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Tutorial on Uncertainty Quantification with Emphasis on...
Transcript of Tutorial on Uncertainty Quantification with Emphasis on...
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Tutorial on Uncertainty Quantification withEmphasis on Polynomial Chaos Methods
Mohamed Iskandarani Ashwanth Srinivasan Carlisle Thacker,Shuyi Chen Chiaying Lee,
University of MiamiOmar Knio Ihab Sraj Alen Alexandrian Justin Winokur,
Duke UniversityYoussef Marzouk Patrick Conrad,
MIT
October 3, 2013
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Outline
Uncertainty Quantification (UQ)
What is Polynomial Chaos
Forward Propagation and Analysis
Bayesian Inference
UQ-Initial Conditions
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
What is UQ
Uncertainty Quantification revolves around:• Identification: What are the uncertainty sources?,• Characterization: aleatoric (intrisically random) or
epistemic (fixed but have unknown values)Characterization may be scale dependent
• Forward Propagation: Propagate input uncertaintythrough numerical model to calculate output uncertainty
• Inverse Propagation: Use observations/experiments tocorrect input uncertainties
• Sensitivity Analysis: Which uncertainties contribute themost to output uncertainties
• Reduction: Improve forecast by assimilating observationsUQ assesses confidence in model predictions and allowsresource allocation for fidelity improvements
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Quantifying Ocean Model Uncertainties
• Model equations• Initial Conditions: Observation sparse in space-time• Boundary Conditions
• Momentum, heat and fresh water fluxes• Lateral Boundary Conditions in Regional Models• Bottom boundary conditions
• Parameterization of small scale processes• mixed layer and bottom boundary layer parameters• bulk formula for air-sea fluxes
Predictive simulation requires careful assessment of all sourcesof error and uncertainty
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
UQ Approaches
• Many UQ approaches exist fulfilling specific needs.• Emphasis here will be on representation of uncertain
variables• Emphasis on Forward Propagation which enables analysis
and inverse propagation• Topics centered on Generalized Polynomial Chaos
methods (reflecting the presentor biases and experience).
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
What is Polynomial Chaos (PC)PC combines probabilistic and approximation frameworks toexpress dependency of model outputs on uncertain modelinputs• Series representation:
M(x , t , ξ) ≈ MP.
=P∑
k=0
M̂k (x , t)ψk (ξ) (1)
• ξ: uncertain input characterized by its PDF ρ(ξ)• M(x , t , ξ): model output aka observable• M̂k (x , t): series coefficients• ψk (ξ): basis (shape) functions in ξ-space
• Basic Questions• How to choose ψk ?• How to determine the coefficients M̂k ?• Where to truncate the series?
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Benefit of functional representationWhat can you do with a series?
• Sum series to interpolate in ξ-space• series is computationally (much) cheaper than a complex
model• can sum it millions of time to build histogram or effect
Monte Carlo sampling• Integrate in ξ-space for statistical moments
• Mean: E [M] =∫
Mρ(ξ)dξ =∑
k
M̂k∫ρ(ξ)ψk (ξ)dξ
• Variance: var [M] =∫ (∑
k
M̂kψk (ξ) − E [M]
)2ρ(ξ)dξ
• Differentiate in ξ-space (no adjoint code!)
∂M∂ξ
=∑
k
M̂k∂ψk∂ξ
Series must be reliable to reap benefits
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Example 1 of input uncertainties and ρ(ξ)
• Drag Coefficient is uncertain: CD = αCrefD• α is a multiplicative factor, with α ∈ [αmin, αmax]• Map it to standard interval −1 ≤ ξ ≤ 1α = (αmax − αmin) ξ+12 + αmin
• If all values are equally likely than ρ(ξ) = 12 .• To weigh an area more than others choose a beta
distribution:
ρ(ξ) =(1 + ξ)α(1− ξ)β
2α+β+1B(α + 1, β + 1
E [ξ] =α + 1
α + β + 2
var [ξ] =(α + 1)(β + 1)
(α + β + 2)2(α + β + 3)
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Example 2 of input uncertainties and ρ(ξ)
Uncertainty in Initial Boundary Conditions via EmpiricalOrthogonal Functions perturbations:
u(x ,0, ξ1, ξ2) = u(x ,0) +[√
λ1U1ξ1 +√λ2U2ξ2
](2)
• (λk ,Uk ): are eigenvalues/eigenvectors of covariancematrix obtained from free-run simulation
• u: unperturbed initial condition• u(x ,0, ξ1, ξ2): Stochastic initial condition input• The two independent uncertain variables are the modes
amplitudes: ξ1,2• Uniform distributions: ρ(ξ1,2) = 12
• Gaussian distributions: ρ(ξ1,2) = e−ξ21,2
2√2π
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Polynomial Chaos Basis
ξ-distribution Domain weight ρ(ξ) basis ψk (ξ) parameter
Gauss (−∞,∞) e− ξ
22√
2πHermite none
Gamma (0,∞) ξαe−ξ
Γ(α+1) Laguerre α > 1
Beta [−1,1] (1+ξ)α(1−ξ)β
2α+β+1B(α+1,β+1) Jacobi α, β > 1Uniform [−1,1] 12 Legendre none
• Inner Product in ξ-space:〈ψj , ψk
〉=∫ψk (ξ)ψj(ξ) ρ(ξ)dξ
• Polynomial basis is orthonormal w.r.t. ρ(ξ):〈ψj , ψk
〉= δi,j
• Input parameter domain and distribution often dictate themost convenient basis.
