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Introduction CSPRED UQ CSPUQ Closure
Analysis and Reduction of Chemical Models
under Uncertainty
H. Najm,
R. Berry, and B. Debusschere
Sandia National Laboratories,Livermore, CA, USA
4th International Workshop onModel Reduction in Reacting Flows
June 19-21, 2013San Francisco, CA, USA
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Introduction CSPRED UQ CSPUQ Closure
Acknowledgement
This work was supported by:
DOE Office of Basic Energy Sciences, Div. of Chem. Sci., Geosci., & Biosci.
DOE ASCR Applied Mathematics program.
Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed MartinCompany, for the United States Department of Energy under contract DE-AC04-94-AL85000.
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Introduction CSPRED UQ CSPUQ Closure
Outline
1 Introduction
2 Chemical Model Reduction with CSP
3 Uncertainty Quantification
4 CSP Analysis of Uncertain Chemical Models
5 Closure
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Introduction CSPRED UQ CSPUQ Closure
Uncertainty in Reacting Flow Modeling
Chemical models involve much empiricism
Model uncertainties: choice of species and reactions
Parametric uncertainties:
– Chemical rate constants
– Thermodynamic parameters
Along with parameters in
– turbulence/subgrid models
– mass/energy transport and fluid constitutive laws
– geometry and initial/boundary conditions
Chemical parameters frequently have large uncertainty,
they are commonly measured, computed, or estimated ...
Present focus on parametric uncertainty
– kinetic rate coefficients
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Introduction CSPRED UQ CSPUQ Closure
Uncertainty and Chemical Model Reduction
Typical ingredients in chemical model reduction
A detailed starting chemical kinetic mechanism M0
Operating conditions of interest
Quantities of interest (QoIs) desired with specified accuracy
E ≡ ‖Φ− Φ0‖ < α
Consequences of uncertainty in the detailed model?
Errors in QoIs: acceptable over range of uncertainty
QoIs are uncertain – error measure definition
Probabilistic measures of model fidelity
E ≡ ‖Φ− Φ0‖ ⇒ P(E < α) < ǫ
E ≡ D[p(Φ), p(Φ0)] ⇒ E < α
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Introduction CSPRED UQ CSPUQ Closure
Model Robustness, Error and Uncertainty
Robustness: A reduced model developed based on adatabase of states should not learn "too much" from thedata
Reduced models based on different training/test data
subsets should not vary "much"
Optimally, requirements on reduced model error should bemade in light of uncertainty in detailed model predictions
There is little point in insisting on error bounds much
smaller than uncertainty in the reference data
Measures of reduced model fidelity can include accurateprediction of
the nominal reference solution
uncertainty in specific observables
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Introduction CSPRED UQ CSPUQ Closure
Deterministic Chemical ODE System Analysis
Computational Singular Perturbation (CSP) analysis
Jacobian eigenvalues provide first-order estimates of the
time-scales of system dynamics: τi ∼ 1/λi
Jacobian eigenvectors provide first-order estimates of the
vectors that span the fast/slow tangent spaces
With chosen thresholds, have M “fast" modes
M algebraic constraints define a slow manifold
Fast processes constrain the system to the manifold
System evolves with slow processes along the manifold
CSP Importance indices provide estimates of “importance"
of a given reaction to a given species in each of the
fast/slow subspaces
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Introduction CSPRED UQ CSPUQ Closure
A CSP-based Mechanism Simplification Algorithm
Given a state database D = y1, y2, . . .Choose tolerance ǫ, and a kernel set of active species S0
For all yn ∈ D
Sk=0 = S0;
dok = k + 1
Define new active reactions set:
Rk = Rj|(Iij)slow > ǫSi∈Sk−1
∪ Rj|(Iij)fast > ǫSi∈S
rad
k−1
Define new active species set:
Sk = Si|νij 6= 0Rj∈Rk
while Sk 6= Sk−1
Define local active reactions set: Rn = Rk
Define global active reactions set: R = ∪nRn.
Define global active species set: S = Si|νij 6= 0Rj∈R.
