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1
Monte Carlo Methods
Nicholas Constantine
MetropolisStanislaw Ulam
2
Books about Monte Carlo
• M.H. Kalos and P.A. Whitlock: „Monte Carlo Methods“ (Wiley-VCH, Berlin, 2008 )
• J.M. Hamersley and D.C. Handscomb: „Monte Carlo Methods“ (Wiley & Sons, N.Y., 1964)
• K. Binder and D. Heermann: „Monte Carlo Simulations in Statistical Physics“ (Springer, Berlin, 1992)
• R.Y. Rubinstein: „Simulation and the Monte Carlo Method“ (Wiley & Sons, N.Y., 1981)
3
Simulates an experimental measuring process
with sampling and averaging.
Big advantage:
Systematic improvement by increasing
the number of samples N.
Error goes like:
Monte Carlo Method (MC)
1
N
4
Applications of Monte Carlo
• Statistical partition functions, i.e. averages at constant temperature
• High dimensional integrals
5
Calculation of π
1
1
Choose N pairs
(xi , yi) , i = 1,…,N
of random numbers
xi ,yi{0,1}
1
0
214 dxx
0
x2 + y2 = 1
6
Calculation of π
N
cN 4)(
c = 0
if
c is number of points that fall inside circle.
area c / N
1( )N
N
2 21 1 i iy x c c
applet
7
Calculation of integrals
g(x)
a b
1
1( ) ( ) ( )
b N
iia
g x dx b a g xN
xi
8
Calculation of integrals
random numbers xi [a,b]
homogeneously distributed:
„simple sampling“Good if g(x) smooth.
9
Error of integration
b ah
N
conventional method:
choose N equidistant points distance
error:
22 2
1 1 area h
N T
where T is the
computer time.
one dimension
Δ (N area)2 T -2
10
Error in d dimensions
11 , d dh T N L
Lh T
2
2 2 2 1 ( )d d dNhh T h T
in d dimensions:
error with conventional method:
error with Monte Carlo:1
21
TN
Monte Carlo better for d > 4
11
High dimensional integral
2 2 2( ) ( ) ( )ij i j i j i jr x x y y z z
1 1 1
2
1
1
( ), .., , .., , ..,ij ij n n n
i jn nZr r dx dx dy dy dz dz
Consider n hard spheres of radius R in a 3d box.
Phase space is 3n dimensional: (xi , yi , zi), i = 1,…,n.
Calculate average distance:
with hard sphere constraint2 2 2 2 ( ) ( ) ( )i j i j i jx x y y z z R
12
MC strategy
• Choose particle position.
• If the excluded volume condition is not fulfilled then reject.
• Once the n particles are placed, calculate the average of rij over all pairs (i,j).
13
Not smooth integrals
g(x)
a bxi
need more precision
14
if xi randomly distributed according to p(x).
Good convergence when function smooth.
„importance sampling“
Calculation of integrals
1
1 ( )( )( ) ( ) ( )
( ) ( )
b b Ni
i ia a
g xg xg x dx p x dx b a
p x N p x
( )
( )
g x
p x
15
E(X) = energy of configuration X
Probability for system to be in X is given by
Boltzmann factor:
Canonical Monte Carlo
1 ( )
( )E X
kTeq
T
p X eZ
( )E X
kTT
X
Z e
1( )eqX
p X
ZT is the partition function:
16
Problem of sampling
( ) ( ) ( )eqX
Q T Q X p X The distribution of
energy E around
the average < E >T
gets sharper with
increasing size.
Choosing configurations equally distributed
over energy would be very ineffective.
thermal average
17
M(RT)2 algorithm
N.C. Metropolis, A.W. Rosenbluth,
M.N. Rosenbluth, A.H. Teller and E. Teller (1953)
importance sampling through a Markov chain:
X1 X2 …..
where the probability for a configuration is peq(X)
Markov chain: Xt only depends on Xt -1
18
1. Ergodicity: One must be able to reach any configuration Y after a finite number of steps.
2. Normalization:
3. Reversibility:
Properties of Markov chain
1( )Y
T X Y
Start in configuration X and propose a new
configuration Y with probability T (XY).
( ) ( )T X Y T Y X
19
Transition probability
The proposed configuration Y will be
accepted with probability A(XY) .
Total probability of a Markov chain is:
W(XY) = T(XY) A(XY)
Master equation:
( , )( ) ( ) ( ) ( )
Y Y
dp X tp Y W Y X p X W X Y
dt
20
Properties of W(XY)
• Ergodicity:
• Normality:
• Homogeneity:
0, : ( )X Y W X Y
1( )Y
W X Y
( ) ( ) ( )st stY
p Y W Y X p X
21
E(X) = energy of configuration X
Probability for system to be in X is given by
Boltzmann factor:
Canonical Monte Carlo
1 ( )
( )E X
kTeq
T
p X eZ
( )E X
kTT
X
Z e
1( )eqX
p X
ZT is the partition function:
22
M(RT)2 algorithm
importance sampling through a Markov chain:
X1 X2 …..
where the probability for a configuration is peq(X)Markov chain: Xt only depends on Xt -1
The proposed configuration Y will be
accepted with probability A(XY) .
