Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Overview of Polynomial Chaos Methodsfor Uncertainty Quantification

with Application to HYCOM

Mohamed Iskandarani1 Ashwanth Srinivasan1

Carlisle Thacker12 Omar Knio3

1University of Miami

2Atlantic Oceanographic and Meteorological Laboratory

3Johns Hopkins University

February 2011

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Outline

1 Background.What is polynomial chaos?How does polynomial chaos work?

2 Uncertain inflow through Yucatan Straits.Setup.Results.

3 Omar: advanced techniques.

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Outline

1 Background.What is polynomial chaos?How does polynomial chaos work?

2 Uncertain inflow through Yucatan Straits.Setup.Results.

3 Omar: advanced techniques.

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

What is polynomial chaos?

• Idea of polynomial chaos originated with NorbertWiener in 1938 — before computers.

• It is being used by engineers to assess howuncertainties in a model’s inputs manifest in itsoutputs.

• It can be much more efficient than Monte Carlomethods.

• Can it be useful to oceanographers?

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Why “polynomial”? Why “chaos”?

• “Chaos” simply refers to uncertainty. Nothing to dowith strange attractors?

• Want to compute how uncertainties of a dynamicalsystem’s inputs manifest in its outputs.

• “Polynomial” refers to use of polynomial expansionsto propagate uncertainties.

• Idea is to exploit orthogonality of the polynomials.

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

HYCOM: uncertain inputs.

• Initial conditions.• Boundary conditions.• Forcing.• Parameters.

• Polynomial Chaos can handle only a limited numberof uncertain inputs.

• But it focuses on all likely values of those fewuncertain inputs.

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

HYCOM: uncertain outputs.

• Every field at every time.• Value of a particular field at a particular point and a

particular time.• Derived quantities,

e.g. maximum of the Meridional overturning streamfunction.

• Polynomial Chaos allows focus to be on points ofinterest.

• Not necessary to explore all uncertaintiessimultaneously.

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Simplest case:one uncertain input, one output of interest.

• Call the uncertain input ξ and the output φ.• Uncertainty of ξ is specified via its pdf ρ(ξ).• Want pdf of φ or at least information about how it

varies as ξ varies.• Basic Idea: Express output as a polynomial series.

φ(ξ) = φ0 + φ1P1(ξ) + φ2P2(ξ) + . . .

• Orthogonal polynomials Pk are related to pdf ρ.∫Pj(ξ)Pk (ξ)ρ(ξ)dξ = δj,k

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Simplest case, continued:one uncertain input, one output of interest

φ(ξ) = φ0 + φ1P1(ξ) + φ2P2(ξ) + . . .∫Pj(ξ)Pk (ξ)ρ(ξ)dξ = δj,k

• Is the series guaranteed to converge?• In practice, it must be truncated.• How to compute the coefficients φ0, φ1, φ2, . . .?

φk =1

Nk

∫φ(ξ)Pk (ξ)ρ(ξ)dξ

Nk =∫

P2k (ξ)ρ(ξ)dξ

• Use Gaussian quadrature to evaluate the integrals.

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Simplest case, continued:one uncertain input, one output of interest

• Coefficients:

φk =1

Nk

∫φ(ξ)Pk (ξ)ρ(ξ)dξ

• Gaussian Quadrature:∫φ(ξ)Pk (ξ)ρ(ξ)dξ ≈

∑p

φ(ξp)Pk (ξp)wp

• Quadrature points: ξp• Quadrature weights: wp• Computing outputs φ(ξp) for inputs at quadrature

points ξp requires multiple model runs.• How many quadrature points (runs) are needed?

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Several outputs of interest,one uncertain input.

• Two outputs:

φ(ξ) = φ0 + φ1P1(ξ) + φ2P2(ξ) + . . .

ψ(ξ) = ψ0 + ψ1P1(ξ) + ψ2P2(ξ) + . . .

• Two or more outputs require no more runs than doesone output.

