Front Propagation in the Simplex Algorithm Eli Ben-Naim Los Alamos - - PowerPoint PPT Presentation

Front Propagation in the Simplex Algorithm

Eli Ben-Naim

Los Alamos National Laboratory

Talk, paper available from: http://cnls.lanl.gov/~ebn

• T. Antal, D. Ben-Avraham, E. Ben-Naim, and P.L. Krapivsky, J. Phys. A 41, 465002 (2008)

Physics of Algorithms, Santa Fe, August 31, 2009

with: Tibor Antal (Harvard) Daniel Ben-Avraham (Clarkson) Paul Krapivsky (Boston)

A peculiar spin-flip process

• System: unbounded one-dimensional lattice of spins
• Dynamics:
• Each 1-spin flips independently
• A flip causes all spins to the right to flip as well
• Every 1-spin affects an infinite number of spins!

...00110101... ...00001010... ...01001010... ...01001101... t

σi → 1 − σi, for all i ≥ j

Racz 85 Klee & Minty 72 Pemantle 07

How good is the simplex algorithm?

σi = 0 or 1

Front propagation

• Stable phase = no 1-spins (...000000...)
• Unstable phase = some 1-spins (...101101...)
• Stable phase propagates into unstable phase
• Front = position of leftmost 1

...00110101... ...00001010... ...01001010... ...01001101... t

v

• 1. A problem with no parameters!
• 2. Universal state regardless of initial state

Questions

• What is the speed of the front?
• What is the shape of the front?
• What is the spatial structure of the front?
• What is the time evolution of the front?
• What is the speed of the front?

v = ?

On average, front propagates ballistically

Monte Carlo simulations

20 40 60 80

t

1.7622 1.7624 1.7626

v 20 40 60 80 100

t

50 100 150 200

<x>

vMC = 1.7624 ± 0.0001

x

• Take the first neighbor to the front
• Lifetime of 0 is double that of 1
• Twice more likely that first spin is 0
• Similarly, we expect for all k>0

A simple observation

ρ1 = 1 3 ρk < 1 2 ρk ≡ σk

Depletion of 1 spins, nonuniform density

|10 → X1 with rate = 1 |11 → X0 with rate = 2

• Two assumptions
• 1. Quasi-static: no evolution in front reference frame
• 2. Mean-field: no correlations between spins
• Generalize argument for k=1
• Recursion relation for “density”
• Indeed, there is a depletion of 1 spins

Depletion

Depletion of 1 spins, nonuniform density

σjσk → σjσk

ρk = ρ0 + ρ1 + · · · + ρk−1 2(ρ0 + ρ1 + · · · + ρk−1) + 1

ρk = 1, 1 3, 4 11, 56 145, k = 0, 1, 2, 3, · · · .

0 → 1 1 → 0 dρk dt = 0

dρk dt = (ρ0 + ρ1 + · · · + ρk−1)(1 − ρk) − (1 + ρ0 + ρ1 + · · · + ρk−1)ρk

10 10

1

10

2

10

3

k

2 4 6

Δk

ln k simulation

10 20 30 40

k

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ρk

Approximation Simulation

• Depletion penetrates deep into the front

asymptotically exact

• Depletion of 0-spins grows logarithmically

asymptotically exact

Depletion

∆k ≃ ln k

ρk ≃ 1 2 − 1 2k k ≫ 1

Approximation seems to give accurate picture

• Velocity equals the average size of 1 strings
• Bounds for velocity
• Mean-field: string probability given by product
• Quasi-static approximation: poor estimate for velocity

Velocity and strings

SMF

n

= ρ1ρ2 · · · ρn−1 v = n =

• n

Sn Sn ≡ Prob(11111

n

) vQSA = 1.534070 1 ≤ v ≤ 2

Strong spatial correlations

· · · 0000 | 11111

n

0100 · · · → · · · 0000 00000

n

| 1011 · · ·

v =

• n

n(Sn − Sn−1)

10 20 30 40 50 60 70

n

10

• 12

10

• 10

10

• 8

10

• 6

10

• 4

10

• 2

10

Sn

simulation n

• 10.745

n

• Properly characterized by strings
• Much more likely than Mean-Field suggests

Spatial correlations

Sn ≡ Prob(11111

n

) ∼ n−νλn

Sn

λMC = 0.745 λQSA = 1/2 ν = 1

• For a renewal process, if n and n’ are successive jumps
• For flipping process, successive jumps anti-correlated
• Define “age”= time since last jump
• Velocity is age-dependent
• Density is age-dependent