Total probability of a Markov chain is:
W(XY) = T(XY) A(XY)
23
In stationary state one should have
equilibrium distribution (Boltzmann)
Detailed balance( , )
( ) ( ) ( ) ( )Y Y
dp X tp Y W Y X p X W X Y
dt
0 ( , )
, ( ) ( )stst eq
dp X tp X p X
dt
( ) ( ) ( ) ( )eq eqY Y
p Y W Y X p X W X Y
sufficient condition for detailed balance:
( ) ( ) ( ) ( )eq eqp Y W Y X p X W X Y
24
Metropolis (M(RT)2)
1min( )
( ) ,( )
eq
eq
p YA X Y
p X
1 ( )
( )E X
kTeq
T
p X eZ
1 1( ) ( )
min min( ) ( , ) ( , )E Y E X E
kT kTA X Y e e
Boltzmann:
If energydecreases always accept
increases accept with probability . E
-kTe
25
Glauber dynamics
1
( )
E
kT
E
kT
eA X Y
e
Roy C. Glauber (1963)
(Nobel prize 2005)
also
fulfills
detailed
balance
2626
The Ising Model
• Magnetic Systems
• Opinion models
• Binary mixtures
Ernst Ising
(1900-1998)
Spins
on a
lattice
(1924)
2727
The Ising Model
1 1, , ...,i i N
1, :
N N
i j ii j nn i
E J H
H
Binary variables:
on a graph of N sites
interacting via the Hamiltonian:
28
Order parameter
N
ii
H NTM
10
1lim)( spontaneous magnetization:
ordered phase
disordered phase
critical
temperature
)( cTTM
β = 1/8 (2d)
β ≈ 0.326 (3d)
29
Phase diagram
30
Response functions
cHT
TTH
MT
0,
)(
,
( )v cV H
EC T T T
T
susceptibility:
specific heat:
both diverge at Tc
31
Specific heat
cv TTTC )(
α = 0 (2d)
α ≈ 0.11 (3d)
comparing with experimental
data for binary mixture
32
Susceptibility
cTTT )(
γ = 7/4 (2d)
γ ≈ 1.24 (3d)
numerical data from a finite system
33
Correlation length
)()0()( RRC correlation function:
2
( )R
C R M ae
For T ≠ Tc and
for large R:
where ξ is the
correlation length.
T > Tc
T < Tc
34
Correlation length
cTT
dRRC 2)(
The correlation length diverges at Tc as:
with a critical exponent ν. ν = 1 (2d)
ν ≈ 0.63 (3d)At Tc we have for large R:
with η = 1/4 (2d)
η ≈ 0.05 (3d)
35
Magnetic exponent
1M H
DyAlG in field along [111]
above and below Néel temperature
δ = 15 (2d)
δ ≈ 4.79 (3d)
at Tc
36
d=2 d=3 d=4
α 0 0.11008(1) 0 2 − d / ( d − Δ ϵ )
β 1/8 0.326419(3) 1/2 Δ σ / ( d − Δ ϵ )
γ 7/4 1.237075(10) 1( d − 2 Δ σ ) / ( d − Δ ϵ )
δ 15 4.78984(1) 3 ( d − Δ σ ) / Δ σ
η 1/4 0.036298(2) 0 2 Δ σ − d + 2
ν 1 0.629971(4) 1/2 1 / ( d − Δ ϵ )
ω 2 0.82966(9) 0 Δ ϵ ′ − d
.
Critical exponents
37
Exponent relations
)2(
2
22
d
Exponents are related through:
scaling
hyperscaling
so that only two exponents are independent.
H.E.Stanley, „Introduction to Phase Transitions and
Critical Phenomena“ (Clarendon, Oxford, 1971)
38
Exact solutions
Lars Onsager
1944
2 2
0
1 1ln ( , 0) ln 2cosh(2 ) ln 1 1 sin
2 2TZ T H J d
square lattice:
22sinh 2 cosh 2J J
1 dimension:
2cosh( ) 1 tanh( )N NNZ J J
1( , ) ln 2cosh( )F T H J
1
Bk T
39
Metropolis (M(RT)2)
1min( )
( ) ,( )
eq
eq
p YA X Y
p X
1 ( )
( )E X
kTeq
T
p X eZ
1 1( ) ( )
min min( ) ( , ) ( , )E Y E X E
kT kTA X Y e e
Boltzmann:
If energydecreases always accept
increases accept with probability . E
-kTe
4040
MC of the Ising Model
• Choose one site i (having spin σi).
• Calculate ΔE = E(Y) - E(X) = 2Jσihi .
• If ΔE < 0 then flip spin: σi → - σi .
• If ΔE > 0 flip with probability exp(-ΔE/kT).
Single flip Metropolis:
j
inn of i
h where hi is the local field at site i
new configuration
old configuration
applet
, :
N
i ji j nn
E J
4141
Binary mixtures(lattice gas)
Consider two species A and B distributed with
given concentrations on the sites of a lattice.
EAA is energy of A-A bond.
EBB is energy of B-B bond.
EAB is energy of A-B bond.
Set EAA = EBB = 0 and EAB = 1.
Number of each species is constant.
4242
Kawasaki dynamics
Kyozi Kawasaki
• Choose any A-B bond.
• Calculate ΔE for A-B → B-A.
• Metropolis: If ΔE ≤ 0 flip, else flip with p = exp(-βΔE).
• Glauber: Flip with probability p = exp(- βΔE)/(1+ exp(- βΔE)).
β = 1/kT
43
Other Ising-like models
• Antiferromagnetic models:
• Ising spin glass:
• ANNNI model:
• Metamagnets:
, :ij i j
i j nn
J H
1 2 2, : :
i j i ii j nn i nnn
J J H
, :
( 1)ii j s i
i j nn i
J H H
1 2, : , :
i j i j ii j nn i j nnn i
J J H H
staggered field
random interaction
frustration in x-direction
incomensurate phases
with „Lifshitz point“
tricritical point
44
Antiferromagnetic Ising model
staggered field on square lattice
frustration on triangular lattice
, :
( 1)ii j s i
i j nn i
J H HOn square lattice phase diagram
and critical behavior exactly
identical to ferromagnetic case.
On the triangular lattice the critical temperature is zero
and the ground state is infinitely degenerate (finite entropy).
45
Ising spin glass
frustration
freezing of spins
quenched average
, :ij i j
i j nn
J H 1( ) ( ) ( )
2ij ij ijP J J J J J
Edwards Anderson model (1975):
2
1
10
N
ii
qN
order parameter in
spin glass phase (T<Tc):Fe0.5 Mn0.5 TiO3
46
The ANNNI model
1 2 2, : :
i j i ii j nn i nnn
J J H
2d
3d
Lifshitz point
47
1 2, : , :
( 1)ii j i j i i
i j nn i j nnn i i
J J p h H
Tricritical point
Metamagnet
48
The O(n) model
,
with =1, ,x x yXY i j x i i i i i
i j nn i
J S S H S S S S S
H
1 11
,
with 1, ,..., nn vector i j i i i i i
i j nn i
J S S H S S S S S
H
,
with =1, , ,x x y zHeisenberg i j x i i i i i i
i j nn i
J S S H S S S S S S
H
O(n) model:
n = 1 is the Ising model, n = 2 is the XY-model:
n = 3 is the Heisenberg model:
n = ∞ is the „spherical model“.