• Just save values for all outputs of interest, φ, ψ, . . ..• More coefficients, so more quadrature integrals are

needed.• Quadrature integrals are computationally cheap.• Can examine uncertainty of an entire field.

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

How to use expansion coefficients?

• mean:〈φ〉 =

∫φ(ξ)ρ(ξ)dξ = φ0

• variance:

〈(φ− φ0〉)2 =kmax∑k=1

φ2k

• covariance:

〈(φ− φ0〉)(ψ − ψ0)〉 =kmax∑k=1

φkψk

• Generate a cheap ensemble.

φ(ξ) = φ0 + φ1P1(ξ) + φ2P2(ξ) + . . .

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Two uncertain inputs, one output of interest

• Call the uncertain inputs ξ and χ.• Uncertainties of are specified via joint pdf ρ(ξ, χ).• Now have polynomial series in two variables:

φ(ξ, χ) = φ0 + φ1P1(ξ, χ) + φ2P2(ξ, χ) + . . .

• If uncertain inputs are independent, pdf factors:

ρ(ξ, χ) = ρξ(ξ)ρχ(χ)

• Then 2D quadrature reduces to two 1D quadratures.• Number of quadrature points (runs) is squared.• Curse of dimensionality.• Sparse cubature might provide economy when

exploring consequences of several uncertain inputs.

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Outline

1 Background.What is polynomial chaos?How does polynomial chaos work?

2 Uncertain inflow through Yucatan Straits.Setup.Results.

3 Omar: advanced techniques.

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Problem: How do uncertainties of theYucatan inflow manifest within the Gulf ofMexico?

• Need to quantify inflow uncertainties.• Inflow is characterized by several 2D time-varying

fields.• Computational cost increases dramatically with

number of uncertain parameters.• How to characterize uncertainties of inflow with only

a few parameters?

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

How to characterize uncertainties of inflowwith only a few independent parameters?

• Use multivariate EOFs to characterize 2D spatialpatterns of inflow uncertainty.

• Use corresponding principal components tocharacterize their temporal variability.

• Each mode’s amplitude is assumed to have aGaussian pdf.

• Hermite polynomials — Gauss-Hermite quadrature.• Quadrature points dictate the required HYCOM runs.• Each run is the sum of a "favorite" inflow and its

particular EOF contributions.

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Meridional velocity component of EOF.

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Eigenvalue spectrum.

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Contours of 17 cm SSH from quadratureensemble of 17 cm runs.

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

17 (cm) SSH Contours − Day −−090

Latitude

Long

itude

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

17 (cm) SSH Contours − Day −−150

Latitude

Long

itude

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

17 (cm) SSH Contours − Day −−240

Latitude

Long

itude

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

17 (cm) SSH Contours − Day −−300

Latitude

Long

itude

Figure 1. 17 (cm) SSH contours from all 36 realizations at 90, 150, 240 and 300 days.The contours are coloured based on the realization number. Realization numbers increasefrom cool to hot colors.

1

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Mean sea-surface height.

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Mean SSH(m) − Day090

Latitude

Long

itude

−0.4 −0.2 0 0.2 0.4 0.6

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Mean SSH(m) − Day150

Latitude

Long

itude

−0.4 −0.2 0 0.2 0.4 0.6

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Mean SSH(m) − Day240

Latitude

Long

itude

−0.4 −0.2 0 0.2 0.4 0.6

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Mean SSH(m) − Day300

Latitude

Long

itude

−0.4 −0.2 0 0.2 0.4 0.6

Figure 1. Mean Sea Surface Height (SSH) at 90, 150, 240 and 300 days obtained from the5th order PC expansion.

1

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Standard deviation of sea-surface height.

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −090

Latitude

Long

itude

0 0.2 0.4 0.6 0.8 1

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −150

Latitude

Long

itude

0 0.2 0.4 0.6 0.8 1

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −240

Latitude

Long

itude

0 0.2 0.4 0.6 0.8 1

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −300

Latitude

Long

itude

0 0.2 0.4 0.6 0.8 1

Figure 1. Std dev. of Sea Surface Height (SSH) at 90, 150, 240 and 300 days obtainedwith the 5th order PC expansion

1

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Convergence of series for SSH standarddeviation for day 90.

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −090

Latitude

Long

itude

0.05 0.1 0.15 0.2

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −090

Latitude

Long

itude

0.05 0.1 0.15 0.2

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −090

Latitude

Long

itude

0.05 0.1 0.15 0.2

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −090

Latitude

Long

itude

0.05 0.1 0.15 0.2

Figure 1. Std dev. of Sea Surface Height (SSH) on day 90, obtained with 5th (upper left),4th (upper right), 3rd (lower left) and 2nd (lower right) order PC expansions. At 90 daysthe spread among the realizations is still smooth and lower order approximations are ableto capture many of the features as the 5th order approximation, indicating convergence1

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Convergence of series for SSH standarddeviation for day 300.

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −300

Latitude

Long

itude

0.05 0.1 0.15 0.2 0.25 0.3

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −300

Latitude

Long

itude

0.05 0.1 0.15 0.2

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −300

Latitude

Long

itude

0.05 0.1 0.15 0.2

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Std dev. SSH(m) − Day −300

Latitude

Long

itude

0.02 0.04 0.06 0.08 0.1 0.12 0.14

Figure 1. Std dev. of Sea Surface Height (SSH) on day 300, obtained with 5th (upperleft), 4th (upper right), 3rd (lower left) and 2nd (lower right) order PC expansions. At 300days the spread among the realizations is much more pronounced with many realizationsshedding an eddy. In this case the convergence is much slower than at 90 days.1

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Covariance of SSH with SSH at one point fordays 90, 150, 240, and 300.

*

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Covariance SSH−SSH − Day −090

Latitude

Long

itude

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

*

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Covariance SSH−SSH − Day −150

Latitude

Long

itude

−0.2 −0.1 0 0.1 0.2 0.3

*

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Covariance SSH−SSH − Day −240

Latitude

Long

itude

−2 −1 0 1 2

*

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Covariance SSH−SSH − Day −300

Latitude

Long

itude

−1 0 1 2 3

Figure 1. Covariance between target point located at 86W/24N (indicated by the whitestar) and all other points at 90, 150, 240 and 300 days obtained from the 5th order PCexpansion.

1

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Covariance of u,v-velocity with SSH at onepoint for day 90.

*

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Covariance ssh−vvel − Day −300

Latitude

Long

itude

−0.05 0 0.05 0.1

*

96oW 92oW 88oW 84oW 80oW 18oN

21oN

24oN

27oN

30oN

Covariance ssh−uvel − Day −300

Latitude

Long

itude

−0.06 −0.04 −0.02 0 0.02 0.04 0.06

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Kernel density estimates for mixed-layerdepth at day 90 from artificial ensemble.

55 56 57 58 59 60 61 62 630

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Mixed Layer Depth [m]

Den

sity

PDF of Mixed Layer depth[m] at 86W/24N −− Day 90

39 40 41 42 43 44 45 460

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mixed Layer Depth [m]

Den

sity

PDF of Mixed Layer depth[m] at 86W/24N −− Day 150

14 14.5 15 15.5 16 16.5 17 17.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Mixed Layer Depth [m]

Den

sity

PDF of Mixed Layer depth[m] at 86W/24N −− Day 240

42 44 46 48 50 52 54 56 580

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Mixed Layer Depth [m]

Den

sity

PDF of Mixed Layer depth[m] at 86W/24N −− Day 300

Polynomial Chaos

M.Iskandarani,A.Srinivasan,W.C. Thacker,

O. Knio

Background.What?

How?

Inflow exampleSetup.

Results

Omar

Outline

1 Background.What is polynomial chaos?How does polynomial chaos work?

2 Uncertain inflow through Yucatan Straits.Setup.Results.

3 Omar: advanced techniques.

Background.What is polynomial chaos?How does polynomial chaos work?

Uncertain inflow through Yucatan Straits.Setup.Results.

Omar: advanced techniques.