Scalable algorithms for optimal experimental design for
infinite-dimensional nonlinear Bayesian inverse problems
Alen Alexanderian (Math/NC State), Omar Ghattas (ICES/UT-Austin),
Noémi Petra (Applied Math/UC Merced), Georg Stadler (Courant/NYU)
Data Assimilation and Inverse Problems
Mathematics Institute, University of Warwick
Coventry, United Kingdom
February 23, 2016
The conceptual optimal experimental design problem
Data: d ∈ Rq
Model parameter field: m = {m(x)}x∈D
Bayesian inverse problem:
Use data d to infer the model parameter
field m in the Bayesian sense, i.e., update
the prior state of knowledge on m
Optimal experimental design:
How to design observation system for d to
“optimally” infer m?
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
2 / 32
Some relevant references on OED for PDEs
E. Haber, L. Horesh, L. Tenorio, Numerical methods for experimental design of
large-scale linear ill-posed inverse problems, Inverse Problems, 24:125-137, 2008.
E. Haber, L. Horesh, L. Tenorio, Numerical methods for the design of large-scale
nonlinear discrete ill-posed inverse problems, Inverse Problems, 26, 025002, 2010.
E. Haber, Z. Magnant, C. Lucero, L. Tenorio, Numerical methods for A-optimal
designs with a sparsity constraint for ill-posed inverse problems, Computational
Optimization and Applications, 1-22, 2012.
Q. Long, M. Scavino, R. Tempone, S Wang, Fast estimation of expected
information gains for Bayesian experimental designs based on Laplace
approximations, Comp. Meth. in Appl. Mech. and Eng., 259:24–39, 2013.
Q. Long, M. Scavino, R. Tempone, S. Wang, A Laplace method for
under-determined Bayesian optimal experimental designs, Comp. Meth. in Applied
Mech. and Engineering, 285:849–876, 2015.
F. Bisetti, D. Kim, O. Knio, Q. Long, R. Tempone, Optimal Bayesian experimental
design for priors of compact support with application to shocktube experiments for
combustion kinetics, Intl. Journal for Num. Methods in Engineering, 2016.
X. Huan and Y.M. Marzouk, Simulation-based optimal Bayesian experimental
design for nonlinear systems, Journal of Computational Physics, 232:288-317, 2013.
X. Huan, Y. Marzouk, Gradient-based stochastic optimization methods in Bayesian
experimental design, Intl. Journal for Uncertainty Quantification, 4(6), 2014.
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
3 / 32
How we define an optimal design
ground truth + sensor locations
Given locations xi of possible sensor sites,
choose optimal weights wi for a subset (using
sparsity constraints):
x1 , . . . , xns
design :=
w1 , . . . , wns
MAP solution
Obtain data, solve Bayesian inverse problem:
data + likelihood, prior =⇒ posterior
variance
We seek an A-optimal design:
minimize average posterior variance
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
4 / 32
Bayesian inverse problem in Hilbert space (A. Stuart, 2010)
D: bounded domain
H = L2 (D)
m ∈ H: parameter
Parameter-to-observable map: f : H → R
q
Additive Gaussian noise:
η ∼ N (0, Γnoise )
d = f (m) + η,
Likelihood:
πlike (d|m) ∝ exp
n
o
1
− (f (m) − d)T Γ−1
noise (f (m) − d)
2
Gaussian prior measure: µprior = N (mpr , Cprior )
Bayes Theorem (in infinite dimensions):
dµpost
∝ πlike (d|m)
dµprior
“dµpost ∝ πlike (d|m) dµprior ”
A.M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 2010.
