Dynamical systems perturbed by big jumps: meta stability and non local search

Transcription

1 Dynamical systems perturbed by big jumps: meta stability and non local search Ilya Pavlyukevich Humboldt Universität zu Berlin / Universität Heidelberg Institut für Mathematik Workshop III: Transport Systems in Geography, Geosciences, and Networks May 5 9, 28, IPAM, UCLA Supported by the DFG research projekt Stochastic Dynamics of Climate States

2 . Motivation Greenland ice-core data allows to reconstruct Earth s climate up to 2. years before present. International projects: GRIP (328 m), GISP2 ( m), NGRIP ( m)

3 2. Paleo proxy data. Dansgaard Oeschger events Paleo data proxies: oxygen isotopes, dust, volcanic markers etc. Global climate during the last glacial ( 2 b. p.) has experienced at least 2 abrupt and large amplitude shifts (Dansgaard Oeschger events) T Age (thousands of years before present) rapid warming by 5 C within at most a few decades plateau phase with slow cooling lasting several centuries rapid drop to cold stadial conditions Simulations: Ganopolsky&Rahmstorf, Potsdam Institute for Climate Impact Research

4 3. Paleo proxy data: detailed look. The calcium (Ca) signal from the GRIP ice core: about 8, data points from kyr to 9 kyr before present. Typical interjump time: 2 years, mean waiting time 47 years What triggers the transitions? Langevin equation for climate dynamics dx(t) = U (X(t)) dt + ε dl(t) U double-well potential, wells correspond to the climate states. P. Ditlevsen (Geophys. Res. Lett. 999): histogram analysis of the noise. Noise L has α stable component with α.75.

5 4. The random perturbation L. Lévy processes L is a Lévy process: independent increments L(s) L(t) L(s) B is a Brownian motion: independent increments stationary increments L(t) L(s) d = L(t s) stationary increments L() = stochastically continuous (no fixed jumps) B() = B(t) B(s) N (, t s) right continuous with left limits continuous paths Description via Fourier transform: ϕ t (λ) = Ee iλl(t) L(t + s) = L(t) + (L(t + s) L(t)) d = L(t) + L (s) ϕ t+s (λ) = ϕ s (λ) ϕ t (λ) ϕ t (λ) = Ee iλl(t) = e tφ(λ)

6 Poisson process, intensity c > Ee iλl(t) = e iλk e ct(ct)k = e c(eiλ ) k! k= = exp 5. Examples [ t ] (e iλy )cδ (dy) Compound Poisson [ process, c > ], jumps σ Ee iλl(t) = exp t (e iλy )cσ(dy) General case, Lévy Hinchin Formula Ee iλl(t) [ ( = exp t d 2 λ2 + iµλ + )].5 (e iλy iλy I{ y })ν(dy) d, µ R, ν({}) =, (y 2 )ν(dy) <

7 6. α stable Lévy processes (Lévy flights) Jump measure ν(y) = y +α, α (, 2). ( Ee iλl(t) = exp t (e iλy iλy I{ y }) dy y +α Markov process with a generator Af(x) = [f(x + y) f(x) yf (x)i{ y }] Af(x) = ( ) α/2 f(x) L is a purely jump process. ) dy y +α, fractional Laplacian Control of jumps: J(t) = L(t) L(t ) the jump size at time t >. ( dy N(t, A) = {s (, t] : J(s) A} Poisson t For example: {s (, t] : J(s) } Poisson ( t A y +α ) dy y +α )

8 7. α stable Lévy processes II α =.75 α =.75 Length of sample paths: finite for α (, ) and infinite for α [, 2). Self-similarity: (L(ct)) t = (c /α L(t)) t. Hurst parameter H = α 2. Heavy tails: P(L(t) u) c uα, u. Explicit form of the Fourier transform Ee iλl(t) = e c(α) λ αt, α = α = 2 Cauchy process Brownian motion π + x 2 e x2 2 2π

