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Optimal control of systems governed by PDEs withrandom parameter fields using quadratic approximations
Alen Alexanderian,1 Omar Ghattas,2 Noemi Petra3
Georg Stadler,4 Umberto Villa2
1Department of MathematicsNorth Carolina State University
2Institute for Computational Engineering and SciencesThe University of Texas at Austin
3School of Natural SciencesUniversity of California, Merced
3Courant Institute of Mathematical SciencesNew York University
Computational Methods for Control of Infinite-dimensional SystemsInstitute for Mathematics and its Applications
University of Minnesota, Minneapolis, MinnesotaMarch 14, 2016
Optimal control of systems governed by PDEs withuncertain parameter fields
PDE-constrained control objective
q = q(u(z,m))
where
A(u,m) = f(z)
q: control objective
A: forward operator
m: uncertain parameter field
z: control function
Control of injection wells in porousmedium flow (SPE10 permeability data)
Problem: given the uncertainty model for m, find z that “optimizes” q(u(z,m))
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 2 / 28
Optimization under uncertainty (OUU)
H : parameter space, infinite-dimensional separable Hilbert space
q(z,m): control objective functional
m ∈H : uncertain model parameter field, z: control function
Optimization under uncertainty (OUU):
minz
Em{
q(z,m)
}+ β varm{q(z,m)}
Em{q(z,m)} =
∫H
q(z,m)µ(dm)
varm{q(z,m)} = Em{q(z,m)2} − Em{q(z,m)}2
Main challenges:
Integration over infinite/high-dimensional parameter spaceEvaluation of q requires PDE solves
Standard Monte Carlo approach (Sample Average Approximation) isprohibitive
Numerous (nmc) samples required, each requires PDE solveResulting PDE-constrained optimization problem has nmc PDE constraints
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 3 / 28
Optimization under uncertainty (OUU)
H : parameter space, infinite-dimensional separable Hilbert space
q(z,m): control objective functional
m ∈H : uncertain model parameter field, z: control function
Risk-neutral optimization under uncertainty (OUU):
minz
Em{q(z,m)}
+ β varm{q(z,m)}
Em{q(z,m)} =
∫H
q(z,m)µ(dm)
varm{q(z,m)} = Em{q(z,m)2} − Em{q(z,m)}2
Main challenges:
Integration over infinite/high-dimensional parameter spaceEvaluation of q requires PDE solves
Standard Monte Carlo approach (Sample Average Approximation) isprohibitive
Numerous (nmc) samples required, each requires PDE solveResulting PDE-constrained optimization problem has nmc PDE constraints
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 3 / 28
Optimization under uncertainty (OUU)
H : parameter space, infinite-dimensional separable Hilbert space
q(z,m): control objective functional
m ∈H : uncertain model parameter field, z: control function
Risk-averse optimization under uncertainty (OUU):
minz
Em{q(z,m)}+ β varm{q(z,m)}
Em{q(z,m)} =
∫H
q(z,m)µ(dm)
varm{q(z,m)} = Em{q(z,m)2} − Em{q(z,m)}2
Main challenges:
Integration over infinite/high-dimensional parameter spaceEvaluation of q requires PDE solves
Standard Monte Carlo approach (Sample Average Approximation) isprohibitive
Numerous (nmc) samples required, each requires PDE solveResulting PDE-constrained optimization problem has nmc PDE constraints
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 3 / 28
Optimization under uncertainty (OUU)
H : parameter space, infinite-dimensional separable Hilbert space
q(z,m): control objective functional
m ∈H : uncertain model parameter field, z: control function
Risk-averse optimization under uncertainty (OUU):
minz
Em{q(z,m)}+ β varm{q(z,m)}
Em{q(z,m)} =
∫H
q(z,m)µ(dm)
varm{q(z,m)} = Em{q(z,m)2} − Em{q(z,m)}2
Main challenges:
Integration over infinite/high-dimensional parameter spaceEvaluation of q requires PDE solves
Standard Monte Carlo approach (Sample Average Approximation) isprohibitive
Numerous (nmc) samples required, each requires PDE solveResulting PDE-constrained optimization problem has nmc PDE constraints
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 3 / 28