Valorani et al. CF 2006
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Introduction CSPRED UQ CSPUQ Closure
nHeptane Kinetic Model Simplification with CSP
# sp e c ie s
Err
or
%τ ig
n
100 150 200 250 300 350 400
10›1
100
101
102
T0
= 7 0 0 K
T0
= 8 5 0 K
T0
= 1 1 0 0 K
% Relative error in ignition time vs. simplified model sizes
Control using error tolerances on CSP importance indices
Valorani et al. PCI 2007
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Introduction CSPRED UQ CSPUQ Closure
Elements of Uncertainty Quantification (UQ)
Estimation of model/parametric uncertainty
Expert opinion, data collection
Regression analysis, fitting, parameter estimation
Bayesian inference of uncertain models/parameters
Forward propagation of uncertainty in models
Local sensitivity analysis (SA) and error propagation
Fuzzy logic; Evidence theory — interval math
Probabilistic framework — Global SA / stochastic UQ
Random sampling, statistical methods
Galerkin methods
– Polynomial Chaos (PC) — intrusive/non-intrusive
Collocation methods: PC/other interpolants — non-intrusive
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Introduction CSPRED UQ CSPUQ Closure
Probabilistic Forward UQ & Polynomial Chaos
Representation of Random Variables
With y = f (x), x a random variable, estimate the RV y
Can describe a RV in terms of its density, moments,
characteristic function, or most fundamentally as a function
on a probability space
Constraining the analysis to RVs with finite variance,
enables the representation of a RV as a spectral expansion
in terms of orthogonal functions of standard RVs.
– Polynomial Chaos
Enables the use of available functional analysis methods
for forward UQ
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Introduction CSPRED UQ CSPUQ Closure
Polynomial Chaos Expansion (PCE)
Model uncertain quantities as random variables (RVs)
Given a germ ξ(ω) = ξ1, · · · , ξn – a set of i.i.d. RVs
– where p(ξ) is uniquely determined by its moments
Any RV in L2(Ω,S(ξ),P) can be written as a PCE:
u(x, t, ω) = f (x, t, ξ) ≃P∑
k=0
uk(x, t)Ψk(ξ(ω))
– uk(x, t) are mode strengths
– Ψk() are functions orthogonal w.r.t. p(ξ)
With dimension n and order p: P + 1 =(n + p)!
n!p!
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Introduction CSPRED UQ CSPUQ Closure
Orthogonality
By construction, the functions Ψk() are orthogonal with respect
to the density of ξ
uk(x, t) =〈uΨk〉
〈Ψ2
k〉=
1
〈Ψ2
k〉
∫u(x, t;λ(ξ))Ψk(ξ)pξ(ξ)dξ
Examples:
Hermite polynomials with Gaussian basis
Legendre polynomials with Uniform basis, ...
Global versus Local PC methods
Adaptive domain decomposition of the support of ξ
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Introduction CSPRED UQ CSPUQ Closure
Essential Use of PC in UQ
Strategy:
Represent model parameters/solution as random variables
Construct PCEs for uncertain parameters
Evaluate PCEs for model outputs
Advantages:
Computational efficiency
Sensitivity information
Requirement:
Random variables in L2, i.e. with finite variance
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Introduction CSPRED UQ CSPUQ Closure
Non-intrusive Spectral Projection (NISP) PC UQ
Sampling-based
Relies on black-box utilization of the computational model
Evaluate projection integrals numerically
For any model output of interest φ(x, t;λ):
φk(x, t) =1⟨Ψ2
k
⟩∫
φ(x, t;λ(ξ))Ψk(ξ)pξ(ξ)dξ, k = 0, . . . ,P
Integrals can be evaluated using
– A variety of (Quasi) Monte Carlo methods
– Quadrature/Sparse-Quadrature methods
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Introduction CSPRED UQ CSPUQ Closure
Intrusive PC UQ: A direct non-sampling method
Given model equations:M(u(x, t);λ) = 0
Express uncertain parameters/variables using PCEs
u =P∑
k=0
ukΨk; λ =P∑
k=0
λkΨk
Substitute in model equations; apply Galerkin projection
