( A sister project of CHomP (

68 CAPD::RedHom 32 A subproject of CAPD ( A sister project of CHomP ( Generic homology software based on geometric reductions

69 CAPD::RedHom 32 A subproject of CAPD ( A sister project of CHomP ( Generic homology software based on geometric reductions AS, CR, DMT algorithms

70 CAPD::RedHom 32 A subproject of CAPD ( A sister project of CHomP ( Generic homology software based on geometric reductions AS, CR, DMT algorithms Betti and torsion numbers, homology generators, homology maps, persistence intervals

71 CAPD::RedHom 32 A subproject of CAPD ( A sister project of CHomP ( Generic homology software based on geometric reductions AS, CR, DMT algorithms Betti and torsion numbers, homology generators, homology maps, persistence intervals Z and Z p coefficients

72 CAPD::RedHom 32 A subproject of CAPD ( A sister project of CHomP ( Generic homology software based on geometric reductions AS, CR, DMT algorithms Betti and torsion numbers, homology generators, homology maps, persistence intervals Z and Z p coefficients generic but efficient: for cubical sets, simplicial sets, cubical CW complexes,...

73 CAPD::RedHom 32 A subproject of CAPD ( A sister project of CHomP ( Generic homology software based on geometric reductions AS, CR, DMT algorithms Betti and torsion numbers, homology generators, homology maps, persistence intervals Z and Z p coefficients generic but efficient: for cubical sets, simplicial sets, cubical CW complexes,... written in C++, based on C++ templates and generic programming Authors: P. D lotko, M. Juda, A. Krajniak, MM, H. Wagner,...

74 Rectangular CW-complexes 33

75 Rectangular CW-complexes 34 Theorem. (P. D lotko, T. Kaczynski, MM, T. Wanner, 2010) Consider a rectangular CW-complex given by a rectangular structure Q. Let P and Q denote two arbitrary rectangles in Q with dim Q = 1 + dim P, and define the number α QP as follows. For d = 1 and Q = [a, b] let 1 if P = [a], α QP := 1 if P = [b], 0 otherwise, and for d > 1 set (1) α QP := ( 1) j 1 i=1 dim Q i α Qj P j if P < Q and P j < Q j, 0 otherwise. Then the numbers α QP are incidence numbers for the given rectangular CW-complex.

76 Rectangular CW complex versus cubical approach 35

77 Numerical examples - manifolds 36 T S 1 (S 1 ) 3 S 1 K T T P K dim size in millions H 0 Z Z Z Z Z H 1 Z 3 0 Z 2 + Z 2 Z 4 Z + Z 2 H 2 Z 3 0 Z + Z 2 Z 6 Z 2 2 H 3 Z Z Z 4 Z 2 H 4 Z Linbox::Smith > 600 > CHomP::homcubes RedHom::CR RedHom::CR+DMT

78 Numerical examples - Cahn-Hillard 37 P0001 P0050 P0100 dim size in millions H 0 Z 7 Z 2 Z H 1 Z 6554 Z 2962 Z 1057 H 2 Z 2 Linbox::Smith > 600 > 600 > 600 CHomP::homcubes RedHom::CR RedHom::CR+DMT RedHom::AS

79 Numerical examples - random sets 38 d4s8f50 d4s12f50 d4s16f50 d4s20f50 dim size in millions H 0 Z 2 Z 2 Z 2 Z 2 H 1 Z 2 Z 17 Z 30 Z 51 H 2 Z 174 Z 1389 Z 5510 Z H 3 Z 2 Z 15 Z 71 Z 179 Linbox::Smith 120 > 600 > 600 > 600 CHomP::homcubes RedHom::CR RedHom::CR+DMT

80 Numerical examples - simplicial sets 39 random set S 2 S 5 dim size in millions H 0 Z Z Z H 1 Z H 2 Z 84 Z 0 H 3 0 H 4 Z CHomP::homcubes RedHom::CR+DMT

81 Reduction equivalences 40 Theorem. Assume S is an S-complex resulting from removing an coreduction pair (a, b) in an S-omplex S. Then the chain maps ψ (a,b) : R(S) R(S ) and ι (a,b) : R(S ) R(S) given by c c,a b,a b for k = dim b 1, ψ (a,b) k (c) := c c, b b for k = dim b, c otherwise, and ι (a,b) k (c) := { c c,a b,a b for k = dim b, c otherwise, are mutually inverse chain equivalences.