Some existing approaches for PDE-constrained OUU
Methods based on stochastic collocation, sparse/adaptive sampling, POD, ...Schulz & Schillings, Problem formulations and treatment of uncertainties in aerodynamic design, AIAAJ, 2009.Borzı & von Winckel, Multigrid methods and sparse-grid collocation techniques for parabolic optimalcontrol problems with random coefficients, SISC, 2009.Borzı, Schillings, & von Winckel, On the treatment of distributed uncertainties in PDE-constrainedoptimization, GAMM-Mitt. 2010.Borzı & von Winckel, A POD framework to determine robust controls in PDE optimization, Computingand Visualization in Science, 2011.Gunzburger & Ming, Optimal control of stochastic flow over a backward-facing step using reduced-ordermodeling, SISC 2011.Gunzburger, Lee, & Lee, Error estimates of stochastic optimal Neumann boundary control problems,SINUM, 2011.Kunoth & Schwab, Analytic Regularity and GPC Approximation for Control Problems Constrained byLinear Parametric Elliptic and Parabolic PDEs, SICON, 2013.Tiesler, Kirby, Xiu, & Preusser, Stochastic collocation for optimal control problems with stochastic PDEconstraints, SICON, 2012.Kouri, Heinkenschloss, Ridzal, & Van Bloemen Waanders, A trust-region algorithm with adaptivestochastic collocation for PDE optimization under uncertainty, SISC, 2013.Chen, Quarteroni, & Rozza, Stochastic optimal Robin boundary control problems ofadvection-dominated elliptic equations, SINUM, 2013.Kouri, A multilevel stochastic collocation algorithm for optimization of PDEs with uncertain coefficients,JUQ, 2014.Chen & Quarteroni, Weighted reduced basis method for stochastic optimal control problems with ellipticPDE constraint, JUQ, 2014.Chen, Quarteroni, & Rozza, Multilevel and weighted reduced basis method for stochastic optimalcontrol problems constrained by Stokes equations, Num. Math. 2015.
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 4 / 28
Control of injection wells in a porous medium flow
1
2
3
4
m = mean of log permeability field q = target pressure at production wells
state PDE: single phase flow in a porous medium
−∇ · (em∇u) =
nc∑i=1
zifi(x)
with Dirchlet lateral & Neumann top/bottom BCs
uncertain parameter: log permeability field m
control variables: zi, mass source at injection wells; fi, mollified Dirac deltas
control objective: q(z,m) := 12‖Qu(z,m)− q‖2, q: target pressure
dimensions: ns = nm = 3242, nc = 20, nq = 12
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 5 / 28
Porous medium with random permeability field
Distribution law of m:
µ = N (m, C) (Gaussian measure on Hilbert space H )
Take covariance operator as square of inverse of Poisson-like operator:
C = (−κ∆ + αI)−2 κ, α > 0
C is positive, self-adjoint, of trace-class; µ well-defined on H (Stuart ’10)κα ∝ correlation length; the larger α, the smaller the variance
Random draws for κ = 2× 10−2, α = 4
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 6 / 28
OUU with linearized parameter-to-objective map
Risk-averse optimal control problem (including cost of controls):
minz
Em{q(z,m)}+ β varm{q(z,m)}+ γ ‖z‖2
Linear approximation to parameter-to-objective map about m:
qlin(z,m) = q(z, m) + 〈gm(z, m),m− m〉
gm(z, ·) :=dq(z, ·)dm
is the gradient with respect to m
The moments of the linearized objective:
Em{qlin(z, ·)} = q(z, m),
varm{qlin(z, ·)} = 〈gm(z, m), C[gm(z, m)]〉
qlin(z, ·) ∼ N(q(z, m), 〈gm(z, m), C[gm(z, m)]〉
)Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 7 / 28
Risk-averse optimal control problem with linearizedparameter-to-objective map
State-and-adjoint-PDE constrained optimization problem (quartic in z):
minz∈ZJ (z) :=
1
2‖Qu− q‖2 +
β
2〈gm(m), C[gm(m)]〉+
γ
2‖z‖2
with gm(m) =em∇u · ∇p, where
−∇ · (em∇u) =
nc∑i=1
zifi state equation
−∇ · (em∇p) = −Q∗(Qu− q) adjoint equation
Lagrangian of the risk-averse optimal control problem with qlin:
L (z, u, p, u?, p?) =1
2‖Qu− q‖2 +
β
2
⟨em∇u · ∇p, C[em∇u · ∇p]
⟩+γ
2‖z‖2
+⟨em∇u,∇u?⟩− nc∑
i=1
zi〈fi, u?〉
+⟨em∇p,∇p?