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
5 / 32
The experimental design and a weighted inference problem
design :=
Want wi ∈ {0, 1}
relax
=⇒
x 1 , . . . , x ns
w1 , . . . , wns
0 ≤ wi ≤ 1
threshold
=⇒
wi ∈ {0, 1}
w-weighted data-likelihood:
πlike (d|m; w) ∝ exp
n
o
1
− (f (m) − d)T Wσ (f (m) − d)
2
f : parameter-to-observable map
W := diag(wi )
Wσ :=
1
W
2
σnoise
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
6 / 32
A-optimal design in Hilbert space
Covariance function: cpost (x, y) = Cov {m(x), m(y)}
Z
1
cpost (x, x) dx
average posterior variance =
|D| D
Covariance operator:
Z
[Cpost u](x) =
cpost (x, y)u(y) dy
D
Mercer’s Theorem:
Z
cpost (x, x) dx = tr(Cpost )
D
A-optimal design criterion:
Choose a sensor configuration to minimize tr(Cpost )
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
7 / 32
Relationship to some other OED criteria
For Gaussian prior and noise, and linear parameter-to-observable map:
Maximizing the expected information gain is equivalent to D-optimal design:
1
1
Eµprior Ed|m {Dkl (µpost , µprior )} = − log det Γpost + log det Γprior
2
2
Minimizing Bayes risk of posterior mean is equivalent to A-optimal design:
Eµprior Ed|m km − mpost (d)k2
=
Z Z
km − mpost (d)k2 πlike (d|m) dd µprior (dm)
H Y
= tr(Γpost )
For infinite dimensions, see:
A. Alexanderian, P. Gloor, and O. Ghattas, On Bayesian A- and D-optimal experimental designs
in infinite dimensions, Bayesian Analysis, to appear, 2016.
http://dx.doi.org/10.1214/15-BA969
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
8 / 32
Challenges
The infinite-dimensional Bayesian inverse problem is “merely” an inner
problem within OED
Need trace of posterior covariance operator, which would be intractable in
the large-scale setting
We approximate the posterior covariance by the inverse of the Hessian at the
maximum a posteriori (MAP) point, but explicit construction of the Hessian
is still very expensive
For nonlinear inverse problem, Hessian depends on data, which are not
available a priori
For efficient optimization, we need derivatives of the trace of the inverse
Hessian with respect to the sensor weights
We seek scalable algorithms, i.e., those whose cost (in terms of forward PDE
solves) is independent of the data dimension and the parameter dimension
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
9 / 32
A-optimal design problem: Linear case
Linear parameter-to-observable map: f (m) := Fm
Gaussian prior measure: µprior = N (mprior , Cprior )
Gaussian prior and noise =⇒ Gaussian posterior
2
E
1
1 D −1
mpost = arg min W 1/2 (Fm − d) −1 +
Cprior (m − mprior ), m − mprior
m 2
Γnoise 2
|
{z
}
J (m) := negative log-posterior
Cpost (w) = H−1 (w),
−1
H(w) = F ∗ Wσ F + Cprior
OED problem:
minimize tr
w
∗
F Wσ F +
−1
Cprior
−1 + penalty
A. Alexanderian, N. Petra, G. Stadler, O. Ghattas, A-optimal design of experiments for infinite-dimensional
Bayesian linear inverse problems with regularized `0 -sparsification, SIAM Journal on Scientific Computing,
36(5):A2122–A2148, 2014.