〈ψj , ψk
〉= δi,j
• Wiener-Askey scheme provides a hierarchy of possiblecontinuous PC bases
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Normal Distribution
−6 −4 −2 0 2 4 60
0.2
0.4
0.6
0.8
1
x
PD
F
• Most commonly used input distribution• Support on (−∞,∞)
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Gamma Distribution
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
α=0
α=1
α=2
α=3
x
PD
F
• Useful to represent uncertainties in positive quantities.• Support on (0,∞)
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Beta Distribution
−1 −0.5 0 0.5 10
0.2
0.4
0.6
0.8
1
α=1,β=1
α=2,β=1α=1,β=2
α=2,β=2
x
PD
F
• Useful for uncertainties that varies between set quantities.• Can be tailored to weigh some values more than others• Support on [−1,1]
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Uniform Distribution
−1.5 −1 −0.5 0 0.5 1 1.50
0.2
0.4
0.6
0.8
1
x
PD
F
• Useful for uncertainties with sharp bounds• or not much is known about input distribution• Support on [−1,1]
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Polynomial Chaos Basis• Series: M(x , t , ξ) =
∑Pk=0 M̂k (x , t)ψk (ξ)
• Expectation:
E [ψk ] =∫ψk (ξ)ρ(ξ)dξ = 〈ψk , ψ0〉 = δk ,0
• mean:
E [M] =P∑
k=0
uk (x , t)E [ψk (ξ)] = u0(x , t)
• Variance:
E[(M − E [M])2
]=
P∑k=1
M̂2k (x , t)
• Covariance:
E [ (u − E [u]) (v − E [v ]) ] =P∑
k=1
uk (x)vk (x , t)
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Multidimensional basis
Multi-dimensional basis functions Ψk (ξ1, ξ2, . . . , ξn) are tensorproducts of 1D basis functions:
Ψk (ξ1, ξ2, . . . , ξn) = ψαk1(ξ1)ψαk2
(ξ2) . . . ψαk3(ξn)
• 1D Legendre basis: L0(ξ) = 1, L1(ξ) = ξ, L2(ξ) = 3ξ2−12
• 2D Exampleψ0 = L0(ξ1)L0(ξ2) ψ2 = L0(ξ1)L1(ξ2) ψ5 = L0(ξ1)L2(ξ2) ψ9 = L0(ξ1)L3(ξ2)ψ1 = L1(ξ1)L0(ξ2) ψ4 = L1(ξ1)L1(ξ2) ψ8 = L1(ξ1)L2(ξ2)ψ3 = L2(ξ1)L0(ξ2) ψ7 = L2(ξ1)L1(ξ2)ψ6 = L3(ξ1)L0(ξ2)
• Triangular truncation is common, max order=3
• number of coefficient is P + 1 = (N+p)!N!p!N is the number of stochastic variablesp is the max polynomial degree in 1D
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
How do we determine PC coefficients?
• Series: M(x , t , ξ) =∑P
k=0 M̂k (x , t)ψk (ξ)• Galerkin Projection on ψk basis (minimizes L2-error norm)
M̂k (x , t) = 〈M, ψk 〉 =∫
M(x , t , ξ)ψk (ξ)ρ(ξ)dξ
• Non Intrusive Spectral Projection: Approximate integralnumerically via quadrature
M̂k (x , t) ≈Q∑
q=1
M(x , t , ξq)ψk (ξq)ωq
• ξq/ωq quadrature points/weights• Quadrature requires an ensemble run at ξq.• No code modification is necessary
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Choice of Quadrature
• Gauss quadrature most accurate/point (ψp+1(ξq) = 0) butNaive tensorization cost grows exponentially: pN .