9 8. Object of study. Simple system with Lévy perturbation Small noise (ε ) asymptotics of solutions of SDE X ε x(t) = x U (X ε x(s)) ds + εl(t), ε. L α stable (symmetric) Lévy process (maybe + Brownian motion) Regular n well potential U(x) (smooth, U (m i ), U (s i ), U (x) > x +δ s sn, x ± ) x Meta stable behaviour Transitions between the wells m m n Regular one well potential (smooth, U () > ) U(x) Exit time: σ x (ε) = inf{t : X ε x(t) / [ b, a]} a, b < (b = ) b a x P. Imkeller & I. Pavlyukevich Stoch. Proc. Appl. 6, 26; J. Phys. A: Math. Gen. 39, 26; ESAIM: P&S, 28

10 9. What is known Freidlin, Wentzell (Random Perturbations of Dynamical Systems, 979): Gaussian perturbations: ˆX ε x(t) = x U ( ˆX ε x(s)) ds + εw (t), ε. Perturbation: εw standard Brownian motion of small amplitude (in R d ). Locally infinitely divisible perturbations leading to Gaussian εl ε with jump part [ ] A ε ν(x, dy) f(x) = f(x + εy) f(x) εy, f(x) R d \{} ε Lévy measure ν has all exponential moments. For example, εl ε (t) = επ /ε (t) t, π standard Poisson process. Effects of heavy tails Godovanchuk (Theor. Prob. Appl. 26, 982): Large deviations for Markov processes with heavy jumps. Samorodnitsky, Grigoriou (Stoch. Proc. Appl. 5, 69 97, 23): Tails of solutions of SDEs driven by Lévy processes with power tails.

11 . First exit: Gaussian vs. Lévy U(x) U( b)>u(a) h=u(a) U(x) b ˆX ε x(t) = x x a U ( ˆX ε x(s)) ds + εw (t) Large deviations (Freidlin Wentzell): b X ε x(t) = x x a U (X ε x(s)) ds + εl(t) P(e (2h δ)/ε2 < ˆσ x (ε) < e (2h+δ)/ε2 ) Mean exit time (Kramers, Day, Bovier): ε ( ) π 2h Eˆσ x (ε) U (a) U () exp ε 2 Exponential exit (Day, Bovier) ( ) ˆσx (ε) P Eˆσ x (ε) > u exp ( u) P( ε α δ < σ x (ε) < ε α+δ ) P Eσ x (ε) α [ ε α a α + ] b α ( ) σx (ε) Eσ x (ε) > u exp ( u)

12 . Transitions For small > denote B i = {y : y m i } and τ i x(ε) = inf{t : X ε x(t) j i B j } τ s U(x) s n x m n Theorem. For x B i the following holds as ε : m P(ε α τ i x(ε) > u) e q iu, Eτ i x(ε) ε α q i, q i = P(X ε (τ i x(ε)) B j ) q ij q i, q ij = R\(s i,s i ) (s j,s j ) dy y m i +α, dy y m i +α, i j.

13 Theorem Meta stability Let x (s j, s j ) and t >. Then X ε x ( ) t ε α d Y mj (t), where Y is a Markov chain on {m,..., m n } with a generator Q = (q ij ), q ii = q i. Remark. Y has the invariant measure π(dy) = π j >, n π j δ mj (dy), j= where Q T π =.

14 3. Meta stable behaviour. Gaussian case ˆX ε x(t) = x U ( ˆX ε x(s)) ds + εw (t) Different life times for each well: Eτ i x(ε) exp(v i /ε 2 ) n critical exponents = λ < λ < < λ n such that for Λ (λ i, λ i+ ) the process ˆX ( x ε te Λ/ε2) converges to µ(λ, x) Double well potential: critical time scale: λ ε 2π U () U (m 2 ) exp(2h/ε2 ) Convergence of fin. dim. distrib.: (Kipnis, Newman; Mathieu) ˆX ε (λ ε t) Ŷ (t) h < H m U(x) m 2 H h x Generator of Ŷ ( ), and Ŷ () = { m, if x <, m 2, if x >.

