Sparse Polynomial Chaos Surrogate for ACME Land Model via ... · Polynomial Chaos surrogate for f(...
Transcript of Sparse Polynomial Chaos Surrogate for ACME Land Model via ... · Polynomial Chaos surrogate for f(...
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Sparse Polynomial Chaos Surrogatefor ACME Land Model
via Iterative Bayesian Compressive Sensing
K. Sargsyan1, D. Ricciuto2, P. Thornton2
C. Safta1, B.Debusschere1, H. Najm1
1Sandia National LaboratoriesLivermore, CA
2Oak Ridge National LaboratoryOak Ridge, TN
Sponsored by DOE, Biological and Environmental Research,under Accelerated Climate Modeling for Energy (ACME).
Sandia National Laboratories is a multi-program laboratory operated by Sandia Corporation,a wholly owned subsidiary of Lockheed Martin Corporation,
for the U.S. Department of Energy’s National Nuclear Security Administrationunder contract DE-AC04-94AL85000.
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 1
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OUTLINE
• Surrogates needed for complex models
• Polynomial Chaos (PC) surrogates do well with uncertain inputs
• Bayesian regression provide results with uncertainty certificate
• Compressive sensing ideas deal with high-dimensionality
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 2
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Application of Interest: ACME Land Model
http://www.cesm.ucar.edu/models/clm/
A single-site, 1000-yr simulation takes ∼ 10 hrs on 1 CPUInvolves ∼ 70 input parameters; some dependentNon-smooth input-output relationship
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 3
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Surrogate construction: scope and challenges
Construct surrogate for a complex model f (λ) to enable
• Global sensitivity analysis• Optimization• Forward uncertainty propagation• Input parameter calibration• · · ·
• Computationally expensive model simulations, data sparsity• Need to build accurate surrogates with as few
training runs as possible
• High-dimensional input space• Too many samples needed to cover the space• Too many terms in the polynomial expansion
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 4
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Surrogate construction: scope and challenges
Construct surrogate for a complex model f (λ) to enable
• Global sensitivity analysis• Optimization• Forward uncertainty propagation• Input parameter calibration• · · ·
• Computationally expensive model simulations, data sparsity• Need to build accurate surrogates with as few
training runs as possible
• High-dimensional input space• Too many samples needed to cover the space• Too many terms in the polynomial expansion
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 4
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Surrogate construction: scope and challenges
Construct surrogate for a complex model f (λ) to enable
• Global sensitivity analysis• Optimization• Forward uncertainty propagation• Input parameter calibration• · · ·
• Computationally expensive model simulations, data sparsity• Need to build accurate surrogates with as few
training runs as possible
• High-dimensional input space• Too many samples needed to cover the space• Too many terms in the polynomial expansion
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 4
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Polynomial Chaos surrogate for f (λ)
• Scale the input parameters λi ∈ [ai, bi]
λi =ai + bi
2+
bi − ai2
xi
• Forward function f (·), output u
u(x) = f (λ(x)) ≈K−1∑k=0
ckΨk(x) ≡ g(x)
• Global sensitivity information for free- Sobol indices, variance-based decomposition.
• Bayesian inference useful for finding ck:
P(c|u) ∝ P(u|c)P(c)
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 5
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Polynomial Chaos surrogate for f (λ)
• Scale the input parameters λi ∈ [ai, bi]
λi =ai + bi
2+
bi − ai2
xi
• Forward function f (·), output u
u(x) = f (λ(x)) ≈K−1∑k=0
ckΨk(x) ≡ g(x)
• Global sensitivity information for free- Sobol indices, variance-based decomposition.
• Bayesian inference useful for finding ck:
P(c|u) ∝ P(u|c)P(c)
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 5
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Polynomial Chaos surrogate for f (λ)
• Scale the input parameters λi ∈ [ai, bi]
λi =ai + bi
2+
bi − ai2
xi
• Forward function f (·), output u
u(x) = f (λ(x)) ≈K−1∑k=0
ckΨk(x) ≡ g(x)
• Global sensitivity information for free- Sobol indices, variance-based decomposition.
• Bayesian inference useful for finding ck:
P(c|u) ∝ P(u|c)P(c)
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 5
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Polynomial Chaos surrogate for f (λ)
• Scale the input parameters λi ∈ [ai, bi]
λi =ai + bi
2+
bi − ai2
xi
• Forward function f (·), output u
u(x) = f (λ(x)) ≈K−1∑k=0
ckΨk(x) ≡ g(x)
• Global sensitivity information for free- Sobol indices, variance-based decomposition.