• Rely on Nested Sparse Smolyak Quadrature Tempers thecurse of dimensionality
• Adaptive Quadrature
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Polynomial Chaos Expansions Summary
• Paradigm shift from statistical to combinedprobabilistic/approximation view
• Can quantify approximation error and “convergence” tosolution
• No a-priori restriction/assumption on output statistics• Approach robust to model non-linearity and model
differentiability• Can be done non-intrusively via ensembles.• Multiple independent stochastic variables can be handled
by multi-dimensonal tensorization of 1D basis functionsand quadratures.
• Sampling Challenges for high N or p
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Forward Problem: Parametric Sensitivity
−98 −96 −94 −92 −90 −88 −86 −84 −82 −80 −78
20
22
24
26
28
30
Hurricane Ivan Track
Hurricane Ivan track
Description Rangecritical Richardson # p1 ∈ [0.25, 0.7]background viscosity p2 ∈ [10−4, 10−3]background diffusivity p3 ∈ [10−5, 10−4]drag coefficient factor p4 ∈ [0.2, 1.0]
Table: HYCOM uncertain inputs.
20 40 60 80 100 120 140 160 18024
25
26
27
28
29
30
time (hours)
SS
T (
°C)
µµ ± 2σObserved values
Comparing mean & observed SST. Verticallines show when Ivan enters GoM andwhen it is nearest buoy.
• Legendre basis with p = 5• 210 unknown coefficients• Nested sparse Smolyack Ensemble
size 385 (� 64 = 1, 296 Gaussquadrature)
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Variance Analysis
Ti =Variance due to parameterpi
Total variance
0 50 100 1500
0.2
0.4
0.6
0.8
1Sensitivities for SST
time (hours)
Fra
ctio
n of
var
ianc
e
T1
T2
T3
T4
0 50 100 1500
0.2
0.4
0.6
0.8
1Sensitivities for MLD
time (hours) F
ract
ion
of v
aria
nce
T1
T2
T3
T4
Figure: Evolution of the global sensitivity indices T1, . . . ,T4 for SSTand MLD (bottom). The first vertical line indicates the time thehurricane enters the GOM whereas the second indicates a time atwhich the hurricane is close to the buoy.
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
−95 −90 −85 −80
20
22
24
26
28
30
SST Variance Analysis. T3, t=90 hr
0
0.2
0.4
0.6
0.8
1
−95 −90 −85 −80
20
22
24
26
28
30
SST Variance Analysis. T4, t=90 hr
0
0.2
0.4
0.6
0.8
1
−95 −90 −85 −80
20
22
24
26
28
30
SST Variance Analysis. T3, t=147 hr
0
0.2
0.4
0.6
0.8
1
−95 −90 −85 −80
20
22
24
26
28
30
SST Variance Analysis. T4, t=147 hr
0
0.2
0.4
0.6
0.8
1
Figure: T3 (left) and T4 (right) sensitivity contours for SST. Drag dominatesuncertainty during high winds, otherwise it is background diffusivity.
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0 10 20 30 40 50 600
0.5
1
1.5
2
2.5
3
V (m/s)
CD
×1
03
α = 1.1 α = 1 α = 0.4
~τ = ρaCDV ~VCD = CD0 + CD1(Ts − Ta)
CD0 = a0 + a1Ṽ + a2Ṽ 2
CD1 = b0 + b1Ṽ + b2Ṽ 2
Ṽ = max [ Vmin, min (Vmax,V ) ]
CD is drag coefficientV is wind speed at 10 m.CD saturates for V > Vmax
• Blue circles: aircraft observations• red: wind tunnel• green: drop sondes• magenta: HYCOM fit to COARE 2.5,• Problem: Vmax and CmaxD are not well-known and does CD
decrease for V > Vmax as drop sondes suggest?
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Inverse Modeling Problem
• Perturb CD by introducing 3 control variables (α,Vmax,m)
CD′ = αCD for V < Vmax (3)
CD′ = α[CD + m(V − Vmax)] for V > Vmax (4)
• multiplicative factor 0.4 ≤ α ≤ 1.1• vary Vmax between 20 and 35 m/s• m is a linear slope modeling decrease for V > Vmax with−3.8× 10−5 ≤ m ≤ 0
• Use ITOP data to learn about likely distribution of α, Vmaxand m.