⟩+ 〈Q∗(Qu− q), p?〉
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 8 / 28
Gradient computation for risk averse optimal control
“State problem” for risk-averse optimal control problem with qlin:
⟨em∇u,∇u
⟩=
nc∑i=1
zi〈fi, u〉⟨em∇p,∇p
⟩= −〈Q∗(Qu− q), p〉
for all test functions p and u
“Adjoint problem” for risk-averse optimal control problem with qlin:
⟨em∇p?,∇p
⟩= −β
⟨em∇u · ∇p, C[em∇u · ∇p]
⟩⟨em∇u?,∇u
⟩= −〈Q∗(Qu− q), u〉 − β
⟨em∇u · ∇p, C[em∇u · ∇p]
⟩− 〈Q∗Qp?, u〉
for all test functions p and u
Gradient:∂J∂zj
= γzj − 〈fj , u?〉, j = 1, . . . , nc
Cost of objective = 2 PDE solves; cost of gradient = 2 PDE solves
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 9 / 28
Risk-averse optimal control with linearized objective
−20 0 20 40 60 80 100 120 1400
0.01
0.02
0.03
0.04
dis
trib
uti
on
Θ(z0, ·)Θlin(z0, ·)Θquad(z0, ·)
initial (suboptimal) control z0 distrib. of exact & approx objectives at z0
0 2 4 6 8 10 12 14 160
1
2
3
4
5
dis
trib
uti
on
Θ(zoptlin , ·)
Θlin(zoptlin , ·)
Θquad(zoptlin , ·)
optimal control zoptlin based on qlin distrib. of exact & approx objectives at zopt
lin
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 10 / 28
Risk-averse optimal control with linearized objective
−20 0 20 40 60 80 100 120 1400
0.01
0.02
0.03
0.04
dis
trib
uti
on
Θ(z0, ·)Θlin(z0, ·)Θquad(z0, ·)
initial (suboptimal) control z0 distrib. of exact & approx objectives at z0
0 2 4 6 8 10 12 14 160
1
2
3
4
5
dis
trib
uti
on
Θ(zoptlin , ·)
Θlin(zoptlin , ·)
Θquad(zoptlin , ·)
optimal control zoptlin based on qlin distrib. of exact & approx objectives at zopt
lin
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 10 / 28
Quadratic approximation to parameter-to-objective map
Quadratic approximation to the parameter-to-control-objective map:
qquad(z,m) = q(z, m) + 〈gm(z, m),m− m〉+1
2〈Hm(z, m)(m− m),m− m〉
gm: gradient of parameter-to-objective map
Hm: Hessian of parameter-to-objective map
Observations:
Quadratic approximation does not lead to a Gaussian control objective
However, can derive analytic formulas for the moments of qquad in theinfinite-dimensional Hilbert space setting
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 11 / 28
Analytic expressions for mean and variance with qquad
Mean:
Em{qquad(z, ·)} = q(z, m)+1
2tr[Hm(z, m)
]Variance:
varm{qquad(z, ·)} = 〈gm(z, m), C[gm(z, m)]〉+1
2tr[Hm(z, m)2
]where Hm = C1/2HmC1/2 is the covariance-preconditioned Hessian
Risk averse optimal control objective with qquad:
J(z) = q(z, m) +1
2tr[Hm(z, m)
]+β
2
{〈gm(z, m), C[gm(z, m)]〉+
1
2tr[Hm(z, m)2
]}
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 12 / 28
Randomized trace estimator
Randomized trace estimation:
tr(Hm) ≈ 1
ntr
ntr∑j=1
⟨Hmξj , ξj
⟩=
1
ntr
ntr∑j=1
〈Hmζj , ζj〉
tr(H2m) ≈ 1
ntr
ntr∑j=1
〈Hmζj , C[Hmζj ]〉
where ξj are random Gaussian fields and ζj = C1/2ξj
In computations, we use draws ζj ∼ N (0, C) =: ν
Straightforward to show:∫H
〈Hmζ, ζ〉 ν(dζ) = tr(Hm),
∫H
〈Hmζ, C[Hmζ]〉 ν(dζ) = tr(H2m)
Finite dimensional algorithm: H. Avron and S. Toledo, Randomizedalgorithms for estimating the trace of an implicit symmetric positivesemi-definite matrix, Journal of the ACM, 2011.