• Bayesian inference useful for finding ck:
P(c|u) ∝ P(u|c)P(c)
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 5
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Bayesian inference of PC surrogate: high-d, low-data regime
y = u(x) ≈∑K−1
k=0 ckΨk(x)
Ψk(x1, x2, ..., xd) = ψk1(x1)ψk2(x2) · · ·ψkd (xd)
• Issues:
• how to properly choosethe basis set?
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Dim 1
Dim
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• need to work in underdetermined regimeN < K: fewer data than bases (d.o.f.)
• Discover the underlying low-d structure in the model
• get help from the machine learning community
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 6
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Bayesian inference of PC surrogate: high-d, low-data regime
y = u(x) ≈∑K−1
k=0 ckΨk(x)
Ψk(x1, x2, ..., xd) = ψk1(x1)ψk2(x2) · · ·ψkd (xd)
• Issues:
• how to properly choosethe basis set?
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Dim 1
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• need to work in underdetermined regimeN < K: fewer data than bases (d.o.f.)
• Discover the underlying low-d structure in the model
• get help from the machine learning community
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 6
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Bayesian inference of PC surrogate: high-d, low-data regime
y = u(x) ≈∑K−1
k=0 ckΨk(x)
Ψk(x1, x2, ..., xd) = ψk1(x1)ψk2(x2) · · ·ψkd (xd)
• Issues:
• how to properly choosethe basis set?
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
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9
10
Dim 1
Dim
2
• need to work in underdetermined regimeN < K: fewer data than bases (d.o.f.)
• Discover the underlying low-d structure in the model
• get help from the machine learning community
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 6
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Bayesian inference of PC surrogate: high-d, low-data regime
y = u(x) ≈∑K−1
k=0 ckΨk(x)
Ψk(x1, x2, ..., xd) = ψk1(x1)ψk2(x2) · · ·ψkd (xd)
• Issues:
• how to properly choosethe basis set?
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
Dim 1
Dim
2
• need to work in underdetermined regimeN < K: fewer data than bases (d.o.f.)
• Discover the underlying low-d structure in the model
• get help from the machine learning community
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 6
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Bayesian inference of PC surrogate: high-d, low-data regime
y = u(x) ≈∑K−1
k=0 ckΨk(x)
Ψk(x1, x2, ..., xd) = ψk1(x1)ψk2(x2) · · ·ψkd (xd)
• Issues:
• how to properly choosethe basis set?
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
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10
Dim 1
Dim
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• need to work in underdetermined regimeN < K: fewer data than bases (d.o.f.)
• Discover the underlying low-d structure in the model
• get help from the machine learning community
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 6
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In a different language....
• N training data points (xn, un) and K basis terms Ψk(·)• Projection matrix PN×K with Pnk = Ψk(xn)• Find regression weights c = (c0, . . . , cK−1) so that
u ≈ Pc or un ≈∑
k ckΨk(xn)
• The number of polynomial basis terms grows fast; a p-th order,d-dimensional basis has a total of K = (p + d)!/(p!d!) terms.
• For limited data and large basis set (N < K) this is a sparse signalrecovery problem⇒ need some regularization/constraints.
• Least-squares argminc {||u− Pc||2}
• The ‘sparsest’ argminc {||u− Pc||2 + α||c||0}
• Compressive sensing argminc {||u− Pc||2 + α||c||1}
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 7
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In a different language....
• N training data points (xn, un) and K basis terms Ψk(·)• Projection matrix PN×K with Pnk = Ψk(xn)• Find regression weights c = (c0, . . . , cK−1) so that
u ≈ Pc or un ≈∑
k ckΨk(xn)
• The number of polynomial basis terms grows fast; a p-th order,d-dimensional basis has a total of K = (p + d)!/(p!d!) terms.
• For limited data and large basis set (N < K) this is a sparse signalrecovery problem⇒ need some regularization/constraints.
• Least-squares argminc {||u− Pc||2}
• The ‘sparsest’ argminc {||u− Pc||2 + α||c||0}
• Compressive sensing argminc {||u− Pc||2 + α||c||1}Bayesian Likelihood Prior
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 7
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BCS removes unnecessary basis terms
f (x, y) = cos(x + 4y) f (x, y) = cos(x2 + 4y)
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Order (dim 2)
Ord
er
(dim
1)
−18
−16
−14
−12
−10
−8
−6
−4
−2
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Order (dim 2)
Ord
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(dim
1)
−18
−16
−14
−12
−10
−8
−6
−4
−2
The square (i, j) represents the (log) spectral coefficientfor the basis term ψi(x)ψj(y).
