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The University of Auckland – Applied Mathematics
Bayesian Methods for Inverse Problems :
why and how
Colin FoxTiangang Cui, Mike O’Sullivan (Auckland), Geoff Nicholls (Statistics, Oxford)
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In this talk
• An example in geothermal model calibration
• Inferential solutions to inverse problems, noise and all
• The need to integrate: pixel-wise degraded binary images
• Monte Carlo integration
• Computation (MCMC)
• Output analysis for geothermal problem
• Next talk :: better sampling algorithms
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An inverse problem in geothermal fields
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Schematic of a geothermal field
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HeatCrystallineRock
Convective Magma
Hig
hD
ensity
Cold
Wate
r
Hig
hD
ensity
Cold
Wate
r
Low DensityHot Water
Rock of LowPermeability
TemperatureContour
Geothermal Well
Permeable Rock
Boiling Zone
1
Given near-surface measurements of temperature, pressure, flow, want
• Model calibration :: rock type, porosity, fracture : heat sources
• Predict :: long term (50 year) power generation, land subsidence, etc
• Decide :: robust investment plan
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Wairakei geothermal power generator
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Bayesian formulation of inverse problems
d = Ax + n: data d, image x, measurement noise n, forward map A
image spacedata space
xtrue
xML d
dnf
forwardmap
A
A-1
π (x|d, m) ∝ Pr (d|x, m)Pr (x|m) (Bayes’ rule)
Likelihood: measurement and noise (Physics, instrumentation, probability)
Prior / state space: stochastic modelling, physical laws, previous measurements, expert opinion
Inference based on posterior distribution conditioned on data (computational statistics)
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Bayesian formulation of inverse problems
d = Ax + n: data d, image x, measurement noise n, forward map A
image spacedata space
xtrue
xML d
dnf
forwardmap
A
A-1
π (x|d, m) ∝ Pr (d|x, m)Pr (x|m) (Bayes’ rule)
Likelihood: measurement and noise (Physics, instrumentation, probability)
Prior / state space: stochastic modelling, physical laws, previous measurements, expert opinion
Inference based on posterior distribution conditioned on data (computational statistics)
![Page 8: New The University of Auckland – Applied Mathematics · 2011. 8. 29. · The University of Auckland – Applied Mathematics Bayesian Methods for Inverse Problems : why and how Colin](https://reader033.vdocuments.us/reader033/viewer/2022052103/603d55d20da1d75a15795e8f/html5/thumbnails/8.jpg)
Bayesian formulation of inverse problems
d = Ax + n: data d, image x, measurement noise n, forward map A
image spacedata space
xtrue
xML d
dnf
forwardmap
A
A-1
π (x|d, m) ∝ Pr (d|x, m)Pr (x|m) (Bayes’ rule)
Likelihood: measurement and noise (Physics, instrumentation, probability)
Prior / state space: stochastic modelling, physical laws, previous measurements, expert opinion
Inference based on posterior distribution conditioned on data (computational statistics)
![Page 9: New The University of Auckland – Applied Mathematics · 2011. 8. 29. · The University of Auckland – Applied Mathematics Bayesian Methods for Inverse Problems : why and how Colin](https://reader033.vdocuments.us/reader033/viewer/2022052103/603d55d20da1d75a15795e8f/html5/thumbnails/9.jpg)
Bayesian formulation of inverse problems
d = Ax + n: data d, image x, measurement noise n, forward map A
image spacedata space
xtrue
xML d
dnf
forwardmap
A
A-1
π (x|d, m) ∝ Pr (d|x, m)Pr (x|m) (Bayes’ rule)
Likelihood: measurement and noise (Physics, instrumentation, probability)
Prior / state space: stochastic modelling, physical laws, previous measurements, expert opinion
Inference based on posterior distribution conditioned on data (computational statistics)
![Page 10: New The University of Auckland – Applied Mathematics · 2011. 8. 29. · The University of Auckland – Applied Mathematics Bayesian Methods for Inverse Problems : why and how Colin](https://reader033.vdocuments.us/reader033/viewer/2022052103/603d55d20da1d75a15795e8f/html5/thumbnails/10.jpg)
Stochastic Model
xforward
mapK
datalossP
trueimage
idealdata
noisefreedata
+
n
d=Ax+n : A=PK
actualdata
imagerecoveryalgorithm
x̂
representation / prior density
estimate
+ noise
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Stochastic Model
xforward
mapK
datalossP
trueimage
idealdata
noisefreedata
+
n
d=Ax+n : A=PK
actualdata
imagerecoveryalgorithm
x̂
representation / prior density
estimate
+ noise
if n ∼ fN (·) Pr(d|x,m) = fN (d−Ax)