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 13 / 28
Risk-averse optimal control with quadraticized objective
minz∈Z
1
2‖Qu− q‖22+
1
2ntr
ntr∑j=1
〈ζj , ηj〉+β
2
{〈gm(m), C[gm(m)]〉+ 1
2ntr
ntr∑j=1
∥∥C1/2ηj∥∥2}
with
gm(m) =em∇u · ∇pηj = em(ζj∇u · ∇p+∇υj · ∇p+∇u · ∇ρj)︸ ︷︷ ︸
Hmζj
j ∈ {1, . . . , ntr}
where
−∇ · (em∇u) =∑Ni=1 zifi
−∇ · (em∇p) = −Q∗(Qu− q)−∇ · (em∇υj) = ∇ · (ζjem∇u)
−∇ · (em∇ρj) = −Q∗Qυj +∇ · (ζjem∇p)
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 14 / 28
Risk-averse optimal control with quadraticized objective
minz∈Z
1
2‖Qu− q‖22+
1
2ntr
ntr∑j=1
〈ζj , ηj〉+β
2
{〈gm(m), C[gm(m)]〉+ 1
2ntr
ntr∑j=1
∥∥C1/2ηj∥∥2}
with
gm(m) =em∇u · ∇pηj = em(ζj∇u · ∇p+∇υj · ∇p+∇u · ∇ρj)︸ ︷︷ ︸
Hmζj
j ∈ {1, . . . , ntr}
where
−∇ · (em∇u) =∑Ni=1 zifi
−∇ · (em∇p) = −Q∗(Qu− q)−∇ · (em∇υj) = ∇ · (ζjem∇u)
−∇ · (em∇ρj) = −Q∗Qυj +∇ · (ζjem∇p)
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 14 / 28
Risk-averse optimal control with quadraticized objective
minz∈Z
1
2‖Qu− q‖22+
1
2ntr
ntr∑j=1
〈ζj , ηj〉+β
2
{〈gm(m), C[gm(m)]〉+ 1
2ntr
ntr∑j=1
∥∥C1/2ηj∥∥2}
with
gm(m) =em∇u · ∇pηj = em(ζj∇u · ∇p+∇υj · ∇p+∇u · ∇ρj)︸ ︷︷ ︸
Hmζj
j ∈ {1, . . . , ntr}
where
−∇ · (em∇u) =∑Ni=1 zifi
−∇ · (em∇p) = −Q∗(Qu− q)−∇ · (em∇υj) = ∇ · (ζjem∇u)
−∇ · (em∇ρj) = −Q∗Qυj +∇ · (ζjem∇p)
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 14 / 28
Risk-averse optimal control with quadraticized objective
minz∈Z
1
2‖Qu− q‖22+
1
2ntr
ntr∑j=1
〈ζj , ηj〉+β
2
{〈gm(m), C[gm(m)]〉+ 1
2ntr
ntr∑j=1
∥∥C1/2ηj∥∥2}
with
gm(m) =em∇u · ∇pηj = em(ζj∇u · ∇p+∇υj · ∇p+∇u · ∇ρj)︸ ︷︷ ︸
Hmζj
j ∈ {1, . . . , ntr}
where
−∇ · (em∇u) =∑Ni=1 zifi
−∇ · (em∇p) = −Q∗(Qu− q)−∇ · (em∇υj) = ∇ · (ζjem∇u)
−∇ · (em∇ρj) = −Q∗Qυj +∇ · (ζjem∇p)
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 14 / 28
Risk-averse optimal control with quadraticized objective
minz∈Z
1
2‖Qu− q‖22+
1
2ntr
ntr∑j=1
〈ζj , ηj〉+β
2
{〈gm(m), C[gm(m)]〉+ 1
2ntr
ntr∑j=1
∥∥C1/2ηj∥∥2}
with
gm(m) =em∇u · ∇pηj = em(ζj∇u · ∇p+∇υj · ∇p+∇u · ∇ρj)︸ ︷︷ ︸
Hmζj
j ∈ {1, . . . , ntr}
where
−∇ · (em∇u) =∑Ni=1 zifi
−∇ · (em∇p) = −Q∗(Qu− q)−∇ · (em∇υj) = ∇ · (ζjem∇u)
−∇ · (em∇ρj) = −Q∗Qυj +∇ · (ζjem∇p)
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 14 / 28
Lagrangian for risk-averse optimal control with qquad
L (z, u, p,{υj}ntrj=1, {ρj}ntr
j=1, u?, p?, {υ?
j}ntrj=1, {ρ
?j}ntr
j=1)
=1
2‖Qu− q‖22
+1
2ntr
ntr∑j=1
⟨ζj ,[em(ζj∇u · ∇p+∇υj · ∇p+∇u · ∇ρj)
]⟩+β
2
⟨em∇u · ∇p, C[em∇u · ∇p]
⟩+
β
4ntr
ntr∑j=1
∥∥∥C1/2 [em(ζj∇u · ∇p+∇υj · ∇p+∇u · ∇ρj)]∥∥∥2
+⟨em∇u,∇u?⟩−∑N
i=1 zi〈fi, u?〉
+⟨em∇p,∇p?
⟩+ 〈Q∗(Qu− q), p?〉
+
ntr∑j=1
[⟨em∇υj ,∇υ?
j
⟩+⟨ζje
m∇u,∇υ?j
⟩]+
ntr∑j=1
[⟨em∇ρj ,∇ρ?
j
⟩+⟨Q∗Qυj , ρ
?j
⟩+⟨ζje
m∇p,∇ρ?j
⟩]Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 15 / 28
Adjoint & gradient for risk-averse optimal control w/qquad