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 8
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Iterative Bayesian Compressive Sensing (iBCS)
Iterative BCS: We implement an iterative procedure that allowsincreasing the order for the relevant basis terms while maintaining thedimensionality reduction [Sargsyan et al. 2014], [Jakeman et al. 2015].Combine basis growth and reweighting!
Initial Basis
Iterations
WeightedBCS
Model data
Sparse Basis Final Basis
BasisGrowth
ReweightingNew Basis
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 9
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Basis set growth: simple anisotropic function
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 10
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Basis set growth: ... added outlier term
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 11
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Application of Interest: ACME Land Model
http://www.cesm.ucar.edu/models/clm/
A single-site, 1000-yr simulation takes ∼ 10 hrs on 1 CPUInvolves ∼ 70 input parameters; some dependentNon-smooth input-output relationship
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 12
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FLUXNET experiment
60°N
30°N
0°
30°S
60°S
60°N
30°N
0°
30°S
60°S
180° 180°120°W 60°W 0° 60°E 120°E180° 180°
FLUXNET sites
96 FLUXNET sites covering major biomes and plant functional types
Varying 68 input parameters over given ranges; 5 steady state outputs
Ensemble of 3000 runs on Titan, DoE Leadership Computing Facility atOak Ridge National Lab
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 13
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Sparse PC surrogate and uncertainty decompositionfor the ACME Land Model
Main effect sensitivities : rank input parametersJoint sensitivities : most influential input couplingsAbout 200 polynomial basis terms in the 68-dimensional spaceSparse PC will further be used for• sampling in a reduced space• parameter calibration against experimental data
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33 - bp
31
31 - br_mr
19
19 - froot_leaf
2
2 - mp
16
16 - frootcn3232 - q10_mr
9 - slatop
Site # 31
GPP
33 - bp
9
9 - slatop
19
19 - froot_leaf
31
31 - br_mr
25
25 - fr_fcel2020 - flivewd
Site # 31
TLAI
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 14
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Sparse PC surrogate and uncertainty decompositionfor the ACME Land Model
Main effect sensitivities : rank input parametersJoint sensitivities : most influential input couplingsAbout 200 polynomial basis terms in the 68-dimensional spaceSparse PC will further be used for• sampling in a reduced space• parameter calibration against experimental data
33 - bp
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31 - br_mr
2
2 - mp
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19 - froot_leaf
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16 - frootcn99 - slatop
Site # 31
TOTVEGC
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3131 - br_mr
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35 - cn_s3
3
3 - bp
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32 - q10_mr
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34 - cn_s219
19 - froot_leaf
9 - slatop
Site # 31
TOTSOMC
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 14
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Sparse PC surrogate and uncertainty decompositionfor the ACME Land Model
Main effect sensitivities : rank input parametersJoint sensitivities : most influential input couplingsAbout 200 polynomial basis terms in the 68-dimensional spaceSparse PC will further be used for• sampling in a reduced space• parameter calibration against experimental data
MP
BP
ROOTB_PAR
SLATOP
FLNR
FROOTCN
FROOT_LEAF
LEAF_LONG
BDNR
BR_M
R
Q10_M
R
CN_S2
CN_S3
RF_S2S3
RF_S3S4
R_M
ORT
GPP
TLAI
TOTVEGC
EFLX_LH_TOT
TOTSOMC
10−3
10−2
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 14
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Sparse PC surrogate and uncertainty decompositionfor the ACME Land Model
Main effect sensitivities : rank input parametersJoint sensitivities : most influential input couplingsAbout 200 polynomial basis terms in the 68-dimensional spaceSparse PC will further be used for• sampling in a reduced space• parameter calibration against experimental data
GPPgross primaryproductivity
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Sensitivity
soilpsi_offmpbp
slatopcrit_onset_swileafcn
flnrfrootcnfroot_leaf
fstor2tranbdnr
br_mrq10_mr
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 14
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Sparse PC surrogate and uncertainty decompositionfor the ACME Land Model