Temporal correlations

Strong temporal correlations

nn′ = n2 = v2 n n′ < v2

v = ∞ dτ u(τ) e−τ ck(τ) = σk(τ) ρk = ∞ dτ ck(τ) e−τ

10 20 30 40

k

0.2 0.4 0.6 0.8 1

ck

τ=0 τ=1/4 τ=1/2 τ=1 τ=2 τ=4

1 2 3 4 5 6

τ

1 2 3 4

u

• Young fronts are fast, old fronts are slow!
• Rejuvenation: flip re-invigorates slow fronts
• Shape inversion: new is mirror image of old one

Aging & rejuvenation

A perpetually repeating life-cycle

· · · · · · 0|100000 · · · → · · · 00|11111 . . .

• Problem: infinite hierarchy of equations
• Solution: consider small segments of size L
• Assumption: complete randomness outside segment
• Technically: Approach is exact as
• Evolution equation for all possible 2L-1 states

Small segments

Is this brute-force approach useful?

L → ∞

dP100 dt = −P100 + 3 2P101 + 1 4P110 + 5 4P111 dP101 dt = −3 2P101 + 5 4P110 + 1 4P111 dP110 dt = 1 2P100 − 7 4P110 + 5 4P111 dP111 dt = 1 2P100 + 1 4P110 − 11 4 P111.

• Obtain velocities from steady-state
• Shanks transformation extrapolates to infinity
• Fast convergence for exponential corrections
• Good estimate for the velocity

Shanks transformation

v(m+1)

k

= v(m)

k−1v(m) k+1 − v(m) k

v(m)

k

v(m)

k−1 + v(m) k+1 − 2v(m) k

k v(0)

k

v(1)

k

v(2)

k

v(3)

k

v(4)

k

2 1.500000 3 1.535714 1.418947 4 1.587165 1.826205 1.779225 5 1.629503 1.773099 1.765862 1.764458 6 1.662201 1.766730 1.764592 1.758245 1.762322 7 1.687108 1.765129 1.763533 1.770104 1.765175 8 1.705987 1.764330 1.762272 1.761669 9 1.720251 1.763754 1.761864 10 1.730993 1.763313 11 1.739055

vshanks = 1.76 ± 0.01

Pinned fronts

• Small modification: fixed one spin to 1
• Front does not move, but we can still calculate v!
• Provides excellent approximation: velocity within 1%
• Quasi-static description becomes exact
• Small segment approach exact for all segment lengths
• Some exact results for correlation functions
• Correlations decay slowly

σ0 = 1 σkσk+1 = 1 2 σk+1 σkσk+1 − σkσk+1 ≃ (4k)−1

0111000 0101001 0100110 0111001

vpinned = 1.7753 ± 0.0001

• Now, small segment results are exact
• Shanks transformation converges rapidly and gives

impressive estimates

• Perfect estimate for the velocity

Small segments

k v(0)

k

v(1)

k

v(2)

k

v(3)

k

v(4)

k

1 1. 2 1.333333 1.666666 3 1.5 1.72549 1.769737 4 1.595833 1.750742 1.773156 1.775020 5 1.655039 1.762616 1.774362 1.775178 1.775278 6 1.693228 1.768521 1.774849 1.775239 1.775289 7 1.718565 1.771576 1.775065 1.775267 1.775293 8 1.735709 1.773205 1.775170 1.775280 1.775293 9 1.747473 1.774095 1.775223 1.775287 10 1.755632 1.774593 1.775252 11 1.761337 1.774876 12 1.765350

vshanks = 1.7753 ± 0.0001 vMC = 1.7753 ± 0.0001

Age-dependent densities can now be calculated

Depletion & aging

10 20 30 40

k

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ρk

pinned front propagating front 10 20 30 40 50

k

0.2 0.4 0.6 0.8 1

ck

τ=0 τ=1/4 τ=1/2 τ=1 τ=2 τ=4

c1(τ) = 2 3e−τ c2(τ) = 1 3(2τ − 1)e−τ + e−2τ

Pinned fronts capture all the physics, provide excellent approximation

Summary

• Analysis in a reference frame with the front is useful
• All the hallmarks of nonequilirbium physics
• Depletion
• Strong spatial and temporal correlations
• Aging and rejuvenation
• Mean-field theory explains depletion
• Small segment analysis + extrapolation provides good

estimate for velocity

• Pinning the fronts provides excellent approximation

and reproduces all qualitative features

Outlook Exact analytical solution for the velocity remains an open question, requires exact closure