A1258690579DnhFC

1
Bayesian Computation Approach to Inverse Problems
in Heat Conduction
Principal investigator Prof. Nicholas
Zabaras Presenter Jingbo Wang Materials Process
Design and Control Laboratory Sibley School of
Mechanical and Aerospace Engineering188 Frank H.
T. Rhodes Hall Cornell University Ithaca, NY
14853-3801 Email zabaras_at_cornell.edu Phone
[phone] URL http//www.mae.cornell.edu/zab
aras/
Materials Process Design and Control Laboratory
2
Presentation outline

• Introduction to inverse problems in heat
  conduction
• Introduction to Bayesian computation
• ----- Fundamentals of Bayesian
  statistical inference
• ----- Markov random field (MRF) and
  discontinuity adaptive Markov
• random field (DAMRF)
• ----- Markov chain Monte Carlo (MCMC)
  simulation
• A few examples
• ----- Inverse heat conduction problem
  (IHCP)
• ----- Heat source reconstruction
• Apply Bayesian to computationally intensive
  problems --- an extension to inverse radiation
• Conclusions

Materials Process Design and Control Laboratory
3
Inverse problems in heat conduction
known heat flux
Gh
thermocouples







unknown heat flux
Go












Heat sources
Gg
known temperature
Materials Process Design and Control Laboratory
4
Features of the inverse problem

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

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

• uncertainties

Materials Process Design and Control Laboratory
5
Approaches to the inverse problems
objective
optimization
regularization

• 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

Materials Process Design and Control Laboratory
6
Bayesian computation approach
Advantages

• 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 )

Materials Process Design and Control Laboratory
7
Fundamentals of Bayesian statistics

• Bayes formula

• Bayesian statistics

prior evidence gt posterior probability of a
hypothesis

• Bayesian inference to estimation (regression)
• - interested in distribution of random
  quantities ??1, ?2, ?mT
• - a priori beliefs about P(?)
• - data YY1, Y2, YnT relevant to ?

Likelihood
Priori pdf
Posterior pdf
Materials Process Design and Control Laboratory
8
The likelihood
prior
posterior
Materials Process Design and Control Laboratory
9
The prior

• 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

Materials Process Design and Control Laboratory
10
More complicated models
A hierarchical structure
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
Materials Process Design and Control Laboratory
11
Spatial statistics models
Materials Process Design and Control Laboratory
12
Monte Carlo simulation

• Monte Carlo Principle
• 1. draw an i.i.d. set of samples x(i) i 1N
  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 )

1
N
p

å
x
x
)
(
)
(
d
N
i
x
N

i
1
Materials Process Design and Control Laboratory
13
Markov chain Monte Carlo (MCMC)

• Sampling from a complex distribution using
  Markov chain mechanism

Metropolis-Hastings algorithm
Gibbs sampler
Materials Process Design and Control Laboratory
14
Bayesian formulation for IHCP

• Parameterization of unknown heat flux q0

Unknown vector ?
random

• System input and output relation

direct numerical solver F
Measurement Y
Input ?

• Assumptions
• numerical error much less then
• measurement noise
• ? iid N(0, s2)

simulation noise

• Likelihood function

--- known s
--- unknown s
Materials Process Design and Control Laboratory
15
Bayesian formulation for IHCP (cont.)
Prior distribution modeling
--- Single layer posterior
1
1
H
-

-
-
µ
q
lq
q
T
T
q
H
-
W
Y
Y
p
)
exp(

)
(
exp
)

(
q
Y
)
(
s
2
2
2
--- hierarchical posterior
H
H
q
q
T
-
-
1
)
(
)
(
Y
Y
q
lq
l
l
q
-
-
µ
-
2
/
/
2
T
m
n

exp

exp
)
,
,
(
W
v
v
p
T
T
2
2
v
T
-

-
-
a
a
1
)
1
(
1
-
-
b
l
b
l

exp

exp
v
v
1
0
1
0
T
T
Materials Process Design and Control Laboratory
16
Gibbs sampler and modified Gibbs sampler

• Gibbs sampler

• modified Gibbs sampler

Initialize ?0 For i 0N-1 For j 1m
sample

• Initialize ?(0), ?(0) and vT(0)
• For i 0N-1
• ---For j 1m
• sample
• ---sample u U(0,1)
• ---sample ?() q?(?() ?(i))
• ---if u lt A(?() , ?(i) )
• ?(I1) ?()
• ---else
• ?(I1) ?(i)
• ---sample u U(0,1)
• ---sample vT() qv(vT() vT(i))
• ---if u lt A(vT() , vT(i) )
• vT(I1) vT()
• ---else
• vT(I1) vT(i)

Materials Process Design and Control Laboratory
17
1D IHCP example
Y (d,i?t)
q
x
d
L
--- True q in simulation
q
--- Discretization of q(t)
1.0
dt
?i
?i-1
?i1
t
0
0.5
0.9
1.0
0.1
Materials Process Design and Control Laboratory
18
1D IHCP example (results)
d0.5 sT0.01
d0.5 sT0.001
d0.5 sT0.01 (assume sT unknown )
d0.1 sT0.001
d0.1 sT0.01
Materials Process Design and Control Laboratory
19
Reconstruction of piecewise continuous heat
source
with
0ltxlt0.25
0.25ltxlt0.5
0.5ltxlt0.75
0.75ltxlt1.0
analytical solution to the direct problem
Materials Process Design and Control Laboratory
20
2D heat source reconstruction

• Normalized governing equations

unknown
20
-

)
10
exp(
)
,
,
(
t
t
y
x
f
p
2
125
.
0
2
-

-
2
2
)
725
.
0
(
)
75
.
0
(
y
x
-
exp

2
125
.
0
2
Materials Process Design and Control Laboratory
21
2D heat source reconstruction
true heat source
reconstructed heat source when sT0.005
reconstructed heat source when sT0.02
Materials Process Design and Control Laboratory
22
Inverse Heat Radiation Problem
What g(t) causes measured T?
Materials Process Design and Control Laboratory
23
Direct simulation
A finite element (FE) S4 method framework
Materials Process Design and Control Laboratory
24
Reduced order modeling --- A POD based approach

• reduced order models

• POD eigenfunction problem

• Solution as linear combination of POD
• basis

Materials Process Design and Control Laboratory
25
MCMC algorithm --- a cycle design of single
component update

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

Algorithm
Materials Process Design and Control Laboratory
26
A testing example
Schematic of the example
Profile of testing heat sources
Materials Process Design and Control Laboratory
27
Basis fields
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
Materials Process Design and Control Laboratory
28
Homogeneous temperature solution
Th computed by full model
Th computed by reduced order model
Comparison of reduced order solutions
at thermocouple locations
Materials Process Design and Control Laboratory
29
Heat source reconstruction
MAP estimates of g1 at different magnitude of
noise
Posterior mean of g1 when sT 0.005
MAP estimates of g2 at different magnitude of
noise
Posterior mean of g2 when sT 0.005
Materials Process Design and Control Laboratory
30
Conclusions

• 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

Futures --- Sequential Bayesian filter ---
Stochastic upscaling via Bayesian computation ---
Uncertainty quantification in multiscale
simulation
Materials Process Design and Control Laboratory