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Bayes Theorem: p(θ |T ) ∝ p(T |θ) p(θ)• Likelihood: � = T −M is normally distributed
p(T |θ) =N∏
i=1
1√2πσ2
exp(−(Ti −Mi)2
2σ2
)(5)
• σ2 unknown, treated as hyper-parameter. Assume aJeffreys prior
p(σ2) =
{1σ2
for σ2 > 0,0 otherwise.
(6)
• Uninformed priors for α, Vmax and m:
p({α,Vmax,m}) =
{1
bi−ai for ai ≤ {α,Vmax,m} ≤ bi ,0 otherwise,
(7)
where [ai ,bi ] denote the parameter ranges.
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Bayes Theorem: p(θ |T ) ∝ p(T |θ) p(θ)• Likelihood: � = T −M is normally distributed
p(T |θ) =N∏
i=1
1√2πσ2
exp(−(Ti −Mi)2
2σ2
)(5)
• σ2 unknown, treated as hyper-parameter. Assume aJeffreys prior
p(σ2) =
{1σ2
for σ2 > 0,0 otherwise.
(6)
• Uninformed priors for α, Vmax and m:
p({α,Vmax,m}) =
{1
bi−ai for ai ≤ {α,Vmax,m} ≤ bi ,0 otherwise,
(7)
where [ai ,bi ] denote the parameter ranges.
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Bayes Theorem: p(θ |T ) ∝ p(T |θ) p(θ)• Likelihood: � = T −M is normally distributed
p(T |θ) =N∏
i=1
1√2πσ2
exp(−(Ti −Mi)2
2σ2
)(5)
• σ2 unknown, treated as hyper-parameter. Assume aJeffreys prior
p(σ2) =
{1σ2
for σ2 > 0,0 otherwise.
(6)
• Uninformed priors for α, Vmax and m:
p({α,Vmax,m}) =
{1
bi−ai for ai ≤ {α,Vmax,m} ≤ bi ,0 otherwise,
(7)
where [ai ,bi ] denote the parameter ranges.
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Final Form of Bayes theorem
p({α,Vmax,m}, σ2|T ) ∝
[N∏
i=1
1√2πσ2
exp(−(Ti −Mi)2
2σ2
)]p(σ2) p(α) p(Vmax) p(m)
• Build full posterior with Markov Chain Monte Carlo (MCMC)MCMC requires O(105) estimates of Mi : prohibitive
• Solve for center and spread of posteriorminimization problem requiring access to cost functiongradient and Hessian: Needs an adjoint model
• Rely on Polynomial Chaos expansions to replace HYCOMby a polynomial series that could be either summed forMCMC or differentiated for the gradients.
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Final Form of Bayes theorem
p({α,Vmax,m}, σ2|T ) ∝
[N∏
i=1
1√2πσ2
exp(−(Ti −Mi)2
2σ2
)]p(σ2) p(α) p(Vmax) p(m)
• Build full posterior with Markov Chain Monte Carlo (MCMC)MCMC requires O(105) estimates of Mi : prohibitive
• Solve for center and spread of posteriorminimization problem requiring access to cost functiongradient and Hessian: Needs an adjoint model
• Rely on Polynomial Chaos expansions to replace HYCOMby a polynomial series that could be either summed forMCMC or differentiated for the gradients.
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Final Form of Bayes theorem
p({α,Vmax,m}, σ2|T ) ∝
[N∏
i=1
1√2πσ2
exp(−(Ti −Mi)2
2σ2
)]p(σ2) p(α) p(Vmax) p(m)
• Build full posterior with Markov Chain Monte Carlo (MCMC)MCMC requires O(105) estimates of Mi : prohibitive
• Solve for center and spread of posteriorminimization problem requiring access to cost functiongradient and Hessian: Needs an adjoint model
• Rely on Polynomial Chaos expansions to replace HYCOMby a polynomial series that could be either summed forMCMC or differentiated for the gradients.
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Final Form of Bayes theorem
p({α,Vmax,m}, σ2|T ) ∝
[N∏
i=1
1√2πσ2
exp(−(Ti −Mi)2
2σ2
)]p(σ2) p(α) p(Vmax) p(m)
• Build full posterior with Markov Chain Monte Carlo (MCMC)MCMC requires O(105) estimates of Mi : prohibitive
• Solve for center and spread of posteriorminimization problem requiring access to cost functiongradient and Hessian: Needs an adjoint model
• Rely on Polynomial Chaos expansions to replace HYCOMby a polynomial series that could be either summed forMCMC or differentiated for the gradients.
