Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and...

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Bayesian Computation Approach to Inverse Problems Bayesian Computation Approach to Inverse Problems in Heat Conduction in Heat Conduction Materials Process Design and Control Laborator Materials Process Design and Control Laborator C C O O R R N N E E L L L L U N I V E R S I T Y Principal investigator: Prof. Nicholas Zabaras Principal investigator: Prof. Nicholas Zabaras Presenter: Jingbo Wang Presenter: Jingbo Wang Materials Process Design and Control Laboratory Sibley School of Mechanical and Aerospace Engineering 188 Frank H. T. Rhodes Hall Cornell University Ithaca, NY 14853-3801 Email: [email protected] Phone: (607) 255-9104 URL: http://www.mae.cornell.edu/zabaras/

Transcript of Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and...

Page 1: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

Bayesian Computation Approach to Inverse Bayesian Computation Approach to Inverse Problems in Heat ConductionProblems in Heat Conduction

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y

CCOORRNNEELLLL U N I V E R S I T Y

Principal investigator: Prof. Nicholas ZabarasPrincipal investigator: Prof. Nicholas ZabarasPresenter: Jingbo WangPresenter: Jingbo Wang

Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering

188 Frank H. T. Rhodes HallCornell University

Ithaca, NY 14853-3801

Email: [email protected]: (607) 255-9104

URL: http://www.mae.cornell.edu/zabaras/

Page 2: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y

CCOORRNNEELLLL U N I V E R S I T Y

Presentation outline

• Introduction to inverse problems in heat conductionIntroduction to inverse problems in heat conduction

• Introduction to Bayesian computationIntroduction to Bayesian computation

----- Fundamentals of Bayesian statistical inference----- Fundamentals of Bayesian statistical inference

----- Markov random field (MRF) and discontinuity adaptive Markov ----- Markov random field (MRF) and discontinuity adaptive Markov

random field (DAMRF)random field (DAMRF)

----- Markov chain Monte Carlo (MCMC) simulation----- Markov chain Monte Carlo (MCMC) simulation

• A few examplesA few examples

----- Inverse heat conduction problem (IHCP)----- Inverse heat conduction problem (IHCP)

----- Heat source reconstruction----- Heat source reconstruction

• Apply Bayesian to computationally intensive problems --- an Apply Bayesian to computationally intensive problems --- an

extension to inverse radiationextension to inverse radiation

• ConclusionsConclusions

Page 3: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

Γo

Γg

Γh

* ***

****

** unknown heat flux

known temperature

known heat flux

thermocouples

Inverse problems in heat conduction

CCOORRNNEELLLL U N I V E R S I T Y

CCOORRNNEELLLL U N I V E R S I T Y

Materials Process Design and Control Laboratory

***

** **

**

Heat sources

)( TktTC P

in ,

,),( gTtxT ,g

],,0[ maxtt

,),(

hqntxT

k

on

,h),()0,( 0 xTxT

,),(

0qntxT

k

)(unknown ,0

],,0[ maxtt

on ],,0[ maxtt

in ,

on ].,0[ maxtt

+ f

Page 4: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

Features of the inverse problem

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CCOORRNNEELLLL U N I V E R S I T Y

CCOORRNNEELLLL U N I V E R S I T Y

• ill-posedness well-posedness: --- existence --- uniqueness --- continuous dependence of solutions on measurements

stability

A problem is ill-posed if it is not well-posed

identifiability

• uncertainties

random errors

solution errors

measurement errors

model uncertainties

numerical errors

incomplete mathematical representation

perturbations in systemparameters, initial or boundaryconditions …

discretization error

roundoff error

unsolved length scales

Page 5: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

Approaches to the inverse problems

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CCOORRNNEELLLL U N I V E R S I T Y

optimizationobjective

minimum least-squares error

maximum entropy

Maximum Likelihood

Bayesian inference

Gradient methods

(sensitivity and/or adjoint problems need to be solved)

Monte Carlo methods (importance sampling, rejection sampling, Markov chain MC, simulated annealing)

function specification

Tikhonov regularization

(S) future information

iterative regularization

--- conjugate gradient

--- EM method

convexification method

mollification method

regularization

Page 6: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

Bayesian computation approach

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• probabilistic description of inverse solutions

• quantification of various uncertainties

• data driven in nature

• direct simulation in deterministic space

• prior distribution regularization (spatial statistics

models)

• sampling schemes (MCMC, Latin hypercube …)

Advantages:

Page 7: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

)|()()|( YppYp

Fundamentals of Bayesian statistics

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CCOORRNNEELLLL U N I V E R S I T Y