49
Phase transitions
Mermin-Wagner theorem (1966):
In two dimensions a system with continuous degrees
of freedom and short range interactions has no
phase transition which involves long range order.
Heisenberg model in three dimensions:
50
XY model
David ThoulessMichael Kosterlitz
2 dimensions
( )( ) ( ) , i j i jTs x s x r r x x
Tc
below Tc :
KT-transition1
( )4cT
( )( ) cb T TT e
51
XY model
No symmetry breaking, no long-range order
But still scaling behavior at Tc
specific heat:
susceptibility:
2( ) ( ) ln ( ) , 6q
C T T T q
22( ) ( ) ln ( ) , 0.056cr
T T T r
( ) 1
2 , ( ) cb T TT e
correlation length:
52
XY model
vortex-antivortex pair
At low temperatures the vortices are bound in vortex-antivortex
pairs, which dissociate at the KT transition.
53
Continuous degrees of freedom
Phase space is not discrete anymore.
Monte Carlo move: Choose for site i a new spin:
with small random , i i iS S S S S S
• Calculate ΔE
• Metropolis: If ΔE ≤ 0 flip, else flip with p = exp(-βΔE).
• Glauber: Flip with probability p = exp(- βΔE)/(1+ exp(- βΔE)).applet
5454
Interfaces
5555
Interfaces
f f A+B Asurface tension
A
BA
5656
Ising interface
, :
N
i ji j nn
E JH ++ ++++
++
--
- - - - ---
Fixed boundary conditions:
upper half „+“, lower half „–“Simulate with Kawasaki
dynamics at temperature T.
h
5757
Ising interface
21W i
i
h hN
interface width W:
W
TTc
roughening transition
1i
i
h hN
58
+
_
+
_
, : , :
N N
i j i ji j nn i j nnn
J KEH
Add next-nearest neighbor interaction:
punishes curvature introduces stiffness
Ising interface
13 K 20 K
14 J
5959
Shape of drop
• Start with a block of +1 sites attached to wall of an LL system filled with -1.
• Hamiltonian including gravity g:
• Use Kawasaki dynamics and further do not allow for disconnected clusters of +1.
1 11
1
1
, : , :
with and( )( )
N N
i j i j j ii j nn i j nnn j line j
L Lj
E J K h
j h h h hh h g
L L
H
6060
Shape of drop
L = 257, V = 6613, g = 0.001
after 5107 MC updates
averaged over 20 samples.
Contact angle θ is function
of temperature and vanishes
when approaching Tc . θ
6161
Solid on solid model (SOS)
interface without islands and without overhangs
, :
N
i ji j nn
E h hH
adatoms and
surface growth
6262
Self-affine scaling
W , /L t L f t L z
Family-Vicsek scaling (1985):
ξ is the roughening exponent, z the dynamic exponent.
6363
Self-affine scaling
W , / zL t L f t L
const: W
: W 0
W / z z
t L f u
L t f u u
L u L t L L t
/ zu t L
z
is the growth exponent.
Numerically these laws are verified by data collapse.
0
64
Self-affine scaling
6565
Irreversible growth
• T. Vicsek, „Fractal Growth Phenomena“ (World Scientific, Singapore, 1989
• A.-L. Barabasi and H.E. Stanley, „Fractal Concepts in Surface Growth“ (Cambridge Univ. Press, 1995)
• H.J. Herrmann, „Geometric Cluster Growth Models and Kinetic Gelation“, Phys. Rep. 136, 153 (1986)
6666
Irreversible growth
• Deposition and aggregation patterns
• Fluid instabilities
• Electric breakdown
• Biological morphogenesis
• Fracture and Fragmentation
No thermal equilibrium
6767
Random deposition
6868
Random deposition
Pick a random column.Add a particle on top of that column
1/2
~
~
h t
W t
simplest possible growth model
= 1/2
ξ = 1/2
6969
Random deposition with surface diffusion
1/4
~
~
h t
W t
Particle can move a short distance to find a more stable configuration.
Pick a random column i.Compare h(i) and h(i+1).Particle is added onto whichever is lower. If they are equal, add to column i or i+1 with equal probability.
= 1/4
ξ = 1/2
7070
Restricted Solid On Solid Model (RSOS)
Neighbouring sites may not have a height difference greater than 1.
Pick a random column i.Add a particle only ifh(i) h(i-1) or h(i) h(i+1).Otherwise pick a new column.
c
= 1/3
ξ = 1/2
7171
Growth equations
M. Kardar, G. Parisi and Y.-C. Zhang (1986)
2( , )( , )
2
h x th h x t
t
KPZ equation:
Edwards Wilkinson equation: ( , )( , )
h x th x t
t
S.F. Edwards and D.R. Wilkinson (1982)
= 1/3 and ξ = 1/2
= 1/4 and ξ = 1/2noise
Mehran Giorgio Yi-Cheng
Kardar Parisi Zhang
72
Stochastic Differential Equations
Martin Hairer
Kyoshi Itô
Itô – Stratonowitsch calculus
Ruslan L. Stratonowitsch
7373
Eden model
c
Each site that is a neighbour of an occupied site has an equal probability of being filled.
Make a list of neighbour sites (red).Pick one at random. Add it to cluster (green).Neighbours of this site then become red….
There can be more than one growth site in a column. Can get overhangs.
simple model for tumor growth or epidemic spread
7474
Eden cluster
= 1/3
ξ = 1/2
7575
c
Ballistic depositionParticles fall vertically from above and stick when they touch a particle below or a neighbour on one side.
Pick a column i. Let particle fall till it touches a neighbour. Possible growth sites are indicated in red. Only one possible site per column.
= 1/3
ξ = 1/2
76
Random Walk
k = degree = number of neighbors
Sample a random number z
homogenously between 1 and k
and move the walker
in direction [z] +1,
where [..] is the integer part.
Random Walk
78
Random Walk
The probability p(x,t) to find a random walker
at time t at position x is proportional to the
concentration c(x,t) of non-interacting random
walkers (= diffusion), after ensemble averaging. p
( , ) ( , )
c x tD c x t
t
3( , ) 1c x t dx
converges towards stationary solution:
( , ) 0c x t
7979
c
Diffusion Limited Aggregation (DLA)
A particles starts far away from surface.It diffuses until it first touches the surface. If it moves too far away, another particle is started.
