Topological Dynamics: Rigorous Numerics via Cubical Homology

Transcription

1 1 Topological Dynamics: Rigorous Numerics via Cubical Homology AMS Short Course on Computational Topology New Orleans, January 4, 2010 Marian Mrozek Jagiellonian University, Kraków, Poland

2 Outline 2 I. Topological methods in rigorous numerics of dynamical systems Motivation: chaos in Lorenz equations Conley index Interval arithmetic and dynamics of multivalued maps II. Computing homology of spaces and maps Set representation and cubical homology Geometric reduction algorithms Shaving Acyclic subspace algorithm Coreduction algorithm DMT based algorithm Computing homology of continuous maps Persistence III. A sample application: chaos for Hénon map

3 Edward Lorenz 3 during the 1950s became skeptical of the appropriateness of the mathematical models used in meteorology in 1963 published the famous paper: Deterministic Nonperiodic Flow the Lorenz equations: ẋ = σ(y x) ẏ = Rx y xz ż = xy bz Edward Lorenz

4 Ghost solutions 4 Consider the equation z =(αi z )z, z C The only periodic trajectory of this equation is the stationary point at the origin. Consider its Euler discretization Φ h (z) :=z(1 + h(αi z )) For every h>0 this discretization has invariant circles of radius r ± := 1 ± 1 h 2 α 2 h

5 Problem 5 Is some chaotic dynamics present in the Lorenz system? Or maybe it is only present in the numerical scheme? Or maybe the chaotic behaviour is only the consequence of the rounding errors?

6 Isolated invariant sets 6 Let T be continous (T = R) or discrete (T = Z) time. Let ϕ : X T X be a dynamical system. A trajectory of x X is ϕ(x) :={ ϕ(x, t) t T }. a compact set N is an isolating neighborhood iff x bd N ϕ(x) N. N is an isolating neighborhood iff Inv(N,ϕ) :={ x N ϕ(x) N } int N. A compact set S X is called an isolated invariant set if there exists an isolating neighborhood N such that S =Inv(N,ϕ). For continuous time (T = R), a compact set N is an isolating block iff the exit set N := {x N ɛ>0 : ϕ(x, t) N for 0 <t<ɛ} is closed.

7 Conley index 7 Theorem. (Conley and students, 1978) Assume continuous time (T = R). For every isolating neighborhood N of S there exists an isolating block M such that S M N. If M 1 and M 2 are two such blocks, then (M 1 /M 1, [M 1 ]) and (M 2/M 2, [M 2 ]) are homotopy equivalent and, in particular, H (M 1,M 1 ) = H (M 2,M 2 ). Charles Conley For T = R the cohomological Conley index of S and N is Con (N,ϕ) := Con (S, ϕ) :=H (M,M ).

8 An example 8

9 Discrete case (T = Z) 9 Let f : X X be the generator of ϕ, i.e.f(x) :=ϕ(x, 1). A pair of compact sets P =(P 1,P 2 ) is called an index pair for f and an isolated invariant set S iff (i) (positive relative invariance) f(p 2 ) P 1 P 2 (ii) (exit set) P 1 cl(f(p 1 ) \ P 1 ) P 2 (iii) (isolation) S =Inv(cl(P 1 \ P 2 ),f) int(p 1 \ P 2 ) H (P 1,P 2 ) is not an invariant.

10 Leray Functor 10 V 0 - the category of finitely dimensional vector spaces over R the generalized kernel of α V 0 (V,V ) is gker α := ker α n n N the Leray functor L : Endo(V 0 ) Auto(V 0 ): L(V,v) :=(V/gker v, v )

11 Index quadruples and index maps 11 A quadruple P =(P 1,P 2, P 1, P 2 ) is an index quadruple for f and S if (P 1,P 2 ) is an index pair for f and S and ( P 1, P 2 ) is a topological pair such that the map f P :(P 1,P 2 ) x f(x) ( P 1, P 2 ) ι P :(P 1,P 2 ) x x ( P 1, P 2 ) are well defined and ι P is an excision (induces an isomorphism in cohomology) Given an index quadruple, we define the index map as the composition I P := H (f P P) H (ι P ) 1

12 The Conley index for discrete dynamical systems 12 Theorem. (MM,1990,2005) For every isolating neighborhood N of f there exists an index quadruple P such that Inv(N,f) P 1 P 1 N. Moreover, if P and Q are two such quadruples, then L(H (P 1,P 2 ),I P ) = L(H (Q 1,Q 2 ),I Q ). The Conley index of f in N is (CH (N,f),χ(N,f)) := L(H (P 1,P 2 ),I P ).