Adjoint problem for qquad approximation
−∇ · (em∇ρ?j ) = b(j)1 j ∈ {1, . . . , ntr}
− ∇ · (em∇υ?j ) +Q∗Qρ?j = b(j)2 j ∈ {1, . . . , ntr}
− ∇ · (em∇p?)−ntr∑j=1
∇ · (ζjem∇ρ?j ) = b3
−∇ · (em∇u?) +Q∗Qp? −ntr∑j=1
∇ · (ζjem∇υ?j ) = b4
Gradient for qquad approximation
∂L
∂zj= γzj − 〈fj , u?〉, j = 1, . . . , nc
Cost of objective = 2 + 2ntr PDE solves; cost of gradient = 2 + 2ntr PDE solves
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 16 / 28
Risk-averse optimal control with quadraticized objective
−20 0 20 40 60 80 100 120 1400
0.01
0.02
0.03
0.04
dis
trib
uti
on
Θ(z0, ·)Θlin(z0, ·)Θquad(z0, ·)
initial (suboptimal) control z0 distrib. of exact & approx objectives at z0
0 2 4 6 8 10 12 14 160
0.2
0.4
0.6
dis
trib
uti
on
Θ(zoptquad, ·)
Θquad(zoptquad, ·)
optimal control zoptquad based on qquad distrib. of exact & approx objectives at zopt
quad
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 17 / 28
Risk-averse optimal control with quadraticized objective
−20 0 20 40 60 80 100 120 1400
0.01
0.02
0.03
0.04
dis
trib
uti
on
Θ(z0, ·)Θlin(z0, ·)Θquad(z0, ·)
initial (suboptimal) control z0 distrib. of exact & approx objectives at z0
0 2 4 6 8 10 12 14 160
0.2
0.4
0.6
dis
trib
uti
on
Θ(zoptquad, ·)
Θquad(zoptquad, ·)
optimal control zoptquad based on qquad distrib. of exact & approx objectives at zopt
quad
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 17 / 28
Effect of number of trace estimator vectors on distributionof control objective evaluated at optimal controls
0 1 2 3 4 5 6 7 80
0.2
0.4
value of control objective
distribution
ntr = 5ntr = 20ntr = 40ntr = 60
Optimal controls zoptquad computed for each value of trace estimator using quadratic
approximation of control objective, qquad
Each curve based on 10,000 samples of distribution of q(zoptquad,m)
(control objective evaluated at optimal control)
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 18 / 28
Comparison of distribution of control objective for optimalcontrols based on linearized and quadraticized objective
0 1 2 3 4 5 6 7 80
0.2
0.4
0.6
value of control objective
dis
trib
uti
on
Θ(zoptquad, ·)
Θ(zoptlin , ·)
1000 5000 100001
2
3
4
5
sample size
sample
mean
1000 5000 10000
5
10
15
sample size
sample
variance
Comparison of the distributions of q(zoptlin ,m) with q(zopt
quad,m)
β = 1, γ = 10−5 and ntr = 40 trace estimation vectors
KDE results are based on 10,000 samples
Inserts show Monte Carlo sample convergence for mean and variance
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 19 / 28
Some concluding remarks
Optimal control of PDEs with infinite-dimensional parameters
Use of local approximations to parameter-to-objective map
Quadratic approximation is effective in capturing the distribution of thecontrol objective for the target problem