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
10 / 32
A-optimal design problem: Nonlinear case
Nonlinear case: Gaussian prior and noise =⇒
6
Gaussian posterior
Use Gaussian approximation to posterior: for given w and d
Compute the maximum a posteriori probability (MAP) estimate mMAP (w; d)
mMAP (w; d) := arg min J (m, w, d)
m
Gaussian approximation to posterior at MAP point
N mMAP (w; d), H−1 mMAP (w; d), w; d
Problem: data d not available a priori
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
11 / 32
A-optimal designs for nonlinear inverse problems
General formulation: minimize average posterior variance:
tr H−1 mMAP (w; d), w; d
minimize
w
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
12 / 32
A-optimal designs for nonlinear inverse problems
General formulation: minimize expected average posterior variance:
n o
minimize
Ed
tr H−1 mMAP (w; d), w; d
w
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
12 / 32
A-optimal designs for nonlinear inverse problems
General formulation: minimize expected average posterior variance:
n o
minimize Eµprior Ed|m tr H−1 mMAP (w; d), w; d
w
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
12 / 32
A-optimal designs for nonlinear inverse problems
General formulation: minimize expected average posterior variance:
n o
minimize Eµprior Ed|m tr H−1 mMAP (w; d), w; d
+ γP (w)
w
P (w): sparsifying penalty function
In practice: get data from a few training models m1 , . . . , mnd
mi : draws from prior
Training data:
di = f (mi ) + η i ,
Omar Ghattas (UT Austin)
Large-scale OED
i = 1, . . . , nd
February 23, 2016
12 / 32
A-optimal designs for nonlinear inverse problems
General formulation: minimize expected average posterior variance:
n o
minimize Eµprior Ed|m tr H−1 mMAP (w; d), w; d
+ γP (w)
w
P (w): sparsifying penalty function
In practice: get data from a few training models m1 , . . . , mnd
mi : draws from prior
Training data:
di = f (mi ) + η i ,
i = 1, . . . , nd
The problem to solve in practice:
minimize
w
Omar Ghattas (UT Austin)
nd
1 X
tr H−1 mMAP (w; di ), w; di + γP (w)
nd i=1
Large-scale OED
February 23, 2016
12 / 32
Randomized trace estimation
A ∈ Rn×n — symmetric
Trace estimator:
tr(A) ≈
ntr
1 X
z T Az i ,
ntr i=1 i
z i — random vectors
Gaussian trace estimator: z i independent draws from N (0, I)
For z ∼ N (0, I)
E z T Az = tr(A)
2
Var z T Az = 2 kAkF
Clustered eigenvalues =⇒ Good approximation with small ntr
Efficient means of approximating trace of inverse Hessian
M. Hutchinson, A stochastic estimator of the trace of the influence matrix for Laplacian smoothing
splines (1990).
H. Avron and S. Toledo, Randomized algorithms for estimating the trace of an implicit symmetric positive
semi-definite matrix (2011).
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
13 / 32
Optimization problem for computing A-optimal design
OED objective function:
Ψ(w) =
Omar Ghattas (UT Austin)
nd
1 X
tr H−1 w; di )
nd i=1
Large-scale OED
February 23, 2016
14 / 32
Optimization problem for computing A-optimal design
OED objective function:
Ψ(w) =
nd X
ntr
1 X
zk , H−1 w; di ) zk
nd ntr i=1
k=1
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
14 / 32
