Biased Monte Carlo Evelyn Dittmer 27'04'2005

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Biased Monte CarloEvelyn Dittmer27.04.2005
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Overview

• Conventional Monte Carlo Techniques (Metropolis)
• Idea of Biased Monte Carlo
• Application to Chain Molecules
• Correctness of the Sampling Algorithm
• Further BMC Variants

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Conventional Monte Carlo Techniques

• problem sampling from a distribution ?
• expensive evaluation
• sampling as random walk
• transition probability k(o-gtn)
• ? is distribution of the resulting sequence

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Detailed Balance

• constraint for transition probability
• sampled sequence in equilibrium
• ? is stationary density in position space

?(o)k(o-gtn) ?(n)k(n-gto) detailed balance!!
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Monte Carlo Principle
proposal
acceptance
k(o-gtn) p(o-gtn) a(o-gtn)

• p arbitrary proposal distribution
• acceptance allows for detailed balance condition
• ? ? 1/Z exp(-?U) canonical ensemble in position
  space

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Ideal Monte Carlo

• desired acceptance rate a1
• proposal generations via p(o-gtn) produce directly
  a sampling from ?
• acceptance rate 1 is achieved, if

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Metropolis Algorithm

• transition probability equal for all neighboring
  states
• i.e. p(o-gtn)p(n-gto)p symmetric proposal
• a(o-gtn) min(1,?(n)/?(o))
  exp(-?(U(n)-U(o))

t
t1
Step
current state o
current state n
n
generate next proposal n due to probability p
States
o
accept/reject with probability a(o-gtn)
p
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Flow chart
choose molecule at position o
generate new position proposal n
compute energy exp(-?U(n)-U(o))
1
lt1
accept proposal
accept proposal with probability
exp(-?U(n)-U(o))
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Problems

• purely random generated proposals might not obey
  the physics (e.g. occurring overlaps)
• small steps provide better acceptance rate but a
  smaller region of the phase space will be sampled
• large steps lead to low acceptance rates and high
  effort

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Idea of Biased Monte Carlo

• add bias function to proposal matrix
• generated proposals have higher probability to
  get accepted

energy function
bias
new configuration
compute energy
generate proposal
accept/reject
LeeKurKang96
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Introduction of a Bias into the Monte Carlo scheme
Metropolis
Biased Monte Carlo
Ideal Monte Carlo
propo- sal
accep- tance rate
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Spatial Structure of Chain-Molecules
(Biopolymeres)
bond stretching
dihedral
f1
angle bending
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Contributions of the Potential

• bonded potential energyfurther decomposable
  tolow dimensionality gt sampling feasible
• non bonded potential energy(sampling is too
  time-consuming)
• Idea Sampling only from ubond

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Bonded Interactions
i-1
i1
r
?
?
i-2
i

• ubond ustretch(ri,i1) ubend(?i-1,i,i1)
• udihedral(?i-2,i-1,i-i1)growing chain
  segment by segment

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Sampling from a Chain Molecule (Configurational
Biased MonteCarlo CBMC)

• sequence contains molecular configurations
  consisting of l segments, l length of Chain
  Molecule
• proposals are generated iteratively segment by
  segment
• per segment are generated k candidates sampled
  from bonded potential
• one of the k candidates is selected by the means
  of non bonded potential

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Generation of Proposal Segment Candidates

• for each segment generate k proposal candidates
  b1,...,bk
• generating probability is given by bonded forces

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Selection of Proposal Segments

• select one of the proposal segments bn ?
  b1,...,bk
• probability is governed by non bonded
  interactions

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Proposal Configuration (consisting of l Segments)

• for segment i1,...,l renew position by selecting
  one of k candidates
• new configuration of length l

bo
bn
i
i1
i
i1
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Flow-chart
generate k candidates for segment i

b1
bk

Pinon(b1)
Pinon(bk)
select bn ? b1,...,bk
next chain segment ii1
i l
igtl
accept/reject new configuration
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Retracing from selected Segment

• to obey detailed balance, both directions must be
  taken into account k(o-gtn) and k(n-gto) !
• once selected a new segment, we have to retrace
  the old one
• both have an impact on the acceptance probability

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Adjustment of Acceptance
select new proposal segment
retrace old segment
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Algorithm

• generate k candidates
• select new proposal segment
• retrace the old one
• adjust acceptance probability
• accept/reject
• generate next configuration

while il
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Algorithmic Scheme
energy function
generate kproposals
new segment
retrace old segment
select trial
compute energy
accept/reject
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Correctness of the Sampling Algorithm

• detailed balance condition claims
• this is fullfilled for our acceptance probability

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Applications in Literature

• modelling zeolite structure (FalDeem99),
  polypeptides (UlmJor02) by minimizing
  displacement
• modelling copolymere networks, e.g. polyurethanes
  by extended CBMC techniques (PalLas03, PaLaSu92)
• discretization of dihedrals, bias according to
  probability tables (LeeKurKang96)

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Further BMC VariantsForce-Bias

• Generate proposal according to physical forces

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Further BMC VariantsSmart Monte Carlo

• similar to force-bias, but no zero-probabilty in
  case of leaving a certain regiondisplacement
  is governed by sde

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Further BMC VariantsPreferential Sampling

• prefer interesting parts of phase space
• example solute solvent interactions
• define parameter and regeion of interest1.)
  choose molecule m at random2.) if make
  proposal displacement3.) if make proposal
  displacement with probability

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Further BMC VariantsHybrid Monte Carlo

• sampling along the trajectory by means of an
  discrete (symplectic and reversible) integrator
• combination fo MD and MC
• proposal generation
• acceptance

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Summary

• Metropolis not applicable for complex systems
• acceptance 1 not possible
• good acceptance rates acchieved by introducing a
  bias
• chain molecule (CBMC)
• intersting regions of position space
  (Pref. Sampling, F.B., Smart MC)
• sampling the whole phase space (HMC)

Examples
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References

• Understanding Molecular Simulation Frenkel/Smit
  13
• Computer Simulation of Liquids Allen/Tildesley
  7.3
• Molecular Modelling Leach 8.7
• Die Hybride Monte-Carlo-Methode in der
  Molekülphysik, Alexander Fischer, Diploma Thesis