Computational complexity of objective & gradient evaluation is independentof problem dimension (nm = 3242)
objective + gradient cost = ∼125 state PDE solvesobjective/gradient cost for Monte Carlo/SAA: ∼10,000 state PDE solvesdifferences even more striking for nonlinear PDE state problems, since all butone PDE solves are linearized (with SAA all are nonlinear)
For further details, see:
A. Alexanderian, N. Petra, G. Stadler, and O. Ghattas, Mean-variancerisk-averse optimal control of systems governed by PDEs with randomcoefficient fields using second order approximations, submitted.arXiv:1602.07592
Research funded by DOE MMICCs (DE-FC02-13ER26128) and DARPA EQUiPS(W911NF-15-2-0121).
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 20 / 28
Extensions
Alternative risk measures
Higher moments
Higher order approximations of parameter-to-objective map, e.g., third orderTaylor expansion with proper tensor contraction of third order derivativetensor
Highly nonlinear/complex forward problems
Monte Carlo corrections to quadratic approximation
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 21 / 28
Reynolds averaged Navier-Stokes model for a turbulent jet
Governing equations: k-ε closure model
(∇u)u−∇ ·[(ν + νt)
(∇u +∇ut)− p I] = 0
∇ · u = 0
−∇ ·[(ν +
νtσk
)∇k]
+ u · ∇k − 1
2νttr
[(∇u +∇ut)2]+ ε = 0
−∇ ·[(ν +
νtσε
)∇ε]
+ u · ∇ε− 1
2Cε1
ε
kνttr
[(∇u +∇ut)2]+ Cε2
ε2
k= 0
where νt = Cµk2
ε
Quantity of Interest: Jet width at a given point x = x0
Q =
∫Γ0
u · e1dy
ucl(x0)
where Γ0 is the cross-section at x = x0 and ucl(x0) is the centerline velocity
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 22 / 28
RANS model for a turbulent jet
State DOF = 28,379; uncertain parameter DOF = 2626; O(100) damped Newtonsteps per nonlinear state solve (10 min per state solve)
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 23 / 28
Reynolds averaged Navier-Stokes inadequacy model
To model inadequacy of the k-ε closure model we introduce an uncertainprefactor in the turbulent viscosity of the form
νt = emCµk2
ε
Here m ∼ N (0, C) is a spatially correlated normal random field with acovariance operator of the form
C = (δ − γ∇ ·Θ∇)−2
where Θ is an s.p.d tensor that allows for anisotropy in the correlation length
Figure: Samples from m ∼ N (0, C)
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 24 / 28
RANS turbulent jet: Spectrum of the Hessian
Figure: Spectrum of the Hessian Hm (left) and of the prior preconditioned HessianHm = C1/2HmC1/2 (right)
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 25 / 28
RANS turbulent jet: Linear and quadratic approximation
To estimate the expected value of the q.o.i. Q we used a Monte Carloestimator QMC (500 samples, estimator variance 1.49× 10−5).
