Set oriented numerics

Transcription

1 Set oriented numerics Oliver Junge Center for Mathematics Technische Universität München Applied Koopmanism, MFO, February 2016

2 Motivation Complicated dynamics ( chaos ) Edward N. Lorenz,

3 Sensitive dependence on initial conditions Henri Poincaré Science et Méthodes, 1908 Il peut arriver que de petites différences dans les conditions initiales en engendrent de trés grandes dans les phénomènes finaux. Un petite erreur sur les premières produirait une erreur énorme sur les dernières. La prédiction devient impossible et nous avons le phénomène fortuit... Henri Poincaré,

4 Motivation Conformation dynamics Picture: L.E. Kavraki (alanine dipeptide) Dynamics of the distance between two CH 2 groups in an MD simulation of a polymer chain of 100 CH 2 groups (Grubmüller, Tavan, 94).

5 Set oriented numerics M. Dellnitz, A. Hohmann, A subdivision algorithm for the computation of unstable manifolds and global attractors, Num. Math., 75(3): , Related approaches: Hsu ( cell mapping ), Osipenko

6 Attractors X R d state space f : X X continuous (e.g., f is a time-sampling of the flow of an ODE). For x 0 X, consider the iteration Q X compact, the set x k+1 = f (x k ), k = 0, 1, 2,... A Q = f k (Q) k=0 is the global attractor (relative to Q).

7 Attractors Properties of the (relative) global attractor A Q contains all invariant sets in Q A Q contains (part of) the unstable set of an invariant set in Q

8 Set oriented computation of the global attractor input: Q X compact output: sequence B 0, B 1,... of finite collections of compact subsets of Q. Set B 0 = {Q}. Given B k 1, B k is constructed in two steps: Subdivide each cell B B k 1 to cells ˆB ˆB k such that diam( ˆB) θ diam(b), θ (0, 1). Define B k = { ˆB ˆB k : f ( ˆB ) ˆB for some ˆB ˆB k }.

9 y Example: Global attractor of the Hénon-map f (x, y) = (1 ax 2 + y, bx) x

10 Example: Global attractor in the Lorenz system

11 Convergence Let obviously Q k+1 Q k. Q k = B B k B, Theorem (Dellnitz, Hohmann, 97) The sets Q k converge to A Q in Hausdorff-distance as k, i.e. lim h(q k, A Q ) = 0. k Equivalently let Q = k=0,1,... Q k, then Q = A Q.

12 Speed of convergence Theorem (Dellnitz, Hohmann, 97) Let A Q be the global attractor relative to Q with respect to the diffeomorphism f q, q N. If A Q is attractive and hyperbolic and [...] then h(a Q, Q k ) diam(b k )(1 + α + + α k ), where If α < 1 then α = C λq θ min. 1 h(a Q, Q k ) diam(b k ) 1 α.

13 Set oriented computation of chain recurrent sets M. Dellnitz, O. Junge, M. Rumpf, R. Strzodka: The computation of an unstable invariant set inside a cylinder containing a knotted flow, Proc. EQUADIFF, 1999.

14 Software: GAIO Global Analysis of Invariant Objects C-library and Matlab toolbox set collections: set of boxes B(c, r) = {x : x i c i r i, i = 1,..., n} subdivison by bisection storage in a binary tree Root

15 The transfer operator on L 1 : T µ(a) = µ(f 1 (A)) T h (x) = h ( f 1 x ) Df 1 x, properties (Markov operator) (i) T is linear, (ii) T h 0 if h 0, (iii) T h 1 = h 1 for all h 0. T 1 1, hence σ(t ) B 1 (0) fixed points are invariant measures (resp. densities)

16 Theory Example: invariant measure and almost invariant sets in the Lorenz system

17 Numerics Ulam s method partition B 1,..., B n of M (B i connected, positive volume) approximation space: n = span{χ B1,..., χ Bn } projection π n : L 1 n, π n f = n i=1 1 f dm m(b i ) B i approximate Frobenius-Perron operator: T t n := π n T t matrix representation (Tn t ) ij = m(b i Φ t B j ). m(b j )

18 Convergence h n fixed point of T n, Theorem (Li, 76) Let f : [0, 1] [0, 1] be a piecewise C 2 -map, such that inf x f (x) > 2. If f has a unique absolutely continuous invariant measure µ with density h, then h n h 1 0 as n.