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Figure: Fanapi’s JTWC track (black curve) and paths of C-130 flights.The yellow circles on the track represent the typhoon center at00:00 UTC. The circles on the flight paths mark the 119 AXBT drops.The 42× 42 km2 analysis box is also shown.
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10 12 14 16 18 20 22 24 26 28 30 32−600
−500
−400
−300
−200−150−100
−500
09/14 20:35 UTC
Temperature (oC)
De
pth
(m
)
Simulated AXBT (29.54)Observed AXBT (29.43)
10 12 14 16 18 20 22 24 26 28 30 32−600
−500
−400
−300
−200−150−100
−500
09/15 22:58 UTC
Temperature (oC)
De
pth
(m
)
Simulated AXBT (28.80)Observed AXBT (28.83)
10 12 14 16 18 20 22 24 26 28 30 32−600
−500
−400
−300
−200−150−100
−500
09/17 21:87 UTC
Temperature (oC)
De
pth
(m
)
Simulated AXBT (28.67)Observed AXBT (28.50)
10 12 14 16 18 20 22 24 26 28 30 32−600
−500
−400
−300
−200−150−100
−500
09/14 22:44 UTC
Temperature (oC)
De
pth
(m
)
Simulated AXBT (29.35)Observed AXBT (29.55)
10 12 14 16 18 20 22 24 26 28 30 32−600
−500
−400
−300
−200−150−100
−500
09/15 23:67 UTC
Temperature (oC)
De
pth
(m
)
Simulated AXBT (29.07)Observed AXBT (28.75)
10 12 14 16 18 20 22 24 26 28 30 32−600
−500
−400
−300
−200−150−100
−500
09/17 23:96 UTC
Temperature (oC)
De
pth
(m
)
Simulated AXBT (28.75)Observed AXBT (28.31)
Figure: Comparison of HYCOM vertical temperature profiles withAXBT observations on Sep 14 (left), 15 (center) and 17 (right).Temperature averages over the first 50 m are shown in the legend.
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
PC Representation Errors
11 12 13 14 15 16 17 18 19 20 2124
25
26
27
28
29
30
Day of September
SS
T (
oC
)
11 12 13 14 15 16 17 18 19 20 2110−5
10−4
10−3
10−2
Day of September
Rela
tive e
rror
Evolution of the area-averaged SST realizations (blue) and ofthe corresponding PC estimates (red). The normalized rmserror (right panel) remains below 0.1% for the duration of thesimulation.
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Longitude
La
titu
de
09/15 at 0 m
120E 125E 130E 135E15N
20N
25N
30N
2
4
6
8
10x 10
−3
Longitude
La
titu
de
09/15 at 50 m
120E 125E 130E 135E15N
20N
25N
30N
2
4
6
8
10x 10
−3
Longitude
La
titu
de
09/15 at 200 m
120E 125E 130E 135E15N
20N
25N
30N
2
4
6
8
10x 10
−3
Longitude
La
titu
de
09/18 at 0 m
120E 125E 130E 135E15N
20N
25N
30N
2
4
6
8
10x 10
−3
Longitude
La
titu
de
09/18 at 50 m
120E 125E 130E 135E15N
20N
25N
30N
2
4
6
8
10x 10
−3
Longitude
La
titu
de
09/18 at 200 m
120E 125E 130E 135E15N
20N
25N
30N
2
4
6
8
10x 10
−3
Figure: Normalized error between realizations and the correspondingPC surrogates at different depths; Top row: 00:00 UTC Sep 15;bottom row: 00:00 UTC Sep 18.
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Depth Profile of Temperature Statistics
Day of September
Z in m
Mean Temperature
11 12 13 14 15 16 17 18 19 20 21−200
−150
−100
−50
0
18
20
22
24
26
28
30
Day of September
Z in m
Standard deviation
11 12 13 14 15 16 17 18 19 20 21−200
−150
−100
−50
0
0
0.2
0.4
0.6
0.8
1
50m-deep mixed layer2◦C cooling after Fanapi arrivesUncertainties confined to top 50 m.
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
SST Response Surface
Vmax
(m/s)
α
Response surface: 09/17
20 25 30 350.4
0.6
0.8
1
29.1
29.15
29.2
29.25
29.3
Vmax
(m/s)
α
Response surface: 09/18
20 25 30 350.4
0.6
0.8
1
28.6
28.8
29
29.2
Vmax
(m/s)
α
Response surface: 09/19
20 25 30 350.4
0.6
0.8
1
25
26
27
28
Figure: SST response surface as function of α and Vmax , with fixedm = 0. Plots are generated on different days, as indicated. SST’sdependence on Vmax decreases after 09/17.