Witten and Sander, 1981
Tom Witten Len Sander
8080
fractal dimensiondf = 1.7
(in 2d)
DLA clusters
81
Laplacian Growth
0 0c
1c
( , ) 0c x t
( , )gv c x tgrowth velocity
∞
is the concentration of random walkers( , )c x t
82
Laplacian Growth
0 0
1
( , ) 0x t
( , )g xv tgrowth velocity
∞
is the electrostatic potential( , )x t
8383
Electrodeposition
84
Laplacian Growth
( , ) 0c x t
1c
0 0c
growth velocity
nv c
circular geometry
∞
8585
DLA clusters
anisotropy on square lattice:
86
Noise reduction
Put a counter on each site
along the surface of the cluster
and count how many times
a random walker hits this site.
only occupy the site when the
number of hits reaches a
certain threshold.
87
Self similarity
87
105 sites 106 sites
107 sitesoff-lattice
DLA clusters
8888
Electrodeposition
89
Laplacian Growth
( , ) 0p x t
boundary injectionp p
growth velocity
nv p
p (x,t) is the pressure in the Hele-Shaw cell
injectionp
9090
Viscous Fingering
Push a less viscous fluid into a more viscous fluid
inside a horizontal
Hele-Shaw cell.
91
Laplacian Growth
0T
boundary meltT T
meltT
growth velocity
nv T
example: crystal growth inside undercooled melt
92
Nittmann and Stanley, 1987
Snowflakes
DLA with
noise reduction
and
surface tension
on the
hexagonal lattice
real realsimulation simulation
9393
Dielectric breakdown
9494
Dielectric breakdown model (DBM)
0
p
Solve Laplacian field ϕ:
and occupy site
at boundary
with probability:
η = 1same as DLA
9595
Dielectric breakdown model
η = 2
9696
Dielectric breakdown model
η = 4
9797
Dielectric breakdown model
η = 0.1
9898
same as Eden model
Dielectric breakdown model
η = 0
9999
Clustering of clusters
Fractal dimension is df ≈ 1.42 in 2d, df ≈ 1.7 in 3d.
2 /sn s f s t zdynamical scaling:
with dynamical exponent z .
100
Clustering of clusters
101101
Gold colloids
David Weitz, 1984
df = 1.70
Random Walk
103
Application of Random Walk
• Brownian motion
• Diffusion – heat transport - pollutants
• Foraging – search algorithms
• Mathematics (SLE)
• Finance of gambling
• Polymer at theta point
• ...
104
Random Walk
2
4( , )4
xDtN
P x tDt
e
2 ( ) 2x t Dt
( ,0) ( )P x xcentral limit theorem
105
diffusion:( , )
( , )c x t
D c x tt
2 2 3( ) ( , ) 2x t x c x t dx Dt
Random Walk
106
Random Walk
2 ( ) 2x t Dt2 2 ( ) 2length x t Dt mass fractal object with df = 2 in all dimensions ≥ 2
107
Random Walk
108
Variants of Random Walks
2 ( ) 2x t Dtanomalous diffusion:
Levy flights
109
Polymers
• M.Doi, «Introduction to Polymer Physics», Clarendon Press, Oxford, 1995
• P.G. De Gennes, «Scaling Concepts in Polymer Physics», Cornell Univ. Press, 1979
• P.D. Gujrati and A.I. Leonov, «Modelling and Simulations in Polymers», Wiley, 2010
• M.Kröger, «Introduction to Computational Physics», site of IfB, ETH Zürich
110
Self-avoiding walk (SAW)
dilute polymer chains in good solvent
excluded volume effect
N is chain length = number of monomers
= degree of polymerization
solvent
Paul Flory
111
Self-avoiding walkAll configurations of same chain length N
have the same statistical weight.
NN N
generating function:(grand canonical partition function)
number of SAW of length N:
( ) NN
N
Z x x ( )( ) N
N
NN
N
x NZ x x fugacity
1
finite for ln
critical at
infinite for
cNN
cx N
N c
x xNZ
N xx
x x
average chain length
1cx
critical fugacity
1( ) cZ x x x
112
Self-avoiding walk
How to sample correctly self-avoiding walks?Kinetic Growth Walk (KGW): grow a path starting
from one site moving randomly to one of the empty neighbors
Dimerization:
(Alexandrowicz, 1969)
• Start with any SAW of length N
• Put the last monomer of one end in a random direction at the other end.
• If there is no overlap retain this configuration, otherwise reject the move.
• Choose randomly one end and repeat.
113
Self-avoiding walk
reptation algorithmT. Wall and F. Mandel (1975)
Simulate fixed sizes.
114
Self-avoiding walk
reptation algorithm
115
Self-avoiding walk
trapped configurations
• Start with any SAW of length N
• Select one monomer randomly and move it to a possible (i.e. without disrupting the chain)random position.
• If this position is not occupied accept this move, otherwise reject.
• Repeat many times.
116
Self-avoiding walk
kink-jump algorithm
117
Self-avoiding walk
local moves involving
only one monomer:
local moves involving
two monomers:
non-local move:
118
Self-avoiding walk
pivot algorithm
• Start with any SAW of length N.
• Select one monomer randomly as pivot.
• Perform one one branch of this pivot a transformation symmetry.
• If this produces no overlap accept this move, otherwise reject.
• Repeat many times.
N. Madras and A.D. Sokal (1988)
119
Self-avoiding walk
N = 108 with pivot algorithm
120120
Ensemble method
21
1
( )( ) i ji jg r r
NR
N
Define „radius of gyration“ Rg:
Take self-avoiding walk of N monomers.
fd
gN R
121
Self-avoiding walk
fractal dimension df = 4/3 in 2d
3d: df ≈ 1.701..
and df = 2 for d > 4.
