A DSMC-based variance reduction formulation for low-signal ﬂows · 2009. 9. 14. · A DSMC-based variance reduction formulation for low-signal ﬂows H. A. Al-Mohssen [email protected]

A DSMC-based variance reduction formulation for low-signal flows

H. A. Al-Mohssen [email protected] N. G. Hadjiconstantinou [email protected]

Mechanical Engineering Dept., MITSeptember 2009

Motivation

✤ Boltzmann Equation(BE): describes the evolution of PDF f=f(x,c,t)

✤ Direct Simulation Monte Carlo simulates the BE, the uncertainty in "measurement" is:

✤ We want:

✤ Related Work:

๏ Öttinger, 1997: polymer simulation

๏ Chun and Koch, 2005: linearized BE

๏ Hadjiconstantionu, et el. 2004-2009: deviational particles

2

ΣUncertainty �ΣThermal

NSamples⇒ problems in low signal(≡ deviation from equilibrium) flows

(eg. low Ma flows).

ΣUncertainty �Σ�Signal�NSamples

s.t. σ(Signal)→ 0 as Signal → 0

� f� t� c �� f�x� �� f� t�Collision

�12�� ∆1� � ∆2� � ∆1 � ∆2�� f1� f2�c12�Σ��c1��c2

✤ Let 〈R〉 be a property of interest (eg. ux=〈cx〉,〈cx〉 etc.). In general, it can be written as:

Where feq is an arbitrary reference (equilibrium) distribution

✤ An estimate of this quantity (that we will call R ̅) can be calculated by generating samples c i from f(c i)

Notation

3

�R� �� R�c� f �c��c �R�eq �� R �c� feq�c��c and likewise for feq≠f,

� R � 1N��i�1N R�ci�

4

Variance Reduction Approach

I � � � f �c� � g�c�� dc �� g�c� dcI � � f �c� dc � I �

1N �

i�1

Nf �ci�

if � g�c� dc is known deterministically & f �c� � g�c�

(1)

(2)



1N �

i�1

Nf �ci�


(1)

(2)

0 1 2 3 4 5 60

1

2

3

4

5

� f �c� dc0 1 2 3 4 5 6

0

1

2

3

4

5

� � f �c� � g�c�� dc

� g�c� dc



1N �

i�1

Nf �ci�


(1)

(2)

(2) can be estimated more efficiently than (1)



1N �

i�1

Nf �ci�



(1)

(2)

Hadjiconstantinou, Baker, Homolle, Radtke

I � �� f �c� � g�c� dc� �� g�c� dc

Simulate using deviational particles



1N �

i�1

Nf �ci�



(1)

(2)

Hadjiconstantinou, Baker, Homolle, Radtke

I � �� f �c� � g�c� dc� �� g�c� dc

Simulate using deviational particles

This Work

I � �� f �c� dc� � �� g�c� dc� �� g�c� dc

Unmodified DSMC Auxiliary Weighted DSMC

Illustration

�4 �2 0 2 4

0.1

0.2

0.3

0.4

f

c�4 �2 0 2 4

0.1

0.2

0.3

0.4

feq

c

10 20 30 40 50 60Time

2.07�1010

2.08�1010

2.09�1010

2.1�1010

2.11�1010

2.12�1010�Cx4�

Non-Eq Simulation (Regular DSMC)

9

Illustration

10 20 30 40 50 60Time

2.07�1010

2.08�1010

2.09�1010

2.1�1010

2.11�1010

2.12�1010�Cx4�

10


Eq Simulation (Weighted DSMC Auxiliary Simulation)Analytical Eq Value


Eq Simulation (Weighted DSMC Auxiliary Simulation)

Non-Eq Simulation (DSMC VR)

RVR � R � Req � �R�eq

11

Illustration

10 20 30 40 50 60Time

2.07�1010

2.08�1010

2.09�1010

2.1�1010

2.11�1010

2.12�1010�Cx4�

Analytical Eq Value

Formulation

✤ Our Formulation:

๏ Use an unmodified DSMC to directly calculate R ̅

๏ Use an auxiliary simulation to calculate R ̅eq. The auxiliary simulation does not perturb the main DSMC simulation and uses the same samples c i

๏ How can we calculate both R ̅ and R ̅eq from the same set of data?

• Use weights!

12

R �1N��i�1

NR�ci�

Auxiliary Simulation Using Weights

✤ Likelihood ratios (Wi≡W(c i)≡feq(c i)/f(c i)):

✤ As a result:

13

�R�eq �� R �c� feq�c��c �� R�c�� feq �c�f �c� � f �c��c �� R�c��W�c� f �c��c

� Req � 1N��i�1N Wi�R�ci�

RVR � R � Req � �R�eq � 1N �i�1

N �1 �Wi� R�ci� � �R�eq

Rejected

Advection Step

Collision Step

Accepted

Initalize N particles

Select i & jAccept with probability

cr� � cr �MX

Sample equlibrium andnon-equlibrium properties

ci � ci, c j � c j

ci � ci�, c j � c j�

Algorithm Overview

Regular DSMC

Rejected

Advection Step

Collision Step

Accepted



cr� � cr �MX




at ci & Wi � ?? ??

Wi� � ?? ??& Wj

� � ?? ??

Wi� � ?? ??& Wj

� � ?? ??