82 Reduction equivalences 40 Theorem. Assume S is an S-complex resulting from removing an coreduction pair (a, b) in an S-omplex S. Then the chain maps ψ (a,b) : R(S) R(S ) and ι (a,b) : R(S ) R(S) given by c c,a b,a b for k = dim b 1, ψ (a,b) k (c) := c c, b b for k = dim b, c otherwise, and ι (a,b) k (c) := { c c,a b,a b for k = dim b, c otherwise, are mutually inverse chain equivalences.

83 Homology model 41 the reduced S-complex S f the homology model of S as a convenient model to solve the problems of decomposing homology classes on generators.

84 Homology model 41 the reduced S-complex S f the homology model of S as a convenient model to solve the problems of decomposing homology classes on generators. π f : R(S) R(S f ) and ι f : R(S f ) R(S) mutually inverse chain equivalences obtained by composing the maps π (a,b) and ι (a,b).

85 Homology model 41 the reduced S-complex S f the homology model of S as a convenient model to solve the problems of decomposing homology classes on generators. π f : R(S) R(S f ) and ι f : R(S f ) R(S) mutually inverse chain equivalences obtained by composing the maps π (a,b) and ι (a,b). used to transport homology classes between H (S) and H (S f )

86 Homology model 41 the reduced S-complex S f the homology model of S as a convenient model to solve the problems of decomposing homology classes on generators. π f : R(S) R(S f ) and ι f : R(S f ) R(S) mutually inverse chain equivalences obtained by composing the maps π (a,b) and ι (a,b). used to transport homology classes between H (S) and H (S f ) The cost of transporting one generator through π f or ι f in general may be quadratic.

87 Homology model 41 the reduced S-complex S f the homology model of S as a convenient model to solve the problems of decomposing homology classes on generators. π f : R(S) R(S f ) and ι f : R(S f ) R(S) mutually inverse chain equivalences obtained by composing the maps π (a,b) and ι (a,b). used to transport homology classes between H (S) and H (S f ) The cost of transporting one generator through π f or ι f in general may be quadratic. In the case of Free Face Reduction Algorithm and Free Coface Reduction Algorithm it is linear!

88 S q = { s q 1, sq 2,... sq r q }. Homology generators 42

89 Homology generators 42 S q = { s q 1, sq 2,... sq r q }. { [u 1 ], [u 2 ],... [u n ] } generators of the homology group H q (S).

90 Homology generators 42 S q = { s q 1, sq 2,... sq r q }. { [u 1 ], [u 2 ],... [u n ] } generators of the homology group H q (S). Task: decompose [z] H q (X) on homology generators n [z] = x i [u i ] i=1

91 Homology generators 42 S q = { s q 1, sq 2,... sq r q }. { [u 1 ], [u 2 ],... [u n ] } generators of the homology group H q (S). Task: decompose [z] H q (X) on homology generators n [z] = x i [u i ] Linear algebra problem z = i=1 n x i u i + c i=1 with unknown variables x 1, x 2,..., x n Z and c R q+1 (S).

92 z = r q j=1 Homology generators 43 z j s q j, u i = c = r q j=1 s q+1 k = r q j=1 r q+1 u ij s q j, c = r q a kj s q j j=1 ( rq+1 ) a kj y k s q j k=1 k=1 y k s q+1 k

93 z = r q j=1 Homology generators 43 z j s q j, u i = c = r q j=1 s q+1 k = r q j=1 r q+1 u ij s q j, c = r q a kj s q j j=1 ( rq+1 ) a kj y k s q j k=1 k=1 y k s q+1 k Thus, we get a system of r q linear equations with n + r q+1 unknowns r n q+1 z j = u ij x i + a kj y k for j = 1, 2,... r q. i=1 k=1 In case of large S the cost is huge! Solution: transport the problem via π f to homology model and solve it there