New set of equations:G(U(x, t),Λ) = 0
– with U = [u0, . . . , uP]T , Λ = [λ0, . . . , λP]
T
Solving this system once provides the full specification of
uncertain model ouputs
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Introduction CSPRED UQ CSPUQ Closure
Probabilistic Analysis of Uncertain ODE Systems
Handle uncertainties using probability theory
Every random instance of the uncertain inputs provides a
“sample" ODE system
– Uncertainties in fast subspace lead to uncertainty
in manifold geometry
– Uncertainties in slow subspace lead to uncertain
slow time dynamics
Probabilistic measures of importance
Probabilistic comparison of models
One can analyze/reduce each system realization
– Statistics of x(t;λ) trajectories
This can be expensive ⇒ explore alternate means
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Introduction CSPRED UQ CSPUQ Closure
Intrusive Galerkin PC ODE System
du
dt= f (u;λ)
λ =P∑
i=0
λiΨi u(t) =P∑
i=0
ui(t)Ψi
dui
dt=
〈f (u;λ)Ψi〉⟨Ψ2
i
⟩ i = 0, . . . ,P
Say f (u;λ) = λu, then
dui
dt=
P∑
p=0
P∑
q=0
λpuqCpqi, i = 0, · · · ,P
where the tensor Cpqi = 〈ΨpΨqΨi〉/〈Ψ2i 〉 is readily evaluated
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Introduction CSPRED UQ CSPUQ Closure
Dynamical Analysis of the Galerkin PC ODE System
Key questions:
How do the eigenvalues and eigenvectors of the Galerkin
system relate to those of the sampled original system
What can we learn about the sampled dynamics of the
original system from analysis of the Galerkin system
– fast/slow subspaces
– slow manifolds
Can CSP analysis of the Galerkin system be used for
analysis and reduction of the original uncertain system
Stochastic system Jacobian: J
Reformulated Galerkin system Jacobian: J P
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Introduction CSPRED UQ CSPUQ Closure
Key Results
1 The spectrum of J P is contained in the convex hull of the
essential range of the random matrix J.
spect(J P) ⊂ conv(W(J))
2 As P → ∞, the eigenvalues of J P(t) converge weakly, i.e.
in the sense of measures, toward⋃
ω∈Ωspect(J(ω)).
3 J P eigenvalues and eigenpolynomials can be used to
construct polynomial approximations of the random
eigenvalues and eigenvectors.
Sonday et al., SISC, 2011
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Introduction CSPRED UQ CSPUQ Closure
1D Linear Example
x(ξ, t) = a(ξ)x(ξ, t); ξ(ω) ∼ U[−1, 1];
J = a(ξ) ≡
ξ + 1 for ξ ≥ 0,ξ − 1 for ξ < 0.
W(J) = [−2,−1] ∪ [1, 2]; conv(W(J)) = [−2, 2].
LU PC: eigenvalues of J P shown for P = 10, 15, 20, 25, 45
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Introduction CSPRED UQ CSPUQ Closure
A 3D Non-Linear example
Makeev et al., JCP, 2002
u = az − cu − 4duv v = 2bz2 − 4duv
w = ez − fw z = 1 − u − v − w
a = 1.6, b = 20.75 + .45ξ, c = 0.04
d = 1.0, e = 0.36, f = 0.016
u(0) = 0.1, v(0) = 0.2,w(0) = 0.7
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Introduction CSPRED UQ CSPUQ Closure
3D Non-Linear Example; PC order 10
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Introduction CSPRED UQ CSPUQ Closure
3D Non-Linear Example; PC order 10. Eigenvectors.
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Introduction CSPRED UQ CSPUQ Closure
Closure
Probabilistic uncertainty quantification (UQ) framework
Polynomial Chaos representation of random variables
– Utility in forward/inverse UQ
Chemical model reduction under uncertainty
Probabilistic framework
Eigenanalysis of the intrusive Galerkin PC system
– Relationship to the uncertain system dynamics
Work in needed on
Formulation of uncertain manifolds
Stochastic importance criteria
Stochastic model comparison strategies
Uncertain model simplification algorithms
– stochastic & kinetic dimensionality
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