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Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Markov Chain Monte Carlo
0 5 10
x 104
20
25
30
35
Vm
ax
Iteration0 5 10
x 104
−4
−3
−2
−1
0x 10
−5
m
Iteration0 5 10
x 104
0.7
0.8
0.9
1
1.1
α
Iteration
0 5 10
x 104
0.4
0.5
0.6
0.7
0.8
0.9
σ2
Iteration
09/14 − 09/15
0 5 10
x 104
0
0.5
1
1.5
σ2
Iteration
09/15 − 09/16
0 5 10
x 104
0.5
1
1.5
2
2.5
σ2
Iteration
09/17 − 09/18
Figure: Top row: chain samples for Vmax , m and α. Bottom row: chainsamples for σ2 generated for different days, as indicated.
-
10 20 30 400
0.1
0.2
0.3
0.4
Vmax
pd
f
PosteriorPriorMAP
−6 −4 −2 0 2
x 10−5
0
1
2
3x 10
4
m
pd
f
PosteriorPriorMAP
0.9 0.95 1 1.05 1.10
20
40
60
80
α
pd
f
PosteriorPriorMAP
0.4 0.5 0.6 0.7 0.80
5
10
1509/14 − 09/15
σ2
pd
f
0.5 0.6 0.7 0.8 0.90
2
4
6
8
1009/15 − 09/16
σ2
pd
f
0.5 1 1.50
2
4
609/17 − 09/18
σ2
pd
f
1 0.087 3
Figure: Posterior distributions for the drag parameters (top) and thevariance between simulations and observations (bottom). Thenumbers show the Kullback-Liebler divergence quantifying thedistance between 2 prior and posterior pdfs, i.e. the information gain.
-
Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Remarks on posteriors
• Vmax exhibits a well-defined peak at 34 m/s.• Posterior of m resembles prior. Data added little to our
knowledge of m.• α shows a definite peak at 1.03 with a Gaussian
like-distribution.•√σ2 is a measure of the temperature error expected. This
error grows with time from about 0.75◦ to 1◦C.
-
Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Joint posterior PDFs
20 25 30 350.98
1
1.02
1.04
1.06
Vmax
α
5
10
15
20
25
0.98 1 1.02 1.04 1.060.6
0.8
1
1.2
1.4
1.6
σ2
α
100
200
300
400
500
Figure: Left: joint posterior distribution of α (left) and Vmax ; right: jointposterior of α and σ2, generated for Sep 17-Sep 18. Single peaklocated at Vmax = 34 m/s and α = 1.03. The posterior shows a tightestimate for α with little spread around it.
-
Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
0 10 20 30 40 500.5
1
1.5
2
2.5
V (m/s)
CD
×10
3
09/12 − 09/1309/13 − 09/1409/14 − 09/1509/15 − 09/1609/17 − 09/18
Figure: Optimal wind drag coefficient CD using MAP estimate of thethree drag parameters. The symbols refer to AXBT data used in theBayesian inference.
-
Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Variational Form• maximize the posterior density, or equivalently, minimize
the negative of its logarithm
J (α,Vmax ,m, σ21, σ22, σ23, σ24, σ25) =5∑
d=1
[Jd +
(nd2
+ 1)
ln(σ2d )],
(8)where Jd is the misfit cost for day d , the ln(σ2d ) terms comefrom the normalization factors of the Gaussian likelihoodfunctions and from the Jeffreys priors.
• The expression for Jd is:
Jd (α,Vmax ,m, σ2d ) =1
2σ2d
∑i∈Id
[Mi − Ti ]2 , (9)
where Id is the set of nd indices of the observations fromday d .
-
Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Adjoint-Free Gradients
Minimization requires cost function gradients
[∂J∂α
,∂J∂Vmax
,∂J∂m
]=
5∑d=1
1σ2d
∑i∈Id
(Mi − Ti)[∂Mi∂α
,∂Mi∂Vmax
,∂Mi∂m
]Compute them from PC expansion[
∂M∂α
,∂M∂Vmax
,∂M∂m
]=
P∑k=0
M̂k (x , t)[∂ψk∂α
,∂ψk∂Vmax
,∂ψk∂m
].
• ∂ψk∂α easy to compute
• No adjoint model needed• For Hessian just differentiate above again.
-
Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Figure: Posterior probability distributions for (top) drag parametersand (bottom) variances σ2d at selected days using variational methodand MCMC. The vertical lines correspond to the MAP values fromMCMC and optimal parameters found using the variational method.