19 Random perturbations Replace the deterministic evolution x k+1 = f (x k ), k = 0, 1,... by the randomly perturbed evolution x k+1 = f (x k )+ξ k, k = 0, 1,... where ξ k is chosen randomly from some ball B ε (0) according to (e.g) a uniform distribution.

20 Random perturbations Stochastic transition function p : X A [0, 1], where for every x the probability measure p(x, ) determines the distribution of the perturbation ξ k (for all k). Example: p(x, A) = p ε (x, A) = m(a B ε(f (x))). m(b ε (0))

21 Small random perturbations Definition A family p ε of stochastic transition functions is a small random perturbation of (the deterministic system) f, if lim sup ε 0 g(y) p ε (x, dy) g(f (x)) = 0. x X Theorem (Kifer, 86) Let p ε be a small random perturbation of f. For every ε let µ ε be the unique invariant measure of p ε, supported on a neighbourhood of some attractive hyperbolic invariant set of f. Let µ be the corresponding SRB-measure of f. Then (under additional assumptions) µ ε µ weakly as ε 0.

22 Convergence: the hyperbolic case Fix ε, consider p ε, transfer operator P ε, compact on L 2. discretization: P ε P ε d, fixed point uε d Theorem (Dellnitz, J., 99) Let µ be the unique SRB-measure supported on a topologically transitive hyperbolic attractor. Then lim lim ε 0 d uε d µ.

23 Background Almost invariant sets [ 1 ε ε T = ε 1 ε ] the two nodes are (1 ε)-almost invariant eigenvalues : 1 and = 0.98 eigenvectors: (1, 1) at 1 (1, 1) at 0.98 Note that 1 ε = 1 + λ 2. 2

24 Theory Example: invariant measure and almost invariant sets in the Lorenz system

25 Background Spectrum of transfer operator macroscopic dynamics eigenvalue 1 invariant measure eigenvalues λ 1 almost invariant sets A identification via eigenfunction: invariance ratio: A joint nodal domain of eigenfunction m(a Φ t (A)) m(a) λ Dellnitz, Junge: On the approximation of complicated dynamical behavior. SIAM J. Numerical Analysis Froyland, Dellnitz: Detecting and Locating Near-Optimal Almost-Invariant Sets and Cycles, SIAM J. Scientific Computing, 2003.

26 The transfer operator and its generator The infinitesimal generator {T t } t 0 continuous semigroup infinitesimal generator Af := lim t 0 T t f f t for f D(A), D(A) the subspace of X where the limit exists if the vector field F is C 1 Af = div(f F )

27 Numerics Ulam s method for the generator same partition, approximation space, projection n D(A), thus A n := π n A does not work instead define matrix representation lim t 0 (A n ) ij = Tn t f f A n f := lim, f n t 0 t lim t 0 m(b i Φ t B j ), t m(b j ) i j; m(b i Φ t B i ) m(b i ), t m(b i ) otherwise.

28 Numerics Spectral collocation need explicit noise for a well posed problem A ε = ε + A eigenfunctions of A ε are C approximation spaces V n M = S 1 : trigonometric polynomials collocation nodes: uniform grid M = [ 1, 1]: Chebyshev polynomials collocation nodes: Chebyshev grid I n : C V n interpolation in collocation nodes approximate generator I n A ε f, f V n

29 Example 1: A flow on the circle invariant density vector field F (x) = sin(4πx) unique invariant density with C such that f 1 = 1. f (x) = C/F (x)

30 Example 1: A flow on the circle Efficiency L 1 error histogram Ulam s method Ulam for the generator spectral collocation # of evaluations

31 Example 2: An area-preserving cylinder flow The vector field 8 x y x x 10 5 periodic in x, 0 at the y-boundaries

32 Example 2: An area-preserving cylinder flow The fourth eigenvector

33 Example 2: An area-preserving cylinder flow The largest eigenvalues method λ 2 λ 3 λ 4 Ulam (log(λ i )/T ) Ulam (generator) spectral collocation Numerical effort method # of rhs evals time matrix time eigs Ulam sec. 1.0 sec. Ulam (generator) sec. 0.8 sec. spectral collocation sec. 3 sec.

34 Example 3: A volume-preserving 3D flow: the ABC-flow The vector field ẋ = a sin(2πz) + c cos(2πy) ẏ = b sin(2πx) + a cos(2πz) ż = c sin(2πy) + b cos(2πx), with a = 3, b = 2 and c = 1.

35 Example 3: A volume-preserving 3D flow: the ABC-flow Almost invariant sets Ulam s method for the generator Second eigenvector Fifth eigenvector

36 Example 3: A volume-preserving 3D flow: the ABC-flow Almost invariant sets spectral collocation Second eigenvector Fifth eigenvector

37 Example 3: A volume-preserving 3D flow: the ABC-flow Efficiency Ulam s method for the generator boxes Gauss quadrature with 16 nodes on each face rhs evaluations error in the invariant density spectral collocation grid 10 3 rhs evaluations error in the invariant density 10 14

38 Motivation Challenges in controller design highly nonlinear hybrid, quantized, event-based

39 Optimization based controller design time-discrete control system x k+1 = f (x k, u k ), k = 0, 1,..., cost function: c(x, u) > 0 for x T, c(t, u) = 0. accumulated cost: J(x 0, (u k ) k ) = c(x k, u k ) k=0 value function: V (x) = inf (u k ) k J(x, (u k ) k ) optimality principle/bellman operator V (x) = inf {c(x, u) + V (f (x, u))}, V (0) = 0. u U optimal feedback: F (x) = argmin u U {c(x, u) + V (f (x, u))}

40 Quantization of the state information P N P 1 P 2 x [x] Partition P = {P 1,..., P N } of state space X x [x] P quantized state approximation space for the value function: A = span{χ P1,..., χ PN }

41 Discrete problem & solution (I) idea: projection onto A discrete optimality principle on A V ([x]) = inf inf {c(ˆx, u) + V (f (ˆx, u))}, V ([0]) = 0 ˆx [x] u U = shortest path problem on a directed weighted graph solution: Dijkstra s algorithm Theorem (J., Osinga, 04) The discrete value function converges pointwise to the true one.

42 Discrete problem & solution (II) idea: dynamic game, discretization as perturbation discrete optimality principle on A V ([x]) = inf sup {c(ˆx, u) + V (f (ˆx, u))}, V ([0]) = 0 u U ˆx [x] = shortest path problem on a directed weighted hypergraph solution: minmax-dijkstra algorithm [Grüne, J., 08] Theorem (Grüne, J., 07) The discrete value function is a Lyapunov function for the closed loop system.

43 Convergence Theorem (Grüne, J., 08) P i nested sequence of partitions with diam P i 0 c continuous, c(x, u) > 0 for x T V continuous on T Then V Pi V uniformly on sets on which V is continuous. If the set of discontinuities of V has Lebesgue measure 0, then V Pi V in L 1 on compact subsets of the domain of V.

44 Example: An inverted pendulum The model m ϕ u state: x = (ϕ, ϕ) control system: f (x, u) = Φ T (x, u) cost function c(x, u) = T 0 q 1 ϕ 2 (t) + q 2 ϕ 2 (t) dt + Tq 3 u 2

45 Example: An inverted pendulum Value function and feedback trajectory

46 Hybrid systems x k+1 = f c (x k, y k, u k ) y k+1 = f d (x k, y k, u k ) k = 0, 1,... x k X R d compact, y k Y finite 1. no discrete dynamics, control discrete (i.e. U finite) x k+1 = f c (x k, u k ) 2. state dependent switched system x k+1 = f c (x k, y k, u k ) y k+1 = f d (x k ) 3. x controlled by automaton, which is controlled by u x k+1 = f c (x k, y k ) y k+1 = f d (y k, u k )

47 Example: a switched voltage converter The model [Lincoln, Rantzer, 06] ẋ 1 = 1 C (x 2 I load ) ẋ 2 = 1 L x 1 R L x L uv in control: u { 1, 1}.

48 Example: a switched voltage converter Feedback trajectory 0.6 Voltage Current Switch sign Sample

49 Set oriented numerics is good for... computing low-dimensional sets (even in high dimensional state spaces) attractors, invariant sets (dynamical systems) manifolds (mission design) Pareto sets (global multi-objective optimization) computing non-smooth functions on low-dimensional and irregular state spaces value functions (controller design) stability regions transport rates (astrodynamics) incorporating bounded uncertainties in a worst case way software: GAIO (new version 3.0 out soon) book in preparation