Optimization problem for computing A-optimal design
OED objective function:
Ψ(w) =
nd X
ntr
1 X
h zk , H−1 w; di ) zk i
nd ntr i=1
{z
}
|
k=1
yik
Auxiliary variable: yik = H−1 w; di )zk
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
14 / 32
Optimization problem for computing A-optimal design
OED objective function:
Ψ(w) =
nd X
ntr
1 X
h zk , H−1 w; di ) zk i
nd ntr i=1
{z
}
|
k=1
yik
Auxiliary variable: yik = H−1 w; di )zk
OED optimization problem
nd X
ntr
1 X
hzk , yik i + γP (w)
minimize
w
nd ntr i=1
k=1
where
mMAP (w; di ) = arg min J m, w; di , i = 1, . . . , nd
m
H mMAP (w; di ), w; di yik = zk , i = 1, . . . , nd , k = 1, . . . , ntr
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
14 / 32
Application: subsurface flow
Forward problem
−∇·(em ∇u) = f
in D
u=g
on ΓD
e ∇u · n = h
on ΓN
m
u: pressure field
m: log-permeability field (inversion parameter)
Left: true parameter; Right: pressure-field
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
15 / 32
Bayesian inverse problem: Gaussian approximation
MAP point is solution to
minimize J (m, w; d) :=
m
1
1
(Bu − d)T Wσ (Bu − d) + hm − mprior , m − mprior iE
2
2
where
−∇·(em ∇u) = f
in D
u=g
on ΓD
em ∇u · n = 0
on ΓN
B is observation operator
Cameron-Martin inner-product:
E
D
−1/2
−1/2
hm1 , m2 iE = Cprior m1 , Cprior m2 ,
1/2
m1 , m2 ∈ range(Cprior )
Covariance of Gaussian approximation to posterior:
C = H(m, w; d)−1 = [∇2m J (m, w; d)]−1
with m = mMAP (w; d)
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
16 / 32
Optimization problem for A-optimal design
minimize
n
w∈[0,1]
s
nd X
ntr
1 X
hzk , yik i + γP (w)
nd ntr i=1
k=1
where for i = 1, . . . , nd , k = 1, . . . , ntr
−∇·(emi ∇ui ) = f
−∇·(emi ∇pi ) = −B ∗ Wσ (Bui − di )
−1
Cprior
(mi − mpr ) + emi ∇ui · ∇pi = 0
|
{z
}
∇m J (mi )
−∇·(e
mi
∇vik ) = ∇·(yik emi ∇ui )
−∇·(e
mi
∇qik ) = ∇·(yik emi ∇pi ) − B ∗ Wσ Bvik
−1
Cprior
yik + yik emi ∇ui · ∇pi + emi ∇vik · ∇pi + ∇ui · ∇qik = zk
|
{z
}
H(mi )yik
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
17 / 32
The Lagrangian for the OED problem
OED
L
∗
nd
:=
+
∗
∗
∗
∗
∗
w, {ui }, {mi }, {pi }, {vik }, {qik }, {yik }, {ui }, {mi }, {pi }, {vik }, {qik }, {yik }
ntr
1 XX
hzk , yik i + γP (w)
nd ntr i=1 k=1
nd
X
m
∗
∗
∗
e i ∇ui , ∇ui − f, ui − h, ui Γ
N
i=1
nd
X
m
∗
∗
∗ +
e i ∇pi , ∇pi + B Wσ (Bui − di ), pi
i=1
+
nd
X
∗
E
(mi − mpr ), mi
∗ m
+ mi e i ∇ui , ∇pi
i=1
nd ntr
X
X m
∗ m
∗ +
e i ∇vik , ∇vik + yik e i ∇ui , ∇vik
i=1 k=1
nd ntr
X
X m
∗
∗ m
∗ ∗ +
e i ∇qik , ∇qik + yik e i ∇pi , ∇qik + B Wσ Bvik , qik
i=1 k=1
nd ntr
X
X ∗ m
∗
∗ m
∗
m
+
yik e i ∇vik , ∇pi + yik , yik E + yik e i ∇ui , ∇qik + (yik yik e i ∇ui , ∇pi )
i=1 k=1
∗ − zk , yik
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
18 / 32
The adjoint OED problem and the gradient
∗
∗
∗
The adjoint OED problem for {u∗i }, {m∗i }, {p∗i }, {vik
}, {qik
}, {yik
}
∗
∗ mi
∗
B ∗ Wσ Bqik
− ∇·(yik
e ∇pi ) −∇·(emi ∇vik
)=0
−1
∗
∗
∗ m
∗
emi ∇qik
· ∇pi + Cprior
yik
+ yik
e ∇ui · ∇pi + emi ∇ui · ∇vik
=−
1
zk
nd ntr
∗
∗ m
−∇·(emi ∇qik
) − ∇·(yik
e ∇ui ) = 0
B ∗ Wσ Bp∗i − ∇·(m∗i emi ∇pi ) −∇·(emi ∇u∗i ) = b1i
−1
emi ∇p∗i · ∇pi + Cprior
m∗i + m∗i emi ∇ui · ∇pi + emi ∇ui · ∇u∗i = b2i
−∇·(emi ∇p∗i ) − ∇·(m∗i emi ∇ui ) = b3i
Gradient for the OED problem:
g(w) =
nd
X
∗
Γ−1
noise (Bui − di ) Bpi +
i=1
nd X
ntr
X
i=1 k=1
∗
∗
∗
qik
, yik
, and vik
can be eliminated:
∗
∗
qik
= −vik , yik
= −yik ,
Omar Ghattas (UT Austin)
∗
Γ−1
noise Bvik Bqik ,
Large-scale OED
∗
vik
= −qik
February 23, 2016
19 / 32
Computing the OED objective function and its gradient
Solve inverse problems: mi (w), ui (mi (w)) and pi (mi (w)), i = 1, . . . , nd
Solve for (vik , yik , qik ), i = 1, . . . , nd , k = 1, . . . , ntr :
−∇·(emi ∇vik ) = ∇·(yik emi ∇ui )
−∇·(emi ∇qik ) = ∇·(yik emi ∇pi ) − B∗ Wσ Bvik
−1
Cprior
yik + yik emi ∇ui · ∇pi + emi ∇vik · ∇pi + ∇ui · ∇qik = zk
Objective function evaluation: Ψ(w) =
Solve for
(p∗i , m∗i , u∗i ),
1
nd ntr
Pnd Pntr
i=1
k=1
i = 1, . . . , nd
B∗ Wσ Bp∗i − ∇·(m∗i emi ∇pi ) −∇·(emi ∇u∗i )
mi
e
∇p∗i
· ∇pi +
mi
−∇·(e
∇p∗i )
−1
Cprior
m∗i
−
hzk , yik i
+
m∗i em
i ∇ui
∇·(m∗i em
i ∇ui )
· ∇pi + e
mi
∇ui ·
= b1i
∗
∇ui = b2i
= b3i
Gradient:
g(w) =
nd
X
∗
Γ−1
noise (Bui − di ) Bpi −
i=1
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Large-scale OED
nd ntr
X
1 X
Γ−1 Bvik Bvik
nd ntr i=1 k=1 noise
February 23, 2016
20 / 32
Computational cost: the number of PDE solves
r: rank of Hessian of the data misfit
Independent of mesh (beyond a certain resolution)
Independent of sensor dimension (beyond a certain dimension)
Cost (in PDE solves) of evaluating OED objective function & gradient
1
Cost of solving inverse problems ∼ 2 × r × nd × Nnewton
2
Cost of computing OED objective ∼ 2 × r × ntr × nd
3
Cost of computing OED gradient ∼ 2 × ntr × nd + 2 × r × nd
Low-rank approx to misfit Hessian =⇒ Cost in (2) and (3) ∼ 2 × r × nd
OED optimization problem:
Solved via quasi-Newton interior point
Number of quasi-Newton iterations insensitive to parameter/sensor dimension
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
21 / 32
The prior and training models
Prior knowledge: parameter value at few points, correlation
truth
prior mean
prior variance
Prior mean: “smooth” least-squares fit to point measurements
N Z
2
1
αX
mpr = arg min hm, Ami +
δi (x) m(x) − mtrue (x) dx
m 2
2 i=1 D
PN
Prior covariance: Cprior = (A + α i=1 δi )−2 ,
Am = −∇ · (D∇m)
Draws from prior used to generate training data for OED
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
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Computing an optimal design with sparsification
The functions fε
Optimization problem
minimize
w
1
nd X
ntr
1 X
hzk , yik i + γPε (w)
nd ntr i=1
0.8
0.6
k=1
0.4
where
mMAP (w; di ) = arg min J m, w; di
m
H mMAP (w; di ), w; di yik = zk
0.2
0
0
0.2
0.4
0.6
0.8
1
The optimal weight vector w
1
nd = 5, ntr = 20
0.8
Continuation ⇒ `0 -penalty
0.6
Pε (w) :=
ns
X
0.4
fε (wi )
0.2
i=1
0
0
Solve the problem with ε → 0
Omar Ghattas (UT Austin)
Large-scale OED
20
40
60
80
100
February 23, 2016
23 / 32
Effectiveness of the optimal design
Numerical test: how well can we recover the “true model” from “true data”?
Optimal
Random
Truth
Designs with 10 sensors
Comparing MAP point (top row) and
posterior variance (Bottom row)
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
24 / 32
Effectiveness of the optimal design
Numerical test: how well can we recover the “true model” from “true data”?