CCOORRNNEELLLL U N I V E R S I T Y

• Bayesian statistics

)|( YθP)(

)()|(YP

θPθYP

• Bayesian inference to estimation (regression)

- interested in distribution of random quantities θ=[θ1, θ2, … θm]T

- ‘a priori beliefs’ about P(θ) - data Y=[Y1, Y2, … Yn]T relevant to θ

Priori pdf

Likelihood

Posterior pdf

• Bayes’ formula

prior + evidence => posterior probability of a hypothesis

Page 8: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

The likelihood

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CCOORRNNEELLLL U N I V E R S I T Y

Y = F(θ) + ω

For a typical system as,

the likelihood is determined by distribution of ω.eg: ω ~ N(0, σ2)

FYθp T

2))((

21exp{)|( Y FY ))}((

• conditional probability of observation (data) Y given the parameter θ• interface to data

prior posterior

Page 9: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

The prior

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CCOORRNNEELLLL U N I V E R S I T Y

• role of a prior pdf --- incorporate known to a priori information --- regularize the likelihood

• a prior distribution can be “informative” or “improper”

• techniques of prior distribution modeling --- accumulated distribution information --- conjugate prior distributions --- physical constraints --- local uniforms --- spatial statistics models

An example of conjugate prior:if Y|θ,σ ~ N(θ, σ), then θ ~ N(θo, σo), σ ~ inv-Gamma (a,b).

A class π of prior distributions is said to form a conjugate family if the posterior density p(θ|X) is in the class for all X whenever the prior density is in π.

Page 10: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

More complicated models

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CCOORRNNEELLLL U N I V E R S I T Y

* A hierarchical structure

)()(),|()|,( ppYpYp

)()|(),|()|,( ppYpYp

)()|( pYp )|( p Ym Yo , )|( Yop

diminish the effect of poor knowledge on hyper-parameters.

* An augmented formulation

quantifying measurement error from data

* A Expectation-Maximization formulation

more robust formulation (iterative regularization) when there are missing data

! Bayesian is adaptive model --- posteriors can be treated as priors for new data

Page 11: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

Spatial statistics models

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Markov random field (MRF)

• having exponential form• explore the spatial and temporal dependence• close related to Tikhonov regularization

}))((exp{)( ~ ji jiijWp

q

x

t

θi

wi

Neighbors of θ

u,

0

)(2

21u

u, else

DAMRF

Page 12: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

Monte Carlo simulation

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Monte Carlo Principle

1. draw an i.i.d. set of samples {x(i)} i = 1:N from a target density p(x)

2. approximate the target density with the following empirical point-mass function

3. approximate the integral (expectation) I(f) with tractable sums IN( f )

N

ixN x

Nxp

i1

)(1

)(

N

iX

NiN dxxpxffIxf

NfI

1

)()()()(1

)(

Page 13: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

Markov chain Monte Carlo (MCMC)

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CCOORRNNEELLLL U N I V E R S I T Y

Initialize xInitialize x00

For i=0:N-1For i=0:N-1

sample u~U(0,1)sample u~U(0,1)

sample sample xx** ~ ~ q(q(xx**|x|xii) )

if u < A(xif u < A(xii, , xx**)=min)=min{1, p(x{1, p(x**)q(x)q(xii|x|x**)/(p(x)/(p(xii)q(x)q(x**|x|xii))}))}

xxi+1i+1=x=x**

else xelse xi+1i+1=x=xii

• Sampling from a complex distribution using Markov chain mechanism

Metropolis-Hastings algorithm

Gibbs sampler

)|(~ 11

ijj

ij xxpx

Initialize x0

For i = 0:N-1 For j = 1:m sample

Page 14: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

Bayesian formulation for IHCP

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• Parameterization of unknown heat flux q0

m

iii txwq

10 ),( Unknown vector θ

Input θInput θ direct numericalsolver F

direct numericalsolver F

Measurement YMeasurement Y

simulationnoise

Y = F(θ) + ω

• System input and output relation

random

• Likelihood function

FYθp T2

))((2

1exp{)|(

Y FY ))}(( --- known σ

--- unknown σ FYθ,p T2

))((2

1(σ2)-n/2exp{σ2)|(

Y FY ))}((

Assumptions• numerical error much less then measurement noise• ω iid ~ N(0, σ2)

Page 15: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

Bayesian formulation for IHCP (cont.)