122
Yardstick method
coast of
NorwayKoch
curve
123
SAW with attractive interaction
1
1 1
if =1 ,
0 otherwise
N Ni j
ij iji j i
r rE c c
1( ) ,
E E
kT kTN
walksN
p walk e Z eZ
many ground states
124
SAW with attractive interaction
collapse of the polymer below temperature Tt
kT/εθ – point
At temperature Tt , i.e. at the θ – point, the
polymer behaves like a random walk: N ~ (Rg)2
125
SAW with persistence
semi-flexible polymer
Correlation between direction
of first step and nth step decays
exponentially with n and the
characteristic length is
called «persistence length».
By giving more weight to
straight sections than to
curved ones, one can
introduce an effective
stiffness.
126
Polymer melts
→ reptation algorithm
127
Shortest path
weighted graph
128
Edsger Dijkstra
1959
Dijkstra algorithm
greedy algorithm, i.e. tries to find a global optimum through many local optimizations. (No greedy algorithm can solve travelling salesman problem.)
V. Jarník: O jistém problému minimálním [About a certain minimal problem], Práce Moravské Přírodovědecké Společnosti, 6, 1930, pp. 57–63. (in Czech)
similar problem: minimum spanning tree
129
Dijkstra algorithm
1. Assign to every node a tentative distance value: set it to zero for our initial node and to infinity for all other nodes.
2. Set the initial node as current. Mark all other nodes unvisited. Create a set of all the unvisited nodes called the unvisited set.
3. For the current node, consider all of its unvisited neighbors and calculate their tentative distances. Compare the newly calculated tentative distance to the current assigned value and assign the smaller one. For example, if the current node A is marked with a distance of 6, and the edge connecting it with a neighbor B has length 2, then the distance to B (throughA) will be 6 + 2 = 8. If B was previously marked with a distance greater than 8 then change it to 8. Otherwise, keep the current value.
4. When we are done considering all of the neighbors of the current node, mark the current node as visited and remove it from the unvisited set. A visited node will never be checked again.
5. If the destination node has been marked visited (when planning a route between two specific nodes) or if the smallest tentative distance among the nodes in the unvisited set is infinity (when planning a complete traversal; occurs when there is no connection between the initial node and remaining unvisited nodes), then stop. The algorithm has finished.
6. Otherwise, select the unvisited node that is marked with the smallest tentative distance, set it as the new "current node", and go back to step 3.
130
Dijkstra algorithm
131
Dijkstra algorithm
Bellman-Ford algorithm can also handle negative weights.
132132
Simulated Annealing
• P.J.M. Van Laarhoven and E.H. Aarts, „Simulated Annealing: Theory and Applications“ (Kluwer, 1987)
• A. Das and B.K. Chakrabarti (eds.), „Quantum Annealing and Related Optimization Methods“ Lecture Notes No. 679 (Springer, 1990)
133133
Simulated Annealing (SA)
S. Kirkpatrick, C.D. Gelatt and M.P. Vecci, 1983
Given a finite set S of solutions
and a cost function F: S Search a global minimum:s* S* { s S : F(s) F(t) t S }.
Difficult when S is very big (like |S| = n!).
SA is a stochastic optimization technique.
134134
Given n cities σi and the travelling costs c(σi ,σj).
We search for the cheapest trip through all cities.
• S = { permutations of {1,..,n} }• F() = i=1..n c(σi ,σj) für S
Finding the best trajectory is a NP-complete problem,i.e. time to solve grows faster than any polynomial of n.
Travelling Salesman
135135
Define closed configuration on SN : S 2S with N(´) ´ N()
i´ N() j i j
(j)
(i)
´
Travelling Salesman
Traditional optimization algorithm:Improve systematically the costs by exploring close solutions.
If F(σ´) < F(σ) replace σ σ ´ until F(σ) F(t) for all t N(σ).
Problem: One gets stuck in local minima.
Make local changes:
136136
Travelling Salesman
Simulated annealing optimization algorithm:
If F(σ´) ≤ F(σ) replace σ σ ´
If F(σ´) > F(σ) replace σ σ ´
with probability exp(- F / T)
with F = F(σ ´)-F(σ ) > 0
T is a constant (like a temperature)
Go slowly to T 0 in order to find global minimum.
Applet
137137
Slow cooling
T0
T1
T20
Asymptotic convergence is guarantied and
leads to an exponential convergence time.
Different cooling protocols are possible.
node
link
node = site = vertex = agent
link = bond = edge = connection
Complex networks
139139
Complex Networks
• A.L. Barabasi: «Linked», Perseus, New York, 2003
• D.J. Watts: “Six Degrees”, Norton, New York, 2003
• M. Buchanan: “Nexus”, Norton, New York, 2003
• R. Pastor-Santorras and A. Vespigngani: “Evolution and Structure of the Internet”, Cambridge Univ. Press, 2004
• G. Caldarelli: “Scale-free Networks: Complex Webs in Nature and Technology”, Oxford Univ. Press, 2007
• A. Barrat, M. Barthelemy and A. Vespignani, “Dynamical Processes on Complex Networks”, Cambridge Univ. Press, 2008
• M. Newman: «Networks: An Introduction», Oxford Univ. Press, 2010
Technological Networks
• Internet
• Railways
• Electrical grid
• Air traffic
• Gas pipes
• Highways
• WWW
Swiss roads
data from Helmut Honermann, Bundesamt für Raumentwicklung, ARE
142
World Airline Network (WAN)
European Power Grid
144
Natural gas pipeline network
The Internet
Social Networks
• Friendships
• Sexual relations
• Collaborations
• Criminal associations
• Financial institutions
• Gossip
• Elections
• Spam
Friendships in schools
visualization using „pajec“
Survey interviewing 90118 student from 84 schools in US(Add Health Program)
Criminal network
drug traffic
in Tucson
Social Network Analysis (SNA) from Anacapa
149
Financial institutions
Natural Networks
• Brain
• Gels
• Contact networks
• Rivers
• Colloidal Structures
• Protein networks
• Food Webs
151
Brain
152
Contact Network
153
Protein interaction network
Protein-protein
interaction for
schizophrenia
associated proteins
Interaction between
protein can be defined
by how closely located on
a DNA are the genes that
encode the two proteins.
Food web of theNorth Atlantic Ocean
predator → prey
Food Chain Network
Simplest Network
square lattice
each agent has
a unique index
Erdös – Rényi (ER)
Every pair of nodes is connected.