-
Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Uncertainty in Initial Boundary Conditions
Rely on EOFs to characterize uncertainty and reduce thenumber of stochastic variables. For 2 EOFs mode we have:
u(~x ,0, ξ1, ξ2) = u(~x ,0) + α[√
λ1U1ξ1 +√λ2U2ξ2
](10)
• (λk ,Uk ): are eigenvalues/eigenvectors of covariancematrix obtained from free-run simulation
• u: unperturbed initial condition• u(~x ,0, ξ): Stochastic initial condition input• α: multiplicative factor to control size of “kick”
-
Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Figure: First and Second SSH modes from a 14-day series. The 2modes account for 50% of variance during these 14 days.
• Characterize local uncertainty: get perturbation from short,14-day, simulation.
• Uncertainty dominated by Loop Current (LC) dynamics• Mode 1 seems associated with a frontal eddy
-
Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
PC representation• (ξ1, ξ2) independent and uniformly distributed random variables• PC basis: Legendre polynomials of max degree 6, P = 28• Ensemble of 49 realizations for Hermite quadrature
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
ξ 2
ξ1
Figure: Quadrature/Sample points in ξ1, ξ2 space. Center blackcircle corresponds to unperturbed run, while blue circlescorrespond to largest negative and positive perturbations.
-
Col 1: SSH ofrealization (1,1)with weakestfrontal eddy
Col 2: SSH ofunperturbedrealization (4,4)has mediumstrength frontaleddy
Col 3: SSH ofrealization (7,7)has strogestfrontal eddy andearliest LCseparation
Col 4: Loopcurrent edge inensemble
-
SSH stddev(cm) grows intime withmaximum inLC region
-
PC-error: ‖�‖22 =∑
q [η(~x , t , ξq)− ηPC(~x , t , ξq)]2ωq
SSHPC-errors(cm) grow intime withmaxima in LCregion
On day 60PC-error isabout 38% ofstddev
-
T-sectionalong 25N,stddev growsin time withmaximacoincidingwith FrontalEddy duringdays 20–40.
-
PC-error: ‖�‖22 =∑
q [T (~x , t , ξq)− TPC(~x , t , ξq)]2ωq
T PC-errors(cm) grow intime withmaxima in LCregion
On day 60PC-error isabout 50% ofstddev
-
Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Distribution of SSH PC coefficients
-
Figure: Temperature (left) and Salt (right) profiles for extremerealizations at DWH
-
1 2 3 4
1
2
3
4
ξ1
ξ 2
Multi−Index
−1 0 1−1
0
1
ξ1
ξ 2
Realization Stencil
0 5 10 15 200
5
10
15
20
ξ1 order
ξ 2 o
rder
Polynomial Exactness
Full-Tensor (49)
1 2 3 4
1
2
3
4
ξ1
ξ 2
Multi−Index
−1 0 1−1
0
1
ξ1
ξ 2
Realization Stencil
0 5 10 15 200
5
10
15
20
ξ1 order
ξ 2 o
rder
Polynomial Exactness
Classic Smolyak (17)
1 2 3 4
1
2
3
4
ξ1
ξ 2
Multi−Index
−1 0 1−1
0
1
ξ1
ξ 2
Realization Stencil
0 5 10 15 200
5
10
15
20
ξ1 order
ξ 2 o
rder
Polynomial Exactness
Arbitrary Multi-Index (33)Examples of 2D tensorizations
-
Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Varying polynomial order
20 40 60 80 100 120 140 16010
−6
10−5
10−4
10−3
10−2
Simulation Time (hr)
Rel
ativ
e L2
Err
or
p=(5,5,5,5)p=(5,5,7,7)p=(2,2,5,5)p=(2,2,7,7)
Figure: Relative L2 error between the area-averaged SST and theLatin Hypercube Samples.
Simple Truncation P # of realizationsp = (5, 5, 5, 5) 126 385p = (5, 5, 7, 7) 168 513p = (2, 2, 5, 5) 36 73p = (2, 2, 7, 7) 59 169
-
Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Smolyak Projections
• Apply Smolyak’s algorithm directly to construct the PCEinstead of purely generating the quadrature. Thus, the finalprojection becomes a weighted sum of aliasing-freesub-projections. This is an extension of the Smolyak tensorconstruction from quadrature operators to projectionoperators.
• Smolyak projection allows a refinement approach basedon successive inclusion of any admissible multi-index, F ,of quadrature rules while maintaining the representationfree of internal aliasing.