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Materials Process Design and Control Laboratory

Markov Random Field (MRF)

}))((exp{)(~ ji

jiijWp

}2

exp{)( 2/ Wp Tm

2

21

)( uu

else

ji

jin

Wi

ij ~

,0

,1

,

Prior distribution modeling

--- Single layer posterior:

--- hierarchical posterior:

TWYYp T2 )

2

1exp(})(

2

1exp{)|(

Y )(

}2

1exp{}

2

)()(exp{),,( 2//2 W

v

YYvvp Tm

T

Tn

TT

}exp{}exp{ 11

)1(0

1 10 TT vv

H H

H H

Page 16: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

• Gibbs sampler

Gibbs sampler and modified Gibbs sampler

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1

2i

i ii i

b

a a

2

2 21 1

2N N

si s sii ii i p

s s

H Ha W b

s s st t p ji j ik kt i j i k i

Y H W W

),(~| 2iiii N

)|(~ 11

ijj

ij θθpθ

Initialize θ0

For i = 0:N-1 For j = 1:m sample

• modified Gibbs sampler

(i) Initialize θ(0), λ(0) and vT(0)

(ii) For i = 0:N-1 ---For j = 1:m sample ---sample u ~ U(0,1) ---sample λ(*) ~ qλ(λ(*) | λ(i)) ---if u < A(λ(*) , λ(i) ) λ(I+1) = λ(*) ---else λ(I+1) = λ(i) ---sample u ~ U(0,1) ---sample vT

(*) ~ qv(vT(*) | vT

(i)) ---if u < A(vT

(*) , vT(i) )

vT(I+1) = vT

(*) ---else vT

(I+1) = vT(i)

)|(~ 11

ijj

ij θθpθ

Page 17: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

xq

dL

Y (d,iΔt)

--- True q in simulationq

0 0.5 0.9

1.0

--- Normalized governing equation

2

2

xT

tT

1t 0 ,0 1x

0),0( xT 1x0

0

LxxT

)(0

tqxTx

,

,

, 1t 0

1t 0,

t1.0

--- Discretization of q(t)

θiθi-1 θi+1

dt

1D IHCP example

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0.1

Page 18: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

1D IHCP example (results)

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d=0.5 σT=0.01 d=0.5 σT=0.001

d=0.1 σT=0.01 d=0.1 σT=0.001

d=0.5 σT=0.01(assume σT unknown )

Page 19: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

Reconstruction of piecewise continuous heat source

CCOORRNNEELLLL U N I V E R S I T Y

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),())(( txfTxktT

x

5.1

23

4

1

x

xk

0<=x<0.25

0.25<=x<0.5

0.5<=x<0.75

0.75<=x<=1.0

tx

tx

tx

tx

e

ex

ex

e

txf

5.2

)22(

)45(

2

),(

0<=x<0.25

0.25<=x<0.5

0.5<=x<0.75

0.75<=x<=1.0

txetxT),(

with

analytical solution to the direct problem:

Page 20: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

2D heat source reconstruction

CCOORRNNEELLLL U N I V E R S I T Y

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• Normalized governing equations

,2

2

2

2

fy

T

x

T

t

T

,10 x ,10 y ,0t

,01010

yyxx yT

yT

xT

xT

0T , t = 0.

,0t

unknown

insulated

insulated

insulated

insulated

*

*

*

**

**

*

* **

* **

*

*

*

*

*

*

*

*

*

*

*

2125.02

20)10exp(),,(

ttyxf

2

22

125.02

)725.0()75.0(exp{

yx

Page 21: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

2D heat source reconstruction

CCOORRNNEELLLL U N I V E R S I T Y

CCOORRNNEELLLL U N I V E R S I T Y

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true heat source

reconstructed heat source when σT=0.005

reconstructed heat source when σT=0.02

Page 22: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

Inverse Heat Radiation Problem

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CCOORRNNEELLLL U N I V E R S I T Y

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S(gray

boundary)thermocouple

participatingmedia

Vheat source

*

**

*

*

What g(t) causes measured T?

2 ( ) ( )p r

TC k T g t G x x y y z zq

t

4( ) ( )

4 bs I I I r d Is

4b

b

TI

4

14 ( ( ) )

4br I I r s dq

0

1( ) ( ) 0b n sI r s I n I r d n ss s

wT T

Page 23: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

Direct simulation

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A finite element (FE) + S4 method framework

1. Set T(i)guess = T(I-1);

2. Substitute T(i)guess to compute Ib ;

3. Solve intensity eq for I(i);

4. Compute ;

5. Solve temperature eq to update T(i)guess;

6. If the solution converged, set T(i)guess

as T(i) and save I(i); otherwise, go to 2;

7. Go to next time step.

rq

Page 24: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

Reduced order modeling --- A POD based approach

CCOORRNNEELLLL U N I V E R S I T Y

CCOORRNNEELLLL U N I V E R S I T Y

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• POD eigenfunction problem• reduced order models