10 sensor designs
20 sensor designs
1.2
tr Γpost (w)
1.2
Random
Optimal
1
0.8
0.6
0.2
Random
Optimal
1
0.8
0.3
0.4
0.5
0.6
0.6
0.2
0.3
Erel (w)
d = f (mtrue ) + η
Erel (w) :=
Omar Ghattas (UT Austin)
0.4
0.5
0.6
Erel (w)
Γpost (w) := H−1 (mMAP (w; d), w; d)
kmMAP (w; d) − mtrue k
kmtrue k
Large-scale OED
February 23, 2016
25 / 32
Effectiveness of the optimal design
Numerical test: how well we do on average?
10 sensor designs
20 sensor designs
1.2
Ψ(w)
1.2
Random
Optimal
1
0.8
0.6
0.2
Random
Optimal
1
0.8
0.3
0.4
0.5
0.6
0.6
0.2
0.3
Erel (w)
0
0.4
0.5
0.6
Erel (w)
nd
1 X kmMAP (w; di ) − mi k
E rel (w) = 0
nd i=1
kmi k
0
nd
1 X
tr H−1 (mMAP (w; di ), w; di )
Ψ(w) = 0
nd i=1
di generated from nd0 = 50 draws of mi from the prior
(different from the nd = 5 used to solve the OED problem)
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
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OED optimization
obj func/grad evaluation
Scalability with respect to parameter/sensor dimension
300
300
inner CG iters
outer CG
200
inner CG iters
outer CG
200
100
100
500
2,000
3,500
5,000
BFGS iterations
100
50
10
250
500
750
BFGS iterations
100
50
10
500
2,000
3,500
5,000
parameter dimension
Omar Ghattas (UT Austin)
Large-scale OED
10
10
250
500
750
sensor dimension
February 23, 2016
27 / 32
Inversion of a realistic permeability field
m
−∇ · (e ∇u) = f
in D
u=0
on ΓD
e ∇u · n = 0
on ΓN
m
ΓD : at the corners
ΓN : the rest of boundary
Point source at center
Log-permeability field from SPE Comparative Solution Project
Injection well at the center
Production wells at four corners
Parameter (and state) dimension: n = 10, 202
Potential sensor locations: ns = 128
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Large-scale OED
February 23, 2016
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SPE model: the prior
Mean
Omar Ghattas (UT Austin)
Standard deviation
Large-scale OED
Random draw
February 23, 2016
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SPE model: inference with the optimal design
Posterior samples
Truth
Posterior mean
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Large-scale OED
February 23, 2016
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SPE model: Compare optimal vs random design
Optimal
average var = 1.8824
Random
average var = 2.4722
average var = 2.7206
Designs with 32 sensors
Observed data collected using true parameter on high resolution mesh
(n ∼ 2.4 × 105 )
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
31 / 32
Challenges revisited
Infinite-dimensional inference
Proper choice of prior, mass-weighted inner-product
Need trace of posterior covariance (inverse of Hessian, large, dense, expensive
matvecs)
Randomized trace-estimator, Gaussian approximation in nonlinear case
In nonlinear inverse problem, Hessian depends on data (not available a priori)
Random draws from prior to generate training data
Optimal design has combinatorial complexity in general
w-weighted formulation, relax weights 0 ≤ wj ≤ 1, continuation to get 0–1
Conventional OED algorithms intractable for large-scale problems
Infinite-dimensional formulation, gradient based optimization, scalable algorithms
Gaussian assumption (on posterior of inverse problem) remains via linearized
parameter-to-observable map; current worked aimed at higher-order approximation
A. Alexanderian, N. Petra, G. Stadler, and O. Ghattas, A fast and scalable method for A-optimal design of
experiments for infinite-dimensional Bayesian nonlinear inverse problems, SIAM Journal on Scientific
Computing, 38(1):A243–A272, 2016. http://dx.doi.org/10.1137/140992564
Omar Ghattas (UT Austin)
Large-scale OED
February 23, 2016
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