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Stochastic Model
xforward
mapK
datalossP
trueimage
idealdata
noisefreedata
+
n
d=Ax+n : A=PK
actualdata
imagerecoveryalgorithm
x̂
representation / prior density
estimate
+ noise
if n ∼ fN (·) Pr(d|x,m) = fN (d−Ax)
Prior distribution πpr(x) uses physical laws, expert knowledge, stochastic modelling
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Stochastic Model
xforward
mapK
datalossP
trueimage
idealdata
noisefreedata
+
n
d=Ax+n : A=PK
actualdata
imagerecoveryalgorithm
x̂
representation / prior density
estimate
+ noise
if n ∼ fN (·) Pr(d|x,m) = fN (d−Ax)
Prior distribution πpr(x) uses physical laws, expert knowledge, stochastic modelling
Focus of inference is posterior distribution for x given d
π (x|d) ∝ Pr (d|x) πpr (x)
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Solutions to Inverse Problem = Summary Statistics
Posterior distribution π (x|d) encapsulates all information about x
Regularized Inversion - Modes
x̂MLE = arg max Pr (d|x) x̂MAP = arg max π (x|d)
(least-squares, Moore-Penrose inverse, Tikhonov regularization, Kalman filtering, Backus-
Gilbert, Prussian-hat cleaning)
Inferential Solutions
“Answers” are expectations over the posterior π (x|d)
Eπ [f (x)] =∫
Xf (x) π (x|d) dx
If
f (x) = indicator function that image shows cancer
E [f (x)] is posterior probability (based on measurements, prior) that patient has cancer.
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Two modes over 100×100 pixel image
x
π(x|d)
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Two modes over 100×100 pixel image
x
π(x|d)
if maximum value (at mode) is twice the value of the lower local maximum
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Two modes over 100×100 pixel image
x
π(x|d)
if maximum value (at mode) is twice the value of the lower local maximum
width of global mode is half the width of the lower local mode
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Two modes over 100×100 pixel image
x
π(x|d)
if maximum value (at mode) is twice the value of the lower local maximum
width of global mode is half the width of the lower local mode
in each of the coordinate directions x1, x2, · · · , x100×100
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Two modes over 100×100 pixel image
x
π(x|d)
if maximum value (at mode) is twice the value of the lower local maximum
width of global mode is half the width of the lower local mode
in each of the coordinate directions x1, x2, · · · , x100×100
there is 210000−1 ≈ 103010 times more probability mass in the lower, broader, peak than around
the mode
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Two modes over 100×100 pixel image
x
π(x|d)
if maximum value (at mode) is twice the value of the lower local maximum
width of global mode is half the width of the lower local mode
in each of the coordinate directions x1, x2, · · · , x100×100
there is 210000−1 ≈ 103010 times more probability mass in the lower, broader, peak than around
the mode ∫X
π (x|d) dx =∫
Xπ (x|d) dx1 dx2 · · · dx10000
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Mode vs Mean
true noisy
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MLE
θ = 0
θ =
0.1
25θ
= 0
.25
θ =
0.3
75θ
= 0
.5
MAP
θ =
0.6
75
mean sample MPM
Pixel-wise noise on binary image, Ising prior with “lumping” constant θ
F and Nicholls “Exact MAP states and expectations from perfect sampling: Greig, Porteous and Seheult
revisited” 2001
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Monte Carlo integration + importance sampling
If x(1), . . . , x(n) ∼ π (·|d)
Eπ [f (x)] ≈ 1n
n∑i=1
f(x(i)
)Construct x(1), . . . , x(n) as iterates of an ergodic map
Deterministic iteration: x(n+1) = M(x(n)
)π(x) =
∑M(y)=x
π(y)|M ′(y)|
(Frobenius-Perron)
Stochastic Iteration: Markov chain with transition kernel p (x, y)
∫X
π (x) p (x, y) dx =∫
Xπ (y) p (y, x) dx (global balance)
π (x) p (x, y) = π (y) p (y, x) (detailed balance)
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Monte Carlo integration + importance sampling
If x(1), . . . , x(n) ∼ π (·|d)
Eπ [f (x)] ≈ 1n
n∑i=1
f(x(i)
)Construct x(1), . . . , x(n) as iterates of an ergodic map
Deterministic iteration: x(n+1) = M(x(n)
)π(x) =
∑M(y)=x
π(y)|M ′(y)|
(Frobenius-Perron)
Stochastic Iteration: Markov chain with transition kernel p (x, y)
∫X
π (x) p (x, y) dx =∫
Xπ (y) p (y, x) dx (global balance)
π (x) p (x, y) = π (y) p (y, x) (detailed balance)
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Sampling Algorithms in Physics
Hamiltonian Dynamics
Equate unknowns x to ‘position’ variables q with potential energy
E(q) = −T log (π(x))− T log (Z)
Auxiliary ’momentum’ variables p with kinetic energy K(p) = 12‖p‖
2.
