Tutorial on Uncertainty Quantification with Emphasis on ...carthe.org/tutorials_pdf/2013_10/Iskandarani_Mohamed_UQ_tutorial.pdfEmphasis on Polynomial Chaos Methods Mohamed Iskandarani

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


Outline

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


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


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).


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?


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


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)


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


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


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 (−∞,∞)


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,∞)


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]


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]


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)


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


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


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


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


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)


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.


−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


0

0.2

0.4

0.6

0.8

1

−95 −90 −85 −80

20

22

24

26

28

30


0

0.2

0.4

0.6

0.8

1

−95 −90 −85 −80

20

22

24

26

28

30


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.

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?


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.


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.


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.

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.

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

)


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

)


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

)


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

)


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

)


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.


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.

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.


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.


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


20 25 30 350.4

0.6

0.8

1

28.6

28.8

29

29.2

Vmax

(m/s)

α


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.


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.


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.


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.


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.


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 .


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.


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 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”


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


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


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


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


Arbitrary Multi-Index (33)Examples of 2D tensorizations


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


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.


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.


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

Tutorial on Uncertainty Quantification with Emphasis on ...carthe.org/tutorials_pdf/2013_10/Iskandarani_Mohamed_UQ_tutorial.pdfEmphasis on Polynomial Chaos Methods Mohamed Iskandarani

Documents