( ) ( )

1

1 eNi i

Vie

U U dvN

1

( ) ( ) ( )TK

h Ti i

i

T t r a t r

1

( ) ( ) ( )IK

h Ii i

i

I t r s b t r s

1

( ) 1TK

jj ji i j j T

i

daM H a S Q g t j K

dt

1 1

1I IK K

ji i ji i j Ii i

A b B b D j K

2( )Tj p jV

M C dv T T

ji j iVH k dv

Tj jrVS dvq

( )Tj jV

Q G x x y y z z dv 4

{ ( ) }I I I Iji i j i jVA s d dv

4 4{( ) }I I

ji i jVB d d dv

4

( )I Ij b b jV

D I I d dv

• Solution as linear combination of POD basis

Page 25: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

MCMC algorithm --- a cycle design of single component update

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• implicit likelihood MH sampler• increasing acceptance probability single component update

Algorithm:

( ) ( 1) ( ) ( ) ( )2

1 1( ) exp{ ( )}

22i i i

j j j j j jqjqj

q

},...,,,...,,{ )()(1

)1(1

)1(2

)1(1

)1( im

ij

ij

iiij

Initialize Initialize θθ00

For i=0:N-1For i=0:N-1

For j=1:mFor j=1:m

sample u~U(0,1)sample u~U(0,1)

sample sample θθ**jj ~ ~ qqjj((θθ **

jj | | θθ ii

-j-j , , θθ ii

jj ) )

if u < A(if u < A(θθ iijj , , θθ **

jj ))

θθi+1i+1==θθ**

else else θθi+1i+1==θθii

Page 26: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

A testing example

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x

z y

1m

1m

1m

g(t)(0.5m, 0.5m, 0.5m)

800K

800K

800K

800K

800K

O

800K

***

12

3

o t

g(t)

400kW/m3

0.05s0.01s 0.04s

80kW/m3

o t

g(t)

0.02s 0.04s 0.05s

160kW/m3

80kW/m3

g1(t)

g2(t)

Schematic of the example

Profile of testing heat sources

Page 27: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

Frame 001 09 Mar 2004 Frame 001 09 Mar 2004

Basis fields

CCOORRNNEELLLL U N I V E R S I T Y

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Materials Process Design and Control Laboratory

Frame 001 09 Mar 2004 Frame 001 09 Mar 2004 Frame 001 09 Mar 2004 Frame 001 09 Mar 2004

Frame 001 09 Mar 2004 Frame 001 09 Mar 2004 Frame 001 09 Mar 2004 Frame 001 09 Mar 2004 Frame 001 09 Mar 2004 Frame 001 09 Mar 2004

Frame 001 09 Mar 2004 Frame 001 09 Mar 2004

Frame 001 09 Mar 2004 Frame 001 09 Mar 2004 Frame 001 09 Mar 2004 Frame 001 09 Mar 2004

1st, 3rd and 6th Basis of Th

1st, 3rd and 6th Basis of Ih

along direction s =[0.9082483 0.2958759 0.2958759]

1st, 3rd and 6th Basis of Ih

along direction s =[-0.9082483 0.2958759 0.2958759]

Page 28: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

Homogeneous temperature solution

CCOORRNNEELLLL U N I V E R S I T Y

CCOORRNNEELLLL U N I V E R S I T Y

Materials Process Design and Control Laboratory

Th computed by full model Th computed by reduced order model

Comparison of reduced order solutions atthermocouple locations

Page 29: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

Heat source reconstruction

CCOORRNNEELLLL U N I V E R S I T Y

CCOORRNNEELLLL U N I V E R S I T Y

Materials Process Design and Control Laboratory

MAP estimates of g1 at different magnitude of noise

MAP estimates of g2 at different magnitude of noise

Posterior mean of g1 when σT =0.005

Posterior mean of g2 when σT =0.005

Page 30: Bayesian Computation Approach to Inverse Problems in Heat Conduction Materials Process Design and Control Laboratory Principal investigator: Prof. Nicholas.

• Bayesian inference treats the inverse problem as a well-posed problem in an expanded stochastic space• Bayesian inference approach provides statistical distribution as well as point estimates of inverse solution• In seeking point estimates, Bayesian approach regularizes the ill-posedness of inverse problem through prior distribution modeling• MCMC samplers provide accurate estimates for the statistics of inverse solution• Bayesian computation is applicable to complex inverse problems via reduced-order modeling

Conclusions

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Futures--- Sequential Bayesian filter--- Stochastic upscaling via Bayesian computation--- Uncertainty quantification in multiscale simulation