Hamiltonian H(q, p) = E(q) + K(p) has canonical distribution
P (q, p) =1
ZPexp (−E(q))
1ZK
exp (−K(p))
Hamiltonian dynamics leave P (q, p) invariant + stochastic transitions for ergodicity
Langevin diffusion
dx = s dB +s2
2∇ log (π(x)) dt
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Metropolis-Hastings algorithm
1. given state xt at time t generate candidate state x′ from a proposal distribution q (.|xt)
2. With probability α(xt → x′
)= min
(1,
π(x′)q (xt|x′)π(xt)q (x′|xt)
)set Xt+1 = x′
otherwise Xt+1 = xt
3. Repeat
q (.|xt) can be any distribution that ensures the chain is irreducible and aperiodic.
Pros
• Provably convergent to any property, e.g. ‘best’ estimate, uncertainty in estimate
• State space can be continuous, discrete, stochastic, or variable dimension
Cons
• Can be (very) slow
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Metropolis-Hastings algorithm
1. given state xt at time t generate candidate state x′ from a proposal distribution q (.|xt)
2. With probability α(xt → x′
)= min
(1,
π(x′)q (xt|x′)π(xt)q (x′|xt)
)set Xt+1 = x′
otherwise Xt+1 = xt
3. Repeat
q (.|xt) can be any distribution that ensures the chain is irreducible and aperiodic.
Pros
• Provably convergent under mild requirements
• State space can be continuous, discrete, stochastic, or variable dimension
Cons
• Can be (very) slow
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Results for geothermal field
Time [day]
Time [day]
p[b
ar]
(Wellhead)
h[k
J/kg]
10
10
20
20
30
30
40
40
50
50
60
60
70
70
80
80
90
90
2000
1400
1600
1800
2200
40
50
60
70
80
90
4
Wellhead pressure (top) and flowing enthalpy (bottom).
Black line is estimated result, triangles and squares are observed data.
Tiangang Cui, Bayesian Inference for Geothermal Model Calibration MEng Auckland, 2005
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Output analysis for geothermal field
MCMC Updates
Log-lik
elihood
Lag
Auto
corr
ela
tion
0
0
0
1
0.5
30,000 60,000
10,0005,000
-15
-20
-25
-30
-35
φ [-]
m [-]
Srl [-]
log(k [m2])
p0 [bar]
ps [bar]
0
0
0
00
0
00 0.1
0.2
0.2 0.3
0.4
0.4
0.6
0.7
0.8
0.8 0.9 1
10 20 30 40 50
100 120 140 160
-15.15 -15.05 -14.95
10,000
4,000
4,000
4,000
4,000
4,000
8,000
8,000
8,000
log10(k [m2])p0 [bar]
ps
[bar]
10
20
30
40
50
110120
130140
150160
-15.15
-15.05
-14.95
Sv0
Sls
0 0.1 0.2 0.3
0.7
0.8
0.9
1
5
Autocorrelation and MCMC output trace. Histogram for the marginal distribution of param-
eters. Scatter plot of the joint marginal distribution for log(k), p0 and ps. Histogram for the
marginal distribution of Sv0 and Sls and scatter plot of their joint marginal distribution.
Tiangang Cui, Bayesian Inference for Geothermal Model Calibration MEng Auckland, 2005
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Inferred iso-temperature surface
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Conclusions
1. Bayesian methods can solve substantial real-world inverse problems
2. Time to convergence depends on efficient simulation of the forward map, fast MCMC
algorithms, good proposal distributions
3. Computation required is feasible
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