94 X, Y cubical complexes Homology of cubical maps 44

95 Homology of cubical maps 44 X, Y cubical complexes g : X Y is cubical if maps elementary cubes to elementary cubes. g induces chain map g # : R(X) R(Y )

96 Homology of cubical maps 44 X, Y cubical complexes g : X Y is cubical if maps elementary cubes to elementary cubes. g induces chain map g # : R(X) R(Y ) examples of interest: inclusions and projections

97 Homology of cubical maps 44 X, Y cubical complexes g : X Y is cubical if maps elementary cubes to elementary cubes. g induces chain map g # : R(X) R(Y ) examples of interest: inclusions and projections U := { [u 1 ], [u 2 ],... [u m ] } and W := { [w 1 ], [w 2 ],... [w n ] } bases of H q (X) and H q (Y )

98 Homology of cubical maps 44 X, Y cubical complexes g : X Y is cubical if maps elementary cubes to elementary cubes. g induces chain map g # : R(X) R(Y ) examples of interest: inclusions and projections U := { [u 1 ], [u 2 ],... [u m ] } and W := { [w 1 ], [w 2 ],... [w n ] } bases of H q (X) and H q (Y ) To find the matrix of g ecompose g # (u i ) on generators in W

99 Homology of cubical maps 44 X, Y cubical complexes g : X Y is cubical if maps elementary cubes to elementary cubes. g induces chain map g # : R(X) R(Y ) examples of interest: inclusions and projections U := { [u 1 ], [u 2 ],... [u m ] } and W := { [w 1 ], [w 2 ],... [w n ] } bases of H q (X) and H q (Y ) To find the matrix of g ecompose g # (u i ) on generators in W Using the diagram g # R(X) R(Y ) π f ι f R(Y f ) we can solve the problem in the homology model Y f, where it is much simpler.

100 Computing Homology of Maps 45 In principle, computing homology of a map f : X Y is a three step procedure: (1) Find a finite representation of f (2) Use it to build the chain map (3) Compute the map in homology from the chain map

101 Computing Homology of Maps 45 In principle, computing homology of a map f : X Y is a three step procedure: (1) Find a finite representation of f (2) Use it to build the chain map (3) Compute the map in homology from the chain map

102 Combinatorial representations 46 Let X = X be a full cubical set with representation X and let f : X X be a continuous map. We say that F : X X is a representation of f if f(q) int F(Q).

103 Combinatorial representations 46 Let X = X be a full cubical set with representation X and let f : X X be a continuous map. We say that F : X X is a representation of f if f(q) int F(Q). and for each x X the set F (x) := { F(Q) x Q X } is acyclic.

104 Combinatorial representations 46 Let X = X be a full cubical set with representation X and let f : X X be a continuous map. We say that F : X X is a representation of f if f(q) int F(Q). and for each x X the set F (x) := { F(Q) x Q X } is acyclic. A representation for a given X may not exist. However, under a sufficiently good subdivision, the represntation always exists.

105 Combinatorial representations 46 Let X = X be a full cubical set with representation X and let f : X X be a continuous map. We say that F : X X is a representation of f if f(q) int F(Q). and for each x X the set F (x) := { F(Q) x Q X } is acyclic. A representation for a given X may not exist. However, under a sufficiently good subdivision, the represntation always exists. Theorem. (Allili, Kaczynski, 2000, Kaczynski, Mischaikow, MM 2004) If F is a representation of f, then there is an algorithm which transforms F into a chain map whose homology is the homology of f.

106 Graph approach (Granas and L. Górniewicz, 1981) 47

107 The homology map algorithm (Mischaikow, MM, Pilarczyk 2005) 48 (1) Construct a representation F of f : X Y. (2) Construct the graph G of F := F. (3) If the homologies of the values of F are not trivial, refine the grid and go to 1. (4) Apply shaving to X, Y and G in such a way that the shaved G is the graph of an acyclic mv map F : X Y (5) Find the homologies of the projections p : G X and q : G Y. (6) Return H (q)h (p) 1