15 4. Doubts and questions X ε x(t) = x U (X ε x(s)) ds + εl(t), ε. Why α stable (non-gaussian, heavy-tail) noise? Why small noise?

16 5. A natural system with small noise Is there a Gaussian system with a priori small noise? Simulated annealing (Kirkpatrick et al., Geman&Geman, Černy, 985): Ẑ(t) = z σ(t) temperature, σ(t) as t. U (Ẑ(s)) ds + σ(s)dw (s), ANNEALING: gradual cooling of steel (copper, glass etc.) in order to induce softenness, to relieve internal stresses, to refine the crystalline structure. I. Pavlyukevich: Stoch. Proc. Appl., 28; J. Phys. A: Math. and Theor., 27

17 6. Gaussian diffusion: long time behaviour Stochastic optimisation: look for a global minimum m of U. ˆX(t) = x U ( ˆX(s)) ds + εw (t) Generator A ε f = ε2 2 f U f.5 Invariant measure: A εµ = µ ε (dx) = c(ε)e 2U(x)/ε2.5 dx µ ε δ m, ε The spectrum { Λ ε k, Ψε k } k of A ε is discrete. Λ ε =, Ψ ε (x) = Λ ε exp( Θ/ε 2 ) SPECTRAL GAP (Friedman, Day, Bovier et al.) E x f( ˆX ε (t)) = f, L 2 (µ ε )(x) + e Λε t f, Ψ ε L 2 (µ ε )Ψ ε (x) + Law( ˆX ε (t)) µ ε, t

18 Ẑ z (t) = z INTUITION: 7. Gaussian simulated annealing U (Ẑz(s)) ds + Ẑ(t) ˆX σ(t) (t) FOURIER EXPANSION : σ(s)dw (s), σ(t) = ( ) /2 θ. ln(λ + t) E,z f(ẑ(t)) f, L 2 (µ σ(t) ) + e Λσ(t) t f, Ψ σ(t) L 2 (µ σ(t) )Ψ σ(t) + CONVERGENCE: µ σ(t) δ m, Λ σ(t) t e Θ/σ(t)2 t = t (t + λ) Θ/θ {, θ > Θ,, θ < Θ, t +, θ > Θ Z(t) m. This works, see Chiang&Hwang&Sheu, Holley&Kusuoka&Stroock and others HOWEVER: very slow cooling, local search, unknown critical rate Θ...

19 8. Modifications: big jumps Szu and Hartley (Phys. Lett. A 22, 987): fast simulated annealing Discrete schema with Cauchy jumps ξ k X kh = X (k )h + U (X (k )h )h + ξ k (k )h U = U(X kh ) U(X (k )h ) Metropolis acceptance probability: Accept always if U Accept with probability exp ( U kh) if U > Non-local search Fast algebraic cooling Convergence??? Probably works??? Applications to image recognition

20 9. Simulated annealing with α stable processes Analogously to Gaussian simulated annealing: Z(t) = z U (Z(s)) ds + dl(s) (λ + s) θ, λ >, θ >. Convergence of Z(t) as t? Time-homogeneous case: (s j,s j ) ( ) t X ε ε α Y (t) Y Markov chain on {m,..., m n } with generator Q = (q ij ) dy q ij = y m i +α, q dy ii = q i = y m i +α R\(s i,s i ) Law(Y (t)) π, Q π = Z(t) X /(λ+t)θ (t) Y ( t ) (t + λ) αθ { π, αθ <, no convergence, αθ >.

21 Slow cooling: convergence Theorem 3. [slow cooling] Let αθ <. Then Eτ i (λ) λαθ q i, P(Z(τ i (λ)) B j ) q ij q i, λ, τ s U(x) s n m n x Z(t) π, t. m

22 2. Fast cooling: trapping Theorem 4. [trapping] Let αθ >. Then P(τ i (λ) < ) = O(λ αθ ), λ, Eτ i (λ) =

23 22. Gaussian vs. α stable simulated annealing Simulated annealing with α stable process: allows to determine the measure π, i.e. the sizes of potential wells the search is non-local polynomially fast cooling BUT: how to find the global minimum of U?

