overview of polynomial chaos methods for uncertainty...
TRANSCRIPT
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 =
∫P2
k (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 quadraturepoints 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.