108 The homology map algorithm (Mischaikow, MM, Pilarczyk 2005) 48 (1) Construct a representation F of f : X Y. (2) Construct the graph G of F := F. (3) If the homologies of the values of F are not trivial, refine the grid and go to 1. (4) Apply shaving to X, Y and G in such a way that the shaved G is the graph of an acyclic mv map F : X Y (5) Find the homologies of the projections p : G X and q : G Y. (6) Return H (q)h (p) 1 Pilarczyk (2005) implementation satisfactorily fast for a class of practical problems remains computationally most expensive part in applications in dynamics preserving the acyclicity of values when applying reductions is computationally expensive

109 Coreduction model approach, MM Using coreductions construct homology models of X, Y, and G Using homology models find the homology of the projections p : G X and q : G Y Compute the inverse of p and return q p 1 No need to preserve the acyclicity under reductions significantly faster than the previous graph approach

110 Graph approach versus coreduction model approach 50 Set Emb Size CHomP RedHom speedup Dim 10 6 homcubes CR z2torus8.map z2torus12.map z2torus16.map z2torus19.map

111 Alternative based on Čech structures (outline) 51 choose a Čech structure X on X for Q X take F(Q) as a convex enclosure of f(q) obtained via rigorous numerics F : K(X ) K(X F(X )) acts as a simplicial map the homology of f is computed when K(X ) K(X F(X )) induces an isomorphism this is guaranteed when the enclosure is good enough

112 Alternative based on Čech structures (outline) 51 choose a Čech structure X on X for Q X take F(Q) as a convex enclosure of f(q) obtained via rigorous numerics F : K(X ) K(X F(X )) acts as a simplicial map the homology of f is computed when K(X ) K(X F(X )) induces an isomorphism this is guaranteed when the enclosure is good enough

113 Alternative based on Čech structures 52 Example: consider f : C z z 2 C

114 Alternative based on Čech structures 52 Example: consider f : C z z 2 C

115 Cubical persistence via coreductions and inclusions 53 Assume field coefficients Compute the maps induced in homology by inclusions Find the compositions and ranks of the respective matrices βq i,j := rank ι i,j q, Compute the number of (i, j)-persistence intervals from formula pi q (i, j) = ( β i,j 1 q β i 1,j 1 q ) ( β i,j q β i 1,j q ).

116 Direct coreduction approach to cubical persistence 54 levelwise coreductions

117 Direct coreduction approach to cubical persistence 54 levelwise coreductions seperate queue for BFS on each level

118 Direct coreduction approach to cubical persistence 54 levelwise coreductions seperate queue for BFS on each level selection always from the lowest level non-empty queue

119 Direct coreduction approach to cubical persistence 54 levelwise coreductions seperate queue for BFS on each level selection always from the lowest level non-empty queue result: coreductions for all sublevel sets together in O(n log n) time

120 Direct coreduction approach to cubical persistence 54 levelwise coreductions seperate queue for BFS on each level selection always from the lowest level non-empty queue result: coreductions for all sublevel sets together in O(n log n) time Complexity of finding persistence intervals on the plane is O(n log n)

121 Timings 55 Grid Levels classical RedHom RedHom approach (*) CR-incl. CR-direct (*) - implementation of Edelsbrunner-Letscher-Zomorodian algorithm for cubical sets by V. Nanda

122 References 56 MM, P. Pilarczyk, N. Żelazna, Homology algorithm based on acyclic subspace, Computers and Mathematics with Applications (2008). MM, B. Batko, Coreduction homology algorithm, Discrete and Computational Geometry (2009). K. Mischaikow, MM, P. Pilarczyk, Graph Approach to the Computation of the Homology of Continuous Maps, Foundations of Computational Mathematics (2005). MM, Čech Type Approach to Computing Homology of Maps, Discrete and Computational Geometry (2010). P. D lotko, T. Kaczynski, MM, T. Wanner, Coreduction Homology Algorithm for Regular CW-Complexes, Discrete and Computational Geometry (2011). MM, T. Wanner, Coreduction Homology Algorithm for Inclusions and Persistence, Computers and Mathematics with Applications (2010). S. Harker, K. Mischaikow, MM, V. Nanda, H. Wagner, M. Juda, P. D lotko, The Efficiency of a Homology Algorithm based on Discrete Morse Theory and Coreductions, Proceedings CTIC 2010 (2010). A. Krajniak, MM, T. Wanner, Direct coreduction approach to persistence, in preparation.