24 23. Simulated annealing with stable-like processes In progress: I.P., J. Comp. Phys., 27 V (t) = v U (V (s )) ds + dh(v (s ), s) (λ + s) θ, λ >, θ > H a stable-like process with the jump measure dy ν x (dy) = y +α(x) U(x) A α(x) s s n x a s s n x E m n E m n m α(x) = { A, U(x) E, trapping, a, U(x) > E, exit, m < a < A < 2, aθ < < Aθ.

25 24. An example of non-local search The lowest minmum: U(4.9, 9.9) =.46 The second { lowest minimum: U( 9.7,.) =.85.8, if U(x), α(x) = θ =.75.., if U(x) >, Look for a local minimum: U(m) V k = V k U(V k )h ξh k (V k ) (λ+(k )h) θ k 2 6, λ = 4, V [ 2, 2] 2, h =. Success ratio: 96 out of.

26 25. Numerical example in R Stable-like simulated annealing a =., A =.8, θ =.75 λ N first k = 5 4 k = 5 5 N k k N k k Gaussian SDE θ = 3 λ N first k = 5 4 k = 5 5 N k k N k k Fast simulated annealing λ N first k = 5 4 k = 5 5 N k k N k k Gaussian simulated annealing θ = 3 λ N first k = 5 4 k = 5 5 N k k N k k

27 26. Numerical example in R 4. Shekel s function S,4 (y) = i= c i + y a i 2, y R4. Stable-like simulated annealing a =.2, A =.9, θ =.6 λ N first k = 5 4 k = 5 5 N k k N k k Gaussian SDE θ = λ N first k = 5 4 k = 5 5 N k k N k k Fast simulated annealing λ N first k = 5 4 k = 5 5 N k k N k k Gaussian simulated annealing θ = λ N first k = 5 4 k = 5 5 N k k N k k

28 27. Numerical example in R 6. Hartman s function 4 6 H 4,6 (y) = c i exp a ij (y i p ij ) 2,, y R 6. i= j= Stable-like simulated annealing a =.5, A =.9, θ =.6 λ N first k = 5 4 k = 5 5 N k k N k k Gaussian SDE θ = 6 λ N first k = 5 4 k = 5 5 N k k N k k Fast simulated annealing λ N first k = 5 4 k = 5 5 N k k N k k Gaussian simulated annealing θ = 6 λ N first k = 5 4 k = 5 5 N k k N k k

29 Pictures Page Slides/greenland.gif Page 2 From: Ganopolski, A. and Rahmstorf, S., Abrupt glacial climate changes due to stochastic resonance. Physical Review Letters 88(3), 385+, 22. Page 3 From: Ditlevsen, P. D., Observation of α stable noise induced millenial climate changes from an ice record, Geophysical Research Letters 26(), , 999. Page Page 8 From: H. Szu, Automated fault recognition by image correlation neutral network technique. IEEE Transactions on Industrial Electronics, 4(2):97 28, 993. References. Imkeller, P. and Pavlyukevich, I. First exit times of SDEs driven by stable Lévy processes, Stochastic Processes and their Applications 6 (4), 6 642, Imkeller, P. and Pavlyukevich, I. Metastable behaviour of small noise Lévy driven diffusions, ESAIM: Probability and Statistics, Imkeller, P. and Pavlyukevich, I. Lévy flights: transitions and meta stability, Journal of Physics A: Mathematical and General 39, L237 L246, Pavlyukevich, I. Simulated annealing for Lévy driven jump diffusions, Stochastic Processes and their Applications 8, 7 5, Pavlyukevich, I. Cooling down Lévy flights, Journal of Physics A: Mathematical and Theoretical 4, , Pavlyukevich, I. Lévy flights, non-local search and simulated annealing, Journal of Computational Physics 226 (2), , 27 Ilya Pavlyukevich, Humboldt Universität zu Berlin, Institut für Mathematik, Rudower Chaussee 25, 2489 Berlin, Germany pavljuke@math.hu-berlin.de pavljuke 28