157
Adjacency Matrix
158
Adjacency Matrix
Complex Structures of Networks
• Scale-free
• Small world
• Modular
Properties of networks
• Degree distribution• Clustering Coefficient• Shortest Path• Assortativity• Betweenness Centrality• Cliquishness• Connectivity Correlation• Cutting bonds, bottlenecks• ......
161
Degree
degree k is the number
of connections of a site
directed networks:
Scale-free networks
WWW: 2.4out 2.1in
( )P k k
degree k is the number of connections of a site
A network is called «scale-free» if
its distribution of degrees follows:
internet: 2.4
163
Scale-free networkscollaborations
movie actors: 2.3 scientific collaborations:
(bipartite network of authors and papers
taken from arXiv cond-mat)
Scale-free networksBarabasi-Albert model:
preferential attachment:attach new sites to a site
of degree k with probability
proportional to k
→ = 3
„configuration model“:attach new sites to a site
of degree k with probability
proportional to k-α
slope = - 3
(1998)
165
Small-world propertyclustering coefficient
2number of connections between neighbors
( 1)C
k k
shortest path
chemical distance between two sitesl
A network has «small-world property», when
0.5 < C ≤ 1 and l does not grow faster than log N.
«six degrees of separation»
Duncan Watts Steven Strogatz
Stanley Milgram
166
Small-world propertyD.J. Watts and S.H. Strogatz, Nature, 1998
Start with regular lattice
with degree k and rewire a
fraction p of its connections.
k
Npl
k
kpC
ln
ln)1(
14
23)0(
p
167
Betweeness centrality
,
2 number of shortest paths from to through
1 2 number of shortest paths from to ( )
i j v
i j v
N N i jg v
Measures the load during traffic.
Assortativity
where M is the number of edges and ji and ki the degrees of the nodes connected by the edge i. The range of A is between ‐1 and 1. A positive value indicates that nodes with similar degrees are connected, while a negative value indicates that nodes with low degree are attached to nodes with high degree and vice versa.
222
2
)(2
)(4
i iii ii
i iii ii
kjkjM
kjkjMA
The assortativity A describes the correlation between the degree of the nodes at each side of an edge:
Questions
• Propagation of information or epidemics
• Robustness against attack
• Synchronization of oscillators
• Clogging, gridlock of traffic
• Basins of attraction, cycles
• Optimization dynamics
• Self-repair, complementation
Apollonian packing
Apollonian network
• scale-free
• small world
• Euclidean
• space-filling
• matching
Applications
• Systems of electrical supply lines
• Friendship networks
• Computer networks
• Force networks in polydisperse packings
• Highly fractured porous media
• Networks of roads
Degree distribution
n-1
n- 2
2
1
2
3 3
3 3 2
3
scale
( , )
-free: (
3 2
3
)
( )
n
m
P k k
W k kk n k
n-1
n
n+1
3 2
3 3 2
1 ln3
1 1.585l
2
2
n
number of sites at generation ( , ) number of vertices of degree
cummulative distribution ( ) ( , ) /
n
nk k
N nm k n k
W k m k n N
Small-world properties
clustering coefficient
2number of connections between neighbors
( 1)C
k k
shortest path
chemical distance between two sitesl
0.828C lnl N
Percolation threshold
10−5
10−4
10−3
10−2
1/N
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
pc
0.17
,
,
1 2 31
at : and simultaneously c
3
bond peronnected
porou
col
s media Archie's la
ation
w
c
vc
p P P P
p L vL N
epidemics
random failure
Electrical conductance
1S
2S{
• from center to all sites of generation n:
• from
}= number of outputs ofsystemeach site
1 1 2 21 3 , 2n n S S S S 1.24current distribution ( )P I I
1 2 to : 0 4 . 5zP P
2 3z
fluxfuses =
malicious attack
Ising model
coupling constant
correlation length diverges at
free energy, entropy, specific heat are smooth
magnetization
n
n c
T
J n
J T
m e T
opinion
Apollonian networks
• Extremely stable against random failure.
• Very fragile against malicious attacks.
• Only one opinion
Evolution of epidemics with healingon a mobile population
Optimizing through rewiring
185
Numerical Methods
186
Finding the solution (root) of an equation:
f (x) = 0is equivalent to the optimization problem
of finding the minimum (or maximum) of F(x)
given by:
Solving equations
0( )d
F xdx
187
Newton method
1 0 1 0 0( ) ( ) ( ) '( ) 0f x f x x x f x
1
( )
'( )n
n nn
f xx x
f x
Be x0 a first guess, then linearize around x0:
188
Secant method
1
1
( ) ( )( )' n n
nn n
f x f xf x
x x
1 11
( )( )
( ) ( )n
n n n nn n
f xx x x x
f x f x
If derivative of f is not known analytically:
189
Secant method
190
Secant method
Might not converge
191
Bisection method
Take two starting values x1 and x2
with f (x1) < 0 and f (x2) > 0.
Define mid-point xm as xm = (x1 + x2) / 2 .
If sign(f (xm)) = sign(f (x1))
then replace x1 by xm
otherwise replace x2 by xm.
192
Bisection method
193
Regula falsi
Take two starting values x1 and x2
with f (x1) < 0 and f (x2) > 0.
Approximate f by a straight line between
f (x1) and f (x2) and calculate its root as:
xm = (f (x1) x2 – f (x2) x1) / (f (x1) – f (x2)).
If sign(f (xm)) = sign(f (x1)), then
replace x1 by xm otherwise replace x2 by xm.
194
Regula falsi
195
Regula falsi
196
Be a N-dimensional vector.
N-dimensional equations
0)( xf
0)( xF
x
System of N coupled equations:
corresponding to the N-dimensional
optimization problem:
197
N-dimensional Newton method
,
( )( ) i
i jj
f xJ x
x
11
( )n n nx x J f x
Define the Jacobi matrix:
Must be non-singular and
also well-conditioned
for numerical inversion.
198
System of linear equations
b11 x1 +. .. . + b1N xN = c1
. . . . . .