We then used the analytical expression presented before to compute theexpectation of the linearized Qlin and quadratic Qquad approximation of theq.o.i.
We finally considered an approximation Qquad of the Qquad that uses a lowrank approximation HLRm of the Hessian instead of the the true Hessian Hm.The main advantage is that now the trace of C1/2HLRm C1/2 can becomputed analytically and that sampling Qquad does not require to solve anyincremental forward/adjoint problem.
QMC E[Qlin] E[Qquad] E[Qquad]1.2044 1.2010 1.2125 1.2112
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 26 / 28
RANS turbulent jet: Variance reduction MC
The linear and quadratic approximations of Q are effective (and easy tocompute) surrogates for variance reduction in MC simulations
In fact, by linearity of expectation, for Q = {Qlin, Qquad, Qquad} and
Y = Q− Q we have
E[Q] = E[Q− Q] + E[Q] = E[Y ] + E[Q]
Since Var[Y ]� Var[Q], many fewer samples are required to achieve thesame variance as that of the Monte Carlo estimator
Q Ylin Yquad Yquad
Variance 7.4670E-03 1.3426E-04 3.7904E-05 3.2583E-05Number of Samples 7467 134 38 33
Table: Variance of the Monte Carlo estimators for Q and the correction Y . The secondrow indicates the number of samples necessary to obtain a mean square error of 10−6
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 27 / 28
Reynolds Averaged Navier-Stokes - Variance Reduction MC
n samples QMC Y MClin + E[Qlin] Y MC
quad + E[Qquad] Y MCquad + E[Qquad]
50 1.2157 1.2086 1.2108 1.2097100 1.2129 1.2079 1.2112 1.2101150 1.2107 1.2088 1.2105 1.2095200 1.2078 1.2092 1.2100 1.2090250 1.2063 1.2091 1.2101 1.2091300 1.2055 1.2087 1.2102 1.2092350 1.2036 1.2086 1.2103 1.2092400 1.2036 1.2083 1.2104 1.2093450 1.2030 1.2082 1.2103 1.2092500 1.2044 1.2084 1.2102 1.2092
n samples V ar[QMC ] V ar[Y MClin ] V ar[Y MC
quad] V ar[Y MCquad]
50 1.6866E-04 5.3357E-06 4.7332E-07 4.1531E-07100 7.4525E-05 1.6918E-06 2.9706E-07 2.5409E-07150 5.6089E-05 1.3427E-06 2.3033E-07 1.9537E-07200 4.2830E-05 1.0260E-06 2.3183E-07 2.0324E-07250 3.3459E-05 7.2285E-07 1.6875E-07 1.4784E-07300 2.6100E-05 5.3132E-07 1.3172E-07 1.1483E-07350 2.2158E-05 4.4029E-07 1.1049E-07 9.4659E-08400 1.8295E-05 3.4960E-07 8.9182E-08 7.6542E-08450 1.6088E-05 2.8972E-07 7.8000E-08 6.6891E-08500 1.4934E-05 2.6852E-07 7.5808E-08 6.5166E-08
Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 28 / 28