Algorithm Overview

Regular DSMC

+

VR DSMC

✤ For every particle at c i there will be on average particles at c i. If we have x particles at c i there will be (at c i)

๏ particles in an Equilibrium simulation

๏ particles in a Non-equilibrium simulations

✤ Since we advance particles per the Non-equilibrium rules each particle at needs to simulate:

✤ The final collision rules

๏ Accepted

๏ Rejected

Conditional Weight Update Rules

16

x Pci�ci�x Peq:ci�ci�

Pci�ci��

Peq:ci�ci�

Pci�ci�

Wi &Wj � Wi Wj

Wi � Wi1 �Wj c�r1 � c�r

&Wj � Wj1 �Wi c�r1 � c�r

�

✤ Problem:

๏ These weight update rules are not stable ⇒ loss of Variance Reduction

✤ Solution:

๏ From definition Wi=feq(c i)/f(c i) ⇒ we need knowledge of PDF

๏ Re-construct the PDF from samples, this is a standard numerical method known as Kernel Density Estimation. Specifically, for every particle at c replace

✤ Implementation:

๏ For accepted particles

Stability

17

Wi � W�i �

1�� Si �� j�Si Wj (average weights within a sphere of radius ε in velocity space)

f �ci� � f��ci� � � K�c� � c� f �c�� d c� f

��ci� � f �ci� as K��c� � ∆��c�since

Rejected

Advection Step

Collision Step

Accepted


at ci & Wi �feq�ci�f �ci�


cr� � cr �MX


Wi� �Wi

�1�Wj cr� ��1�cr� � & Wj

� �Wj�1�Wi cr� ��1�cr� �



Wi � Wi�Wj�

, Wj � Wi�Wj�

Final Algorithm

(1)W�i �

1�� Si �� j�Si Wj

estimate Wi�

& Wj�

using (1)

�

�

�

�

�

��

�

0.10 0.500.20 0.300.15�

0.20

0.10

0.05

0.02

0.01

ErrrelError

Results: Bias (error) vs. ε in 0D

19

ε/Cmp

NCell �3

Cmp3 >>1

0.0 0.5 1.0 1.5 2.0Time �Τ�

1.005

1.010

1.015

1.020

1.025

1.030�cx4�

�

�

Error �� Exact

Appx. Sol.

SS

Results: 1D Transient Couette Flow

20

0.2 0.4 0.6 0.8 1.0

x

L

0.990

0.995

1.000

1.005

1.010

1.015

T

T0

0.2 0.4 0.6 0.8 1.0

x

L

�0.020

�0.015

�0.010

�0.005

qy

qy,0

0.2 0.4 0.6 0.8 1.0

x

L

0.990

0.995

1.000

1.005

1.010

1.015

1.020

Ρ

Ρ0

0.2 0.4 0.6 0.8 1.0

x

L

�0.004

�0.002

0.002

uy

c0

0.2 0.4 0.6 0.8 1.0

x

L

0.990

0.995

1.000

1.005

1.010

1.015

T

T0

0.2 0.4 0.6 0.8 1.0

x

L

�0.020

�0.015

�0.010

�0.005

qy

qy,0

0.2 0.4 0.6 0.8 1.0

x

L

0.990

0.995

1.000

1.005

1.010

1.015

1.020

Ρ

Ρ0

0.2 0.4 0.6 0.8 1.0

x

L

�0.004

�0.002

0.002

uy

c0

0.2 0.4 0.6 0.8 1.0

x

L

0.990

0.995

1.000

1.005

1.010

1.015

T

T0

0.2 0.4 0.6 0.8 1.0

x

L

�0.020

�0.015

�0.010

�0.005

qy

qy,0

0.2 0.4 0.6 0.8 1.0

x

L

0.990

0.995

1.000

1.005

1.010

1.015

1.020

Ρ

Ρ0

0.2 0.4 0.6 0.8 1.0

x

L

�0.004

�0.002

0.002

uy

c0

NCell=500

Kn=1.0

~1% Relative Error

Relative Sampling Uncertainty

21

� � � � � � � � � � ��

�

�

�

�

�

�

�

�

��

1�10�4 5�10�4 0.001 0.005 0.010 0.050 0.100

Uwall

c0

0.1

1

10

100

1000

Σu

DSMC

VR-DSMC

≈ 5 cm/s

1,000,000 less samples for the same uncertainty at 5cm/s!

Stability

22

Σ2 �Wi� �Σ2 �Wi�Kn�10,��Si ��0

�� Si ��

10 10 10 10 10 10 10 10 10 10 10 10 10

3

33 3 3 3 3 3 3 3 3 3 3

1

1

11 1 1 1 1 1 1

11

0.3

0.3

0.30.3

0.30.3

0.30.3

0.3

0.1

0.1

0.10.1

0.1

0.1

0.15 10 15 20

1.0

2.0

3.0

1.5

For stability at a fixed ε Kn� � �� Si ��

� NCell

Conclusions

✤ Variance reduction using likelihood ratios is viable and exciting

๏ Main advantage: the DSMC simulation is never perturbed

✤ Small increase in computational cost

๏ Need to find NN of some particle ⇒ total cost scales as O(NCell Log(NCell))

✤ Stability Issues:

๏ KDE introduces bias that is a function of ε

๏ for low Kn NCell↑ for a Stable and Accurate solution

✤ Looking forward:

๏ Other collision Models

• BGK

• Maxwell

๏ Improve bias for a given NCell

23

Q&AMore info including sample code:http://web.mit.edu/husain/www

Appendix

DSMC with weights: Scattering probability

26

✤ DSMC is a set of probabilistic steps

✤ Start by selecting the same number of candidate particles:

Candidates= NEffNCel(NCell-1)MXσΔt/VCell

✤ if we choose particles of velocity classes c i and c j with weights Wi and Wj respectively there will be:

(NEff)2WiWj CijσΔt/VCell Collisions

✤ to correctly account for this we use the following collision probabilities:

Pi �Wj cijMX

Pj �Wi cijMX

Documents

A DSMC-based variance reduction formulation for low-signal ﬂows · 2009. 9. 14. · A DSMC-based variance reduction formulation for low-signal ﬂows H. A. Al-Mohssen [email protected]