. . . . . .
bN1 x1 + . . . + bNN xN = cN
1x B c
solution:
Bx c
199
System of linear equations
0
( )( ) i
j
f xf x Bx c J B
x
1 11 ( )n n nx x B Bx c B c
apply Newton method:
exact solution in one step
11
( )n n nx x J f x
200
N-dimensional secant method
,
( ) ( )( ) i j j i
i jj
f x h e f xJ x
h
jj xh
If the derivatives are not known analytically:
where hj should be chosen as:
being ε the machine precision,
i.e. ≈ 10-16 for a 64 bit computer.
201
Other techniques
0 1( ) ( , ), , ...,i i jf x x g x j i i N Relaxation method:
Start with xi (0) and iterate: xi (t+1) = gi (xj (t)).
Gradient methods:1. Steepest descent
2. Conjugate gradient
202
Ordinary differential equations
( , )dy
f y tdt
dNN
dt
First order ODE, initial value problem:
with y (t0) = y0
examples:
( )room
dTT T
dt
radioactive decay
coffee cooling
203
Euler method
20 0 0
20 0 0 1 1
( ) ( ) ( ) ( )
( , ) ( ) ( )
dyy t t y t t t O t
dt
y t f y t O t y t y
explicit forward integration algorithm
choose small Δt, Taylor expansion:
0 , ( ) n n nt t n t y y t convention:
204
Euler method
21 ,( ) ( )n n n ny y t f y t O t
Start with y0 and iterate:
This is the simplest finite difference method.
Since the error goes with Δ2 t one needs a very
small Δt and that is numerically very expensive.
205
Euler method
206
The order of a method
If the error at one time-step is O(Δn t) the method is „locally of order n“. To consider a fixed time interval T one needs T /Δt time-steps so that the total error is:
1( ) ( )n nTO t O t
t
and therefore the method is „globally of order n-1“.
The Euler method is globally of first order.
207
The Newton equation
2
2 ( )
d xm F x
dt
( )
dxv
dtdv F x
dt m
2nd order ODE
Transform in a
system of two
coupled ODEs
of first order.
208
Euler method for N coupled ODEs
1 1( , ..., , ) , , ...,ii N
dyf y y t i N
dt
21 1( ) ( ) ( ( ), ..., ( ), ) ( )i n i n i n N n ny t y t t f y t y t t O t
0nt t n t
N coupled ODEs
of first order:
with
iterate with a small Δt :
209
Runge - Kutta method
210
References for R-K method
• G.E. Forsythe, M.A. Malcolm and C.B. Moler, „Computer Methods for Mathematical Computations“ (Prentice Hall, Englewood Cliffs, NJ, 1977), Chapter 6
• E. Hairer, S.P. Nørsett and G. Wanner, “Solving Ordinary Differential Equations I” (Springer, Berlin, 1993)
• W.H. Press, B.P.Flannery, S.A Teukolsky and W.T. Vetterling, “Numerical Recipes” (Cambridge University Press, Cambridge, 1988), Sect. 16.1 and 16.2
• J.C. Butcher, „The Numerical Analysis of Ordinary Differential Equations“ (Wiley, New York, 1987)
• J.D. Lambert, „Numerical Methods for Ordinary Differential Equations“ (John Wiley & Sons, New York, 1991)
• L.F. Shampine, „Numerical Solution of Ordinary Differential Equations“ (Chapman and Hall, London, 1994)
211
2nd order Runge - Kutta method
212
2nd order Runge - Kutta method
1 1
2 2( ) ( ) ( ( ), )i i iy t t y t t f y t t
31 1
2 2( ) ( ) ( ( ), ) ( )i i iy t t y t t f y t t t t O t
Extrapolate using Euler method
to the value halfway, i.e. at t + Δt / 2:
Evaluate derivative in Euler method at this value:
213
Fourth order Runge - Kutta
1
2 1
3 2
4 3
( , )
( / 2, / 2)
( / 2, / 2)
( , )
n n
n n
n n
n n
k f y t
k f y k t t
k f y k t t
k f y k t t
531 2 41 ( )
6 3 3 6n n
kk k ky y t O t
RK4
applet
214
RK4
• k1 is the slope at the beginning of the interval.
• k2 is the slope at the midpoint of the interval, using slope k1 to determine the value of y at the point tn+Δt/2 using Euler method.
• k3 is again the slope at the midpoint, but now using the slope k2 to determine the y-value.
• k4 is the slope at the end of the interval, with its y-value determined using slope k3.
1 2 3 42 2
6slope
k k k k Then use Euler method with
215
q-stage Runge - Kutta method
11
q
n n i ii
y y t k
1
1
( , )i
i n ij j n ij
k f y t k t t
1
( , )q
i n ij j n ij
k f y t k t t
iteration:
with
explicit
implicit:
and α1 = 0
216
q-stage Runge - Kutta method
q = 1 is the Euler method (unique)
Butcher array or Runge-Kutta tableau:
2 21
3 31 32
1 2 1
1 1
,
2
....
....
q q q q q
q q
resumes all
parameters
of the general
Runge-Kutta
217
0
½ ½
½ 0 ½
1 0 0 1
Example RK4
its stage is: q = 4
1 1 1 1
6 3 3 6
Butcher array:
RK4 is of order p = 4
218
Order of Runge - Kutta method
1
1
( ) ( ) ( )q
pi i
i
y t t y t t k O t
The R-K method is of order p if:
One must choose the αi, βij and ωi for i,j [1, q] such
that the left hand side = 0 in O(Δtm) for all m ≤ p.
11
11
1
( ),
( ) ( ) ( )!
mpm p
mm y t t
d fy t t y t t O t
m dt
Taylor expansion:
219
Order of Runge - Kutta method
11
11 1
1
( ),!
mq pm
i i mi m y t t
d fk t
m dt
1 1 1( , ) ( , )n n n nf y t f y t
up to O(Δt p+1)
Example q = p = 1:
gives Euler method.
Example q = p = 2:1 1 2 2
1
2nn
dfk k f t
dt
where index „n“ means „at (yn,tn)“.