• A larger number of polynomials can be integrated than ispossible with a classical dimensional truncation /quadrature using the same ensemble, The 513 HYCOMrealizations yields 402 coefficient with Smolyak projectioncompared to 168 using Smolyak quadrature.
-
Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Adaptive Projections
• Rewrite projection as tensor products of projectiondifferences: (
∆k1 ⊗ ...⊗∆kd)
U,
• The L2 norm of this difference can be readily used todefine an error indicator for multi-index k,
�(k) = ||(∆k1 ⊗ ...⊗∆kd
)U||
The indicator represents the variance surplus due to the ksub-projection.
• The surplus is computed for each k ∈ F and thesub-projection with the highest �(k) is selected forsubsequent refinement, which generally consists ofinclusion of admissible forward neighbors.
-
0 20 40 60 80 100 120 140 16010
−6
10−5
10−4
10−3
Simulation Time (hr)
Rel
ativ
e L2
Err
or
Adaptive: t=60hr, Iteration 5Full Database: Pseudo−SpectralFull Database: Direct
Figure: Relative L2 difference between the PCE of the averaged SSTand the LHS sample. Plotted are curves generated with (i) theadaptive Smolyak projection adapted at t = 60 hr, (ii) the Smolyakprojection with the full database, and (iii) Smolyak classicalquadrature using the full database. For the adapted solution, therefinement is stopped after iteration 5, leading to 69 realizations anda PCE with 59 polynomials. The full 513 database curves have 402polynomials for the pseudo-spectral construction and 168polynomials for the Smolyak quadrature.
-
Uncertainty Quantification (UQ) What is Polynomial Chaos Forward Propagation and Analysis Bayesian Inference UQ-Initial Conditions
Publications• A. Srinivasan, J. Helgers, C. B. Paris, M. LeHenaff, H. Kang, V. Kourafalou,
M. Iskandarani, W. C. Thacker, J. P. Zysman, N. F. Tsinoremas, and O. M. Knio.Many task computing for modeling the fate of oil discharged from the deep waterhorizon well blowout. In Many-Task Computing on Grids and Supercomputers(MTAGS), 2010 IEEE Workshop on, pages 1–7, November, 2010. IEEE.
• W. C. Thacker, A. Srinivasan, M. Iskandarani, O. M. Knio, and M. Le Henaff.Propagating oceanographic uncertainties using the method of polynomial chaosexpansion. Ocean Modelling, 43–44, pp 52–63, 2012.
• A. Alexanderian, J. Winokur, I. Sraj, M. Iskandarani, A. Srinivasan,W. C. Thacker, and O. M. Knio, Global sensitivity analysis in an ocean generalcirculation model: a sparse spectral projection approach, ComputationalGeosciences, 16, 757–778, 2012.
• I. Sraj, M. Iskandarani, A. Srinivasan, W. C. Thacker, A. Alexanderian, C. Leeand S. S. Chen and O. M. Knio, Bayesian Inference of Drag Parameters UsingAXBT Data from Typhoon Fanapi, Monthly Weather Review, 141, no 7, pp2347–2366, 2013.
• J. Winokur, P. Conrad, I. Sraj, O. M. Knio, A. Srinivasan, W. Carlisle Thacker, Y.Marzouk, and M. Iskandarani, A Priori Testing of Sparse Adaptive PolynomialChaos Expansions Using an Ocean General Circulation Model Database,Computational Geosciences, 2013.
• I. Sraj, M. Iskandarani, A. Srinivasan, W. C. Thacker and O. M. Knio, ComputingModel Gradients from a Polynomial Chaos based Surrogate for an InverseModeling Problem, Monthly Weather Review, in review.
http://dx.doi.org/10.1109/MTAGS.2010.5699424http://dx.doi.org/10.1109/MTAGS.2010.5699424http://dx.doi.org/10.1016/j.ocemod.2011.11.011http://dx.doi.org/10.1016/j.ocemod.2011.11.011http://dx.doi.org/10.1007/s10596-012-9286-2http://dx.doi.org/10.1007/s10596-012-9286-2http://dx.doi.org/10.1175/MWR-D-12-00228.1http://dx.doi.org/10.1175/MWR-D-12-00228.1http://dx.doi.org/10.1007/s10596-013-9361-3http://dx.doi.org/10.1007/s10596-013-9361-3
Uncertainty Quantification (UQ)What is Polynomial ChaosForward Propagation and AnalysisBayesian InferenceUQ-Initial Conditions