Main effect sensitivities : rank input parametersJoint sensitivities : most influential input couplingsAbout 200 polynomial basis terms in the 68-dimensional spaceSparse PC will further be used for• sampling in a reduced space• parameter calibration against experimental data
TLAIleaf area index
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Sensitivity
soilpsi_offbp
slatopleafcn
flnrfrootcn
froot_leafleaf_long
bdnrbr_mr
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 14
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Sparse PC surrogate and uncertainty decompositionfor the ACME Land Model
Main effect sensitivities : rank input parametersJoint sensitivities : most influential input couplingsAbout 200 polynomial basis terms in the 68-dimensional spaceSparse PC will further be used for• sampling in a reduced space• parameter calibration against experimental data
TOTVEGCvegetation carbon
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Sensitivity
mpbp
leafcnflnr
frootcnfroot_leaf
r_mortfstor2tran
bdnrbr_mr
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 14
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Sparse PC surrogate and uncertainty decompositionfor the ACME Land Model
Main effect sensitivities : rank input parametersJoint sensitivities : most influential input couplingsAbout 200 polynomial basis terms in the 68-dimensional spaceSparse PC will further be used for• sampling in a reduced space• parameter calibration against experimental data
EFLX LH TOTlatent heat flux
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Sensitivity
soilpsi_offcn_s3
rootb_parslatop
leafcnflnr
frootcnfroot_leaf
br_mrq10_mr
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 14
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Sparse PC surrogate and uncertainty decompositionfor the ACME Land Model
Main effect sensitivities : rank input parametersJoint sensitivities : most influential input couplingsAbout 200 polynomial basis terms in the 68-dimensional spaceSparse PC will further be used for• sampling in a reduced space• parameter calibration against experimental data
TOTSOMCsoil organicmatter carbon
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Sensitivity
cn_s3slatop
leafcnflnr
bpfrootcn
froot_leaffstor2tran
br_mrq10_mr
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 14
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SummarySurrogate models are necessary for complex models• Replace the full model for both forward and inverse UQ
Uncertain inputs• Polynomial Chaos surrogates well-suited
Limited training dataset• Bayesian methods handle limited information well
Curse of dimensionality• The hope is that not too many dimensions matter• Compressive sensing (CS) ideas ported from machine learning• We implemented iteratively reweighting Bayesian CS algorithm that
reduces dimensionality and increases order on-the-fly.
Open issues• Computational design. What is the best sampling strategy?• Overfitting still present. Cross-validation techniques help.
Software: employed SNL-CA UQ library UQTk (www.sandia.gov/uqtoolkit)
K. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 15
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Literature
• M. Tipping, “Sparse Bayesian learning and the relevance vector machine”, JMachine Learning Research, 1, pp. 211-244, 2001.
• S. Ji, Y. Xue and L. Carin, “Bayesian compressive sensing”, IEEE Trans. SignalProc., 56:6, 2008.
• S. Babacan, R. Molina and A. Katsaggelos, “Bayesian compressive sensing usingLaplace priors”, IEEE Trans. Image Proc., 19:1, 2010.
• E. J. Candes, M. Wakin and S. Boyd. “Enhancing sparsity by reweighted `1minimization”, J. Fourier Anal. Appl., 14 877-905, 2007.
• A. Saltelli, “Making best use of model evaluations to compute sensitivityindices”,Comp Phys Comm, 145, 2002.
• K. Sargsyan, C. Safta, H. Najm, B. Debusschere, D. Ricciuto and P. Thornton,“Dimensionality reduction for complex models via Bayesian compressive sensing”,Int J for Uncertainty Quantification, 4(1), pp. 63-93, 2014.
• J. Jakeman, M. Eldred and K. Sargsyan, “Enhancing `1-minimization estimates ofpolynomial chaos expansions using basis selection”, J Comp Phys, 289, pp. 18-34,2015.
Thank YouK. Sargsyan ([email protected]) AGU Fall Meeting 2015 Dec 18, 2015 16
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Random variables represented by Polynomial Chaos
X 'K−1∑k=0
ckΨk(η)
• η = (η1, · · · , ηd) standard i.i.d. r.v.Ψk standard polynomials, orthogonal w.r.t. π(η).
Ψk(η1, η2, . . . , ηd) = ψk1(η1)ψk2(η2) · · ·ψkd (ηd)
• Typical truncation rule: total-order p, k1 + k2 + . . . kd ≤ p.Number of terms is K = (d+p)!d!p! .
• Essentially, a parameterization of a r.v. by deterministic spectralmodes ck .
• Most common standard Polynomial-Variable pairs:(continuous) Gauss-Hermite, Legendre-Uniform,(discrete) Poisson-Charlier.