220
Example q = p = 2
1 1 2 2
1 1
2 2n n nn n n
df f fk k f t f t f
dt t y
1
2 21 1 2
221 2
( , )
( , )
( )
n n n
n n
n nnn
k f y t f
k f y t k t t
f ff t f t O t
y t
insert
1 2 2 2 2 21
1 11
2 2 , ,
3 equations for 4 parameters one-parameter family
221
Example q = p = 2
1 2 1 2 21( )n ny y t k k
1
22 22 2
( , )
( , )
n n
n n
k f y t
t tk f y t
222
Order of Runge - Kutta method
To obtain a Runge-Kutta method of given
order p one needs a minimum stage of qmin.
p 1 2 3 4 5 6 7 8 9 10
qmin 1 2 3 4 6 7 9 11 12 – 17 13 - 17
John Butcher
224
Example: Lorenz equation
Edward Norton Lorenz
(1963)
1 2 1
2 1 3 2
3 1 2 3
'
'
'
y = σ y - y
y = y ρ - y - y
y = y y - βy
C.Sparrow: „The Lorenz Equations: Bifurcations, Chaos
and Strange Attractors“ (Springer Verlag, N.Y., 1982)
Is a simplified system of equations describing the 2-dimensional
flow of fluid of uniform depth in the presence of an imposed
temperature difference taking into account gravity, buoyancy,
thermal diffusivity, and kinematic viscosity (friction).
σ Prandtl number
Rayleigh number
σ = 10 , = 8/3
is varied.
Chaos for = 28.
applet
226
Lorenz attractor
227
Example: Lorenz equation
Warwick Tucker
(2002)
Chaotic solutions of the
Lorenz equation exist and
are not numerical artefacts
(14th math problem of Smale).
228
Ordinary differential equations
( , )dy
f y tdt
dNN
dt
First order ODE, initial value problem:
with y (t0) = y0
examples:
( )room
dTT T
dt
radioactive decay
coffee cooling
229
Explicit forward integration
21 ,( ) ( )n n n ny y t f y t O t
Start with y0 and iterate:
This is the simplest finite difference method.
Since the error goes with Δ2 t one needs a very
small Δt and that is numerically very expensive.
Euler method
230
Error reduction
1 21
1 22
1 1 2
22
2
2 2
( ) ( )( )
( ) ( )
( ) ( )
p p
p p
p p p
y t O ty t t
y t O t
t O t
Let us consider a method of order p, .
Be y1 the predicted value for 2Δt
and y2 the predicted value making two steps Δt.
Define difference as δ = y2 – y1
(error = ΦΔt p+1)
12 2
p
231
Error reduction
22 1
2
2 2( ) ( )p
py t t y O t
improve systematically:
22 15
( ) ( )py t t y O t
example RK4:
232
Error reduction
1
1 11 2
1 11 2
1 2
1 1 1 1 22 1 2 1
1 12 1 2
1 12 1
, 1,2
pi
p p
p p
p pi it
p p p p pit t
p p
t t pip p
y y t O t i
t t y t y t y O t
t y t yy O t
t t
General for two different time steps Δt1 and Δt2
233
Richardson extrapolation
Calculate for
various Δti
and extrapolate
0lim i i
tt
yy =
Lewis Fry Richarson
234
Bulirsch - Stoer Method
R. Bulirsch and J. Stoer, „Introduction to
Numerical Analysis“ (Springer, NY, 1992)
Extrapolate with rational function:
0 10
0 1
2, 4, 6,8,12,16, 24,32, 48, 64,96,.
...
...
, ..
n n
k
n k nt m t
n m n
n
p p t p ty y
q q t q t
tt n
n
Choose k and m appropiately.
235
Define the error you want to accept.
Then measure the real error and
define a new
time-step through:
Adaptive time-step Δt
expectedδ
1
1pexpected
new oldmeasured
t t
measuredδ
because δ = ϕ Δt p+1 .
236
Predictor-corrector method
2( ) ( ) ( ) ( )P dyy t t y t t t O t
dt
3
2( ) ( ) [ ( ( )) ( ( ))] ( )C Pt
y t t y t f y t f y t t O t
2
( ( )) ( ( ))( ) ( )
f y t f y t ty t t y t t
idea:
implicit equationmake prediction
using Taylor:
correct by
inserting:
Can be iterated by again inserting corrected value.
237
3th order predictor-corrector
Predict through 3rd order Taylor expansion:
238
3th order predictor- corrector
( ) ( ( ))C
Pdyt t f y t t
dt
( ) ( )C P
dy dyt t t t
dt dt
0
2 2
22 2
3 3
33 3
( )
( )
( )
C P
C P
C P
y t t y c
d y d yt t c
dt dt
d y d yt t c
dt dt
use equation:
define error:
correct:
Procedure
can be
repeated.
Gear coefficients:
c0 = 3/8
c2 = 3/4
c3 = 1/6
Charles William
Gear
239
Be r0 ≡ y. Then one can define the time derivatives:
Predictor:
5th order predictor- corrector
240
5th order predictor- corrector
1 0 1 1 r
(r) r (r ) r r rC P C Pdf f
dt
Corrector:
1st order eq.:
2nd order eq.:2
2 0 2 222
r(r) r (r ) r r rC P C Pd
f fdt
241
1st order equation:
2nd order equation:
Gear coefficients (1971)
242
Comparison of methods
Δt
error
yn+1 - yn
for fixed number of iterations n
4th
5th6th
Predictor - Corrector
Verlet
243
Sets of coupled ODEs
1 1( , ..., , ) , , ...,ii N
dyf y y t i N
dt
Explicit Runge Kutta and Predictor Corrector
methods can be straightforwardly generalized
to a set of coupled 1st order ODEs:
by inserting simultaneously all the values
of the previous iteration.
244
Stiff differential equation
( ) 15 ( ) , 0 , (0) 1y t y t t y '
2( ) ( ) 15 ( ) ( )y t t y t t y t O t
example:
Euler:
Becomes unstable if Δt not small enough:
21
2( ) ( ) ( ( ), ) ( ( ), ) ( )y t t y t t f y t t f y t t t t O t
Use implicit method: (e.g. Adams – Moulton)
245
Stiff sets of equations
'1 1 2
'2 1
1 2
1000
999
(0) 1 , (0) 1
y y y
y y
y y
' ( ) ( ) ( )y t K y t f t This system is called „stiff“
if matrix K has at least one
very large eigenvalue.
example:
' ( ) ( ( ), )y t f y t t This system is called „stiff“ if
Jacobi matrix has at least
one very large eigenvalue.
Solve with with
implicit method.