13 An example 13

14 14 It is possible to characterize the existence of chaotic dynamics in terms of the Conley index.

15 15 Ch. Conley and students, original Conley index for flows J.W. Robbin, D. Salamon, the first Conley index for discrete dynamical systems based on shape theory and the inverse limit functor MM, cohomological Conley index based on Leray functor A. Szymczak, homotopy, Szymczak functor (most general) J. Franks, D. Richeson, a reformulation of Szymczak construction in terms of shift equivalence

16 Interval arithmetic 16 ˆR R fixed, finite set of representable numbers representable intervals: I := {[a, b] a, b ˆR,a b} For {+,,,/} and I,J Idenote by I J the smallest representable interval that contains {a b a I,b J}. Given the endpoints of I and J, one can easily construct the endpoints of I J. first proposed by M. Warmus in 1956 rediscovered by R.E. Moore in 1959

17 The simplest topological tool: Darboux property 17

18 Advanced tool: topological dynamics of multivalued maps 18

19 19 Interval arithmetic combined with classical numerical methods for dynamical systems provides rigorous enclosures of trajectories of dynamical systems in the form of so called combinatorial multivalued representation. There are algorithms which use this combinatorial multivalued representation to rigorously construct index pairs and index quadruples of dynamical system

20 From CAPD to CAPD::RedHom 20

21 From CAPD to CAPD::RedHom 21

22 From CAPD to CAPD::RedHom 22

23 From CAPD to CAPD::RedHom 23

24 From CAPD to CAPD::RedHom 24

25 Motivation 25

26 Goal 26 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

27 Set representation 27 Simplicial complex classical

28 Set representation 28 Simplicial complex classical Cubical set typical in imaging and rigorous numerics very efficient and fast representation (bitmaps)

29 Set representation 29 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

30 Input 30 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!

31 Cubical sets 31 An elementary interval is an interval [k, l] R such that k, l are integers satisfying 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. An elementary cube is full if all I i are nondegenerate. The set A R d is cubical if there exists a finite family A of elementary cubes such that A = A. A cubical set is a full cubical set if it is a union of full elementary cubes. Theorem. (Blass, Holsztyński, 1972) Every polyhedron is homeomorphic to a cubical set.

32 Cubical Chains 32 The group C q of cubical chains is a free group generated by elementary chains of dimension q. Boundary operator : C q C q 1 is given on generators by 0 if Q =[l], Q := [l +1] [l] if Q =[l, l +1]. I P +( 1) dim I I P if Q = I P. Theorem. =0

33 Cubical Homology 33 For an elementary chain c = n i=1 α iq 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). The qth cubical homology group of X is H q (X) :=ker q X / im q X

34 Standard approach 34 Immediate algebraization:

35 Standard approach 35 Immediate algebraization: generate the faces

36 Standard approach 36 Immediate algebraization: generate the faces construct the boundary maps D k =

37 Standard approach 37 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)

38 Geometric reduction algorithms 38 Geometric Reductions Reduce the set so that the representation used is preserved the homology is not changed build chain complex compute homology

39 Shaving 39 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

40 Acyclic subspace 40 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

41 Free face reductions 41 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:

42 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)

43 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 43

44 Coreduction algorithm 44

45 Coreductions for S-complexes 45 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)

46 Augmentible S-complexes 46 Definition. An S-complex is augmentible iff there exists ɛ : S 0 R (augmentation) such that ɛ(t) 0for t S 0 t κ(s, t)ɛ(t) =0for s S 1 Coreductions may be applied to any augmentible S-complexes.

47 Coreduction algorithm 47 Unlike torus, coreductions of Bing s House result in a non-augmentible S- complex.

48 Morse-Forman theory 48 K- the collection of cells of a finite, regular CW complex. Adherence relation: τ σ τ σ and dim τ =dimσ 1 bd σ := { τ τ σ }. cbd σ := { ρ σ ρ }. Definition. (Forman, 1995) f : K Z is a discrete Morse function if for every σ K c + (σ) :=card{ ρ cbd σ f(ρ) f(σ) } 1 and c (σ) :=card{ τ bd σ f(τ) f(σ) } 1. Acellσ Kis critical if c + (σ) =c (σ) =0.

49 Example 49

50 Morse-Froman Homology 50 A discrete Morse function allows us to define paths joining the critical cells and via these paths one defines a boundary operator on the free complex generated by critical cells. Theorem. (Forman, 1995) Homology of the resulting Morse- Forman complex is the same as the homology of the original CW complex. Since often the number of critical cells is significantly smaller than the number of all cells, construction of the Morse-Forman complex may speed up homology computations if we can quickly construct a discrete Morse function.

51 Discrete Morse function by coreductions 51 Theorem. The algorithm runs in linear time.

52 CAPD::RedHom 52 New project: 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 a preliminary version available soon: Authors: P. D lotko, M. Juda, A. Krajniak, MM, H. Wagner,...

53 Numerical examples - manifolds 53 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

54 Numerical examples - Cahn-Hillard 54 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

55 Numerical examples - random sets 55 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

56 Numerical examples - simplicial sets 56 random set Björner set 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

57 Reduction equivalences 57 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 =dimb 1, ψ (a,b) k (c) := c c, b b for k =dimb, c otherwise, and ι (a,b) k (c) := { c c,a b,a b for k =dimb, c otherwise, are mutually inverse chain equivalences.

58 Homology model 58 the reduced S-complex S f thehomology 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!

59 Homology generators 59 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).

60 z = r q j=1 Homology generators 60 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

61 Homology of cubical maps 61 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.

62 Graph approach 62

63 The homology map algorithm 63 (1) Construct a combinatorial enclosure 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

64 Coreduction model approach, MM Using coreductions construct homology models of X, Y,andG 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

65 Graph approach versus coreduction model approach 65 Set Emb Size CHomP RedHom speedup Dim 10 6 homcubes CR z2torus8.map z2torus12.map z2torus16.map z2torus19.map

66 Cubical persistence via coreductions and inclusions 66 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 ).

67 Direct coreduction approach to cubical persistence 67 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)

68 Timings 68 Grid Levels classical RedHom RedHom approach (*) CR-incl. CR-direct (*) - implementation of Edelsbrunner-Letscher-Zomorodian algorithm for cubical sets by V. Nanda

69 Sample application: chaos in Hénon map 69 Consider the Hénon map h : R 2 R 2 given by the formula h(x, y) :=(1+y/5 ax 2, 5bx) at the classical parameter values a =1.4 and b =0.2. Theorem. (T. Kaczynski, K. Mischaikow, MM, 2004) Let S be the maximal invariant subset of the union of constructed connected components C j. Then there exists a semiconjugacy with a subshift of finite type on 8 symbols and topological entropy h =0.28 such that for each periodic sequence θ Σ A with period pρ 1 (θ) contains a periodic orbit with period p. In particular h(s) Theorem. (S. Day, R. Frongilo, R. Trevino, 2008) The classical Hénon map admits an isolated invariant set S semiconjugated to a subshift on 129 symbols and h(s) 0.42.

70 Applications 70 Lorenz equations: K. Mischaikow, A. Szymczak, MM, J. Diff. Equ., 1995, Time series dynamics: K. Mischaikow, J. Reiss, A. Szymczak, MM, Phys. Rev. Lett., Kot-Schaffer map in L 2 ([ π, π]): S. Day, O. Junge, K. Mischaikow, SIAM Dyn. Syst., Hénon map: S. Day, R. Frongilo, R. Trevino, SIAM J. App. Dyn. Sys., Databases for multiparameter systems: Z. Arai, H. Kokubu, W. Kalies, K. Mischaikow, H. Oka, P. Pilarczyk, SIAM J. App. Dyn. Sys., 2009.

71 Conclusion 71 Direct algebraization need not be the quickest way to compute homology of sets and maps. Reduction algorithms operating directly on sets substantially speed up homology copmutations. A universal fastest algorithm probably does not exist and hybrid methods probing data should be used for best performance. There is potential for further improvement in homology computations.

72 References 72 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). M. Juda, MM, Z 2 -Homology of 2-manifolds may be computed in O(nα(n)) time, preprint (2009). P. D lotko, T. Kaczynski, MM, T. Wanner, Coreduction Homology Algorithm for Regular CW-Complexes, Discrete and Computational Geometry (accepted). MM, T. Wanner, Coreduction Homology Algorithm for Inclusions and Persistence, Computers and Mathematics with Applications (accepted). 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). MM, H. Wagner, Coreduction Homology Algorithm for Continuous Maps, in preparation. A. Krajniak, MM, T. Wanner, Direct coreduction approach to persistence, in preparation.