[Wiener, 1938; Ghanem & Spanos, 1991; Xiu & Karniadakis, 2002; Le Maı̂tre & Knio, 2010]
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Bayesian inference of PC surrogate
u '∑K−1
k=0 ckΨk(x) ≡ gc(x)Posterior︷ ︸︸ ︷P(c|D) ∝
Likelihood︷ ︸︸ ︷P(D|c)
Prior︷︸︸︷P(c)
• Data consists of training runsD ≡ {(xi, ui)}Ni=1
• Likelihood with a gaussian noise model with σ2 fixed or inferred,
L(c) = P(D|c) =(
1σ√
2π
)N N∏i=1
exp(− (ui − gc(x))
2
2σ2
)• Prior on c is chosen to be conjugate, uniform or gaussian.• Posterior is a multivariate normal
c ∈ MVN (µ,Σ)
• The (uncertain) surrogate is a gaussian processK−1∑k=0
ckΨk(x) = Ψ(x)Tc ∈ GP(Ψ(x)Tµ,Ψ(x)ΣΨ(x′)T)
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Sensitivity information comes free with PC surrogate,
g(x1, . . . , xd) =K−1∑k=0
ckΨk(x)
• Main effect sensitivity indices
Si =Var[E(g(x|xi)]
Var[g(x)]=
∑k∈Ii c
2k ||Ψk||2∑
k>0 c2k ||Ψk||2
Ii is the set of bases with only xi involved
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Sensitivity information comes free with PC surrogate,
g(x1, . . . , xd) =K−1∑k=0
ckΨk(x)
• Main effect sensitivity indices
Si =Var[E(g(x|xi)]
Var[g(x)]=
∑k∈Ii c
2k ||Ψk||2∑
k>0 c2k ||Ψk||2
• Joint sensitivity indices
Sij =Var[E(g(x|xi, xj)]
Var[g(x)]− Si − Sj =
∑k∈Iij c
2k ||Ψk||2∑
k>0 c2k ||Ψk||2
Iij is the set of bases with only xi and xj involved
-
Sensitivity information comes free with PC surrogate,but not with piecewise PC
g(x1, . . . , xd) =K−1∑k=0
ckΨk(x)
• Main effect sensitivity indices
Si =Var[E(g(x|xi)]
Var[g(x)]=
∑k∈Ii c
2k ||Ψk||2∑
k>0 c2k ||Ψk||2
• Joint sensitivity indices
Sij =Var[E(g(x|xi, xj)]
Var[g(x)]− Si − Sj =
∑k∈Iij c
2k ||Ψk||2∑
k>0 c2k ||Ψk||2
• For piecewise PC, need to resort to Monte-Carlo estimation[Saltelli, 2002].
-
Basis normalization helps the success rate
0 5 10 15 20 25 30 35 40 45Number of measurements
0.0
0.2
0.4
0.6
0.8
1.0Success rate
LU Basis
LU Normalized Basis
-
Input correlations: Rosenblatt transformation
• Rosenblatt transformation maps any (not necessarily independent) set ofrandom variables λ = (λ1, . . . , λd) to uniform i.i.d.’s {xi}di=1[Rosenblatt, 1952].
x1 = F1(λ1)
x2 = F2|1(λ2|λ1)x3 = F3|2,1(λ3|λ2, λ1)...
xd = Fd|d−1,...,1(λd|λd−1, . . . , λ1)0.18
0.2
0.22
0.24
0.4
0.45
0.5
0.3
0.32
0.34
0.36
LabileCellulose
Lig
nin
• Inverse Rosenblatt transformation λ = R−1(x) ensures a well-defined inputPC construction
λi =
K−1∑k=0
λikΨk(x)
• Caveat: the conditional distributions are often hard to evaluate accurately.
-
Strong discontinuities/nonlinearities challenge globalpolynomial expansions
• Basis enrichment [Ghosh & Ghanem, 2005]
• Stochastic domain decomposition• Wiener-Haar expansions,
Multiblock expansions,Multiwavelets, [Le Maı̂tre et al, 2004,2007]
• also known as Multielement PC [Wan & Karniadakis, 2009]
• Smart splitting, discontinuity detection[Archibald et al, 2009; Chantrasmi, 2011; Sargsyan et al, 2011; Jakeman et al, 2012]
• Data domain decomposition,• Mixture PC expansions [Sargsyan et al, 2010]
• Data clustering, classification,• Piecewise PC expansions
-
Piecewise PC expansion with classification
• Cluster the training dataset into non-overlapping subsets D1and D2, where the behavior of function is smoother• Construct global PC expansions gi(x) =
∑k cikΨk(x) using
each dataset individually (i = 1, 2)• Declare a surrogate
gs(x) =
{g1(x) if x ∈∗ D1g2(x) if x ∈∗ D2
∗ Requires a classification step to find out which cluster xbelongs to. We applied Random Decision Forests (RDF).
• Caveat: the sensitivity information is harder to obtain.
0.0: 0.1: 0.2: 0.3: 0.4: 0.5: 0.6: 0.7: 0.8: anm0: 1.0: 1.1: 1.2: 1.3: 1.4: 1.5: 1.6: 1.7: 1.8: 1.9: 1.10: anm1: