Developing fast low-rank tensor methods for solving PDEs with uncertain coefficients

Low-rank tensors for PDEs with
uncertain coefficients
Alexander Litvinenko
Center for Uncertainty
Quantification
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http://sri-uq.kaust.edu.sa/
Extreme Computing Research Center, KAUST
Alexander Litvinenko Low-rank tensors for PDEs with uncertain coefficients

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The structure of the talk
Part I (Stochastic Forward Problem):
1. Motivation
2. Elliptic PDE with uncertain coefﬁcients
3. Discretization and low-rank tensor approximations
Part II (Stochastic Inverse Problem via Bayesian Update):
1. Bayesian update surrogate
2. Examples
Part III (Different Examples relevant for BGS)
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My interests and collaborations
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Motivation to do Uncertainty Quantiﬁcation (UQ)
Motivation: there is an urgent need to quantify and reduce the
uncertainty in multiscale-multiphysics applications.
UQ and its relevance: Nowadays computational predictions are
used in critical engineering decisions. But, how reliable are
these predictions?
Example: Saudi Aramco currently has a simulator,
GigaPOWERS, which runs with 9 billion cells. How sensitive
are these simulations w.r.t. unknown reservoir properties?
My goal is systematic, mathematically founded, develop-
ment of UQ methods and low-rank algorithms relevant for
applications.
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PDE with uncertain coefficient
Consider
− div(κ(x, ω) u(x, ω)) = f(x, ω) in G × Ω, G ⊂ Rd ,
u = 0 on ∂G,
where κ(x, ω) - uncertain diffusion coefficient.
1. Efficient Analysis of High Dimensional Data in Tensor
Formats, Espig, Hackbusch, Litvinenko., Matthies, Zander,
2012.
2. Efficient low-rank approx. of the stoch. Galerkin ma-
trix in tensor formats, Wähnert, Espig, Hackbusch, A.L.,
Matthies, 2013.
3. PCE of random coefficients and the solution of stochas-
tic PDEs in the Tensor Train format, Dolgov, Litvinenko,
Khoromskij, Matthies, 2016.
4. Application of H-matrices for computing the KL expan-
sion, Khoromskij, Litvinenko, Matthies Computing 84 (1-2),
49-67, 2009
0 0.5 1
-20
0
20
40
60
50 realizations of the solution u,
the mean and quantiles
Related work by R. Scheichl, Chr. Schwab, A. Teckentrup, F. Nobile, D. Kressner,...
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Canonical and Tucker tensor formats
[Pictures are taken from B. Khoromskij and A. Auer lecture course]
Storage: O(nd ) → O(dRn) and O(Rd + dRn).
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Karhunen Loève and Polynomial Chaos Expansions
Apply both
Truncated Karhunen Loève Expansion (KLE):
κ(x, ω) ≈ κ0(x) +
L
j=1
κjgj(x)ξj(θ(ω)),
where θ = θ(ω) = (θ1(ω), θ2(ω), ..., ),
ξj(θ) = 1
κj G (κ(x, ω) − κ0(x)) gj(x)dx.
Truncated Polynomial Chaos Expansion (PCE)
κ(x, ω) ≈ α∈JM,p
κ(α)(x)Hα(θ)
ξj(θ) ≈ α∈JM,p
ξ
(α)
j Hα(θ).
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Discretization of elliptic PDE
Ku = f, where
K :=
L
=1
K ⊗
M
µ=1
∆ µ, K ∈ RN×N
, ∆ µ ∈ RRµ×Rµ
,
u :=
r
j=1
uj ⊗
M
µ=1
ujµ, uj ∈ RN
, ujµ ∈ RRµ
,
f :=
R
k=1
fk ⊗
M
µ=1
gkµ, fk ∈ RN
, gkµ ∈ RRµ
.
Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats, Wähnert, Espig,
Hackbusch, Litvinenko, Matthies, 2013.
In [2] we analyzed tensor ranks (compression properties) of the
stochastic Galerkin operator K.
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Numerical Experiments
2D L-shape domain, N = 557 dofs.
Total stochastic dimension is Mu = Mk + Mf = 20, there are
|JM,p| = 231 PCE coefﬁcients
u =
231
j=1
uj,0 ⊗
20
µ=1
ujµ ∈ R557
⊗
20
µ=1
R3
.
Tensor u has 320 · 557 ≈ 2 · 1012 entries ≈ 16 TB of memory.
Instead we store only 231 · (557 + 20 · 3) ≈ 144000 entries
≈ 1.14 MB.
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Level sets
Now we compute level sets
{ui : ui > b · max
i
u},
i := (i1, ..., iM+1)
for b ∈ {0.2, 0.4, 0.6, 0.8}.
The computing time for each b was 10 minutes.
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Part II
Part II: Developing of cheap Bayesian update
surrogate
1. Rosic, Litvinenko, Pajonk, Matthies J. Comp. Ph. 231 (17), 5761-5787, 2013
2. Inverse Problems in a Bayesian Setting, Matthies, Zander, Pajonk, Rosic, Litvinenko. Comp. Meth.for
Solids and Fluids Multiscale Analysis, 2016
Related work by A. Stuart, Chr. Schwab, A. El Sheikh, Y.
Marzouk, H. Najm, O. Ernst

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Numerical computation of Bayesian Update surrogate
Notation: ˆy – measurements from engineers, y(ξ) – forecast
from the simulator, ε(ω) – the Gaussian noise.
Look for ϕ such that q(ξ) = ϕ(z(ξ)), z(ξ) = y(ξ) + ε(ω):
ϕ ≈ ˜ϕ =
α∈Jp
ϕαΦα(z(ξ))
and minimize q(ξ) − ˜ϕ(z(ξ)) 2
L2
, where Φα are known
polynomials (e.g. Hermite).
Taking derivatives with respect to ϕα:
∂
∂ϕα
q(ξ) − ˜ϕ(z(ξ)), q(ξ) − ˜ϕ(z(ξ)) = 0 ∀α ∈ Jp
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Numerical computation of NLBU
∂
∂ϕα
E

q2
(ξ) − 2
β∈J
qϕβΦβ(z) +
β,γ∈J
ϕβϕγΦβ(z)Φγ(z)


= 2E

−qΦα(z) +
β∈J
ϕβΦβ(z)Φα(z)


= 2


β∈J
E [Φβ(z)Φα(z)] ϕβ − E [qΦα(z)]

 = 0 ∀α ∈ J .
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Now, rewriting the last sum in a matrix form, obtain the linear
system of equations (=: A) to compute coefﬁcients ϕβ:



... ... ...
... E [Φα(z(ξ))Φβ(z(ξ))]
...
... ... ...







...
ϕβ
...



 =




...
E [q(ξ)Φα(z(ξ))]
...



 ,
where α, β ∈ J , A is of size |J | × |J |.
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Finally, the assimilated parameter qa will be
qa = qf + ˜ϕ(ˆy) − ˜ϕ(z), (1)
z(ξ) = y(ξ) + ε(ω),
˜ϕ = β∈Jp
ϕβΦβ(z(ξ))
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Example: 1D elliptic PDE with uncertain coeffs
− · (κ(x, ξ) u(x, ξ)) = f(x, ξ), x ∈ [0, 1]
+ Dirichlet random b.c. g(0, ξ) and g(1, ξ).
3 measurements: u(0.3) = 22, s.d. 0.2, x(0.5) = 28, s.d. 0.3,
x(0.8) = 18, s.d. 0.3.
κ(x, ξ): N = 100 dofs, M = 5, number of KLE terms 35, beta distribution for κ, Gaussian covκ, cov.
length 0.1, multi-variate Hermite polynomial of order pκ = 2;
RHS f(x, ξ): Mf = 5, number of KLE terms 40, beta distribution for κ, exponential covf , cov. length 0.03,
multi-variate Hermite polynomial of order pf = 2;
b.c. g(x, ξ): Mg = 2, number of KLE terms 2, normal distribution for g, Gaussian covg , cov. length 10,
multi-variate Hermite polynomial of order pg = 1;
pφ = 3 and pu = 3
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Example: Updating of the parameter
0 0.5 1
0
0.5
1
1.5
0 0.5 1
0
0.5
1
1.5
Figure: Prior and posterior (updated) parameter κ.
Collaboration with Y. Marzouk, MIT, and TU Braunschweig.
Together with H. Najm, Sandia Lab, we try to compare our
technique with his advanced MCMC technique for chemical
combustion eqn.
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Example: updating of the solution u
0 0.5 1
-20
0
20
40
60
0 0.5 1
-20
0
20
40
60
0 0.5 1
-20
0
20
40
60
0 0.5 1
-20
0
20
40
60
0 0.5 1
-20
0
20
40
60
Figure: Original and updated solutions, mean value plus/minus 1,2,3
standard deviations. Number of available measurements {0, 1, 2, 3, 5}
[graphics are built in the stochastic Galerkin library sglib, written by E. Zander in TU Braunschweig]
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Part III: My contribution to MUNA project
MUNA=Management and minimization of Uncertainties in
Numerical Aerodynamics.
1. Quantiﬁcation of airfoil geometry-induced aerodynamic
uncertainties-comparison of approaches, Liu, Litvinenko,
Schillings, Schulz, JUQ 2017
2. Numerical Methods for Uncertainty Quantiﬁcation and
Bayesian Update in Aerodynamics Litvinenko, Matthies,
chapter in Springer book, Vol 122, pp 262-285, 2013.

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Example: uncertainties in free stream turbulence
α
v
v
u
u’
α’
v1
2
Random vectors v1(θ) and v2(θ) model free stream turbulence
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Example: 3sigma intervals
Figure: 3σ interval, σ standard deviation, in each point of RAE2822
airfoil for the pressure (cp) and friction (cf) coefﬁcients.
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Mean and variance of density, tke, xv, zv, pressure
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Example: Kriging and geostat. optimal design
Domain: 20m × 20m × 20m, 25, 000 × 25, 000 × 25, 000 dofs,
4000 measurements.
Log-Permeability. Color scale: showing the 95% conﬁdence interval.
Kriging and spatial design accelerated by orders of magnitude:
Combining low-rank with FFT, Nowak, Litvinenko, Mathemati-
cal Geosciences 45 (4), 411-435, 2013
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Numerics on computer with 16GB RAM:
1. 2D Kriging with 270 million estimation points and 100
measurement values (0.25 sec.),
2. to compute the estimation variance (< 1 sec.),
3. to evaluate the spatial average of the estimation variance
(the A-criterion of geostat. optimal design) for 2 · 1012
estim. points (30 sec.),
4. to compute the C-criterion of geostat. optimal design for
2 · 1015 estim. points (30 sec.),
5. solve 3D Kriging problem with 15 · 1012 estim. points and
4000 measurement data values (20 sec.)
Collaboration with Stuttgart University, hydrology and
geosciences.
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Example from spatial statistics
Goal: To improve estimation of un-
known statistical parameters in a spa-
tial soil moisture ﬁeld, Mississippi
basin, [−85◦ − 73◦] × [32◦, 43◦].
Log-likelihood function with C = e−
|x−y|
θ and Z available
(satellite) data:
L(θ) = −
n
2
log(2π) −
1
2
log |C(θ)| −
1
2
Z C(θ)−1
Z.
Collaboration with statisticians: M. Genton, Y. Sun, R. Huser, H. Rue from KAUST.
n = 512K, matrix setup 261 sec., compression rate 99.98% (0.4 GB against 2006 GB). H-LU is done in
843 sec., error 2 · 10−3
.
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Conclusion
Introduced:
1. Low-rank tensor methods to solve PDEs with uncertain
coefficients,
2. Post-processing in low-rank tensor format, computing level
sets
3. Bayesian update surrogate ϕ (as a linear, quadratic,...
approximation)
4. Quantification of uncertainties in Numerical Aerodynamics
5. Applications in geosciences
6. Estimating unknown coefficients in spatial statistics
(moisture example)
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Thank you
Thank you!
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Possible collaboration
1. Dominic Breit (error estimates for UQ applications to
balance statistical and discretization errors)
2. Gabriel Lord (num. meth. for PDEs with uncertainties;
combination of multiscale methods, UQ techniques and
Bayesian inference for reservoir modeling; low-rank tensor
methods for high-dimensional problems).
3. Lehel Banjai (computation of electromagnetic ﬁelds
scattered from dielectric objects of uncertain shapes;
balancing of the Runge-Kutta time discretization step and
the H-matrix approximation rank in time-dependent PDEs),
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Possible collaboration
1. BGS (CO2 storage, reservoir modeling, spatial statistics in
geology/geophysics),
2. Lyell Institute (subsurface flow under uncertainties, EOR,
Bayesian techniques for data assimilations)
3. EGIS:
3.1. Mike Christie (reservoir modeling under uncertainties,
EOR, seismic wave propagation in uncertain media)
3.2. Vasily Demyanov (uncertainty quantification and
low-rank approximations in geostatistics)
3.3. Dan Arnold (modeling of random geology, multi-scale,
Bayesian inference)
3.4. Ahmed El Sheikh (fast Bayesian update methods,
advanced UQ, surrogates for BU, big data from spatial
statistics).
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My experience since 2002
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Explanation of Bayesian Update surrogate
Let the stochastic model of the measurement is given by
y = M(q) + ε, ε -measurement noise
Best estimator ˜ϕ for q given z, i.e.
˜ϕ = argminϕ E[ q(·) − ϕ(z(·)) 2
2].
The best estimate (or predictor) of q given the
measurement model is
qM(ξ) = ˜ϕ(z(ξ))).
The remainder, i.e. the difference between q and qM, is
given by
q⊥
M(ξ) = q(ξ) − qM(ξ),
Due to the minimisation property of the MMSE
estimator—orthogonal to qM(ξ), i.e. cov(q⊥
M, qM) = 0.
[Thanks to Elmar Zander, TU Braunschweig]
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In other words,
q(ξ) = qM(ξ) + q⊥
M(ξ) (2)
yields an orthogonal decomposition of q.
Actual measurement ˆy, prediction ˆq = ˜ϕ(ˆy). Part qM of q
can be “collapsed” to ˆq. Updated stochastic model q is
thus given by
q (ξ) = ˆq + q⊥
M(ξ) (3)
q (ξ) = q(ξ) + ( ˜ϕ(ˆy) − ˜ϕ(z(ξ))). (4)
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Future plans, Idea N1
Possible collaboration work 1 To develop a low-rank adaptive
goal-oriented Bayesian update technique. The solution of the forward
and inverse problems will be considered as a whole adaptive
process, controlled by error/uncertainty estimators.
z
(y - z) q
f ε
forward update
low-rank and adaptive
y
f z
(y - z)
ε
forward
y q.....
low-rank and adaptive
... q
update
Stochastic forward spatial discret.
stochastic discret.
low-rank approx.
Inverse problem
Errors
inverse operator approx.

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Edge between Green functions in PDEs and covariance
matrices.
Possible collaboration with statistical group, Doug Nychka
(NCAR), Haavard Rue
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Data assimilation techniques, Bayesian update surrogate.
Develop non-linear, non-Gaussian Bayesian update
approximation for gPCE coefﬁcients.
Possible collaboration with Kody Law (OAK NL), Y. Marzouk
(MIT), H. Najm (Sandia NL), TU Braunschweig and KAUST.

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Example: Canonical rank d, whereas TT rank 2
d-Laplacian over uniform tensor grid. It is known to have the
Kronecker rank-d representation,
∆d = A⊗IN ⊗...⊗IN +IN ⊗A⊗...⊗IN +...+IN ⊗IN ⊗...⊗A ∈ RI⊗d ⊗I⊗d
(5)
with A = ∆1 = tridiag{−1, 2, −1} ∈ RN×N, and IN being the
N × N identity. Notice that for the canonical rank we have rank
kC(∆d ) = d, while TT-rank of ∆d is equal to 2 for any
dimension due to the explicit representation
∆d = (∆1 I) ×
I 0
∆1 I
× ... ×
I 0
∆1 I
×
I
∆1
(6)
where the rank product operation ”×” is deﬁned as a regular
matrix product of the two corresponding core matrices, their
blocks being multiplied by means of tensor product. The similar
bound is true for the Tucker rank rankTuck (∆d ) = 2.

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Advantages and disadvantages
Denote k - rank, d-dimension, n = # dofs in 1D:
1. CP: ill-posed approx. alg-m, O(dnk), hard to compute
approx.
2. Tucker: reliable arithmetic based on SVD, O(dnk + kd )
3. Hierarchical Tucker: based on SVD, storage O(dnk + dk3),
truncation O(dnk2 + dk4)
4. TT: based on SVD, O(dnk2) or O(dnk3), stable
5. Quantics-TT: O(nd ) → O(dlogq
n)

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How to compute the variance in CP format
Let u ∈ Rr and
˜u := u − u
d
µ=1
1
nµ
1 =
r+1
j=1
d
µ=1
˜ujµ ∈ Rr+1, (7)
then the variance var(u) of u can be computed as follows
var(u) =
˜u, ˜u
d
µ=1 nµ
=
1
d
µ=1 nµ


r+1
i=1
d
µ=1
˜uiµ

 ,


r+1
j=1
d
ν=1
˜ujν


=
r+1
i=1
r+1
j=1
d
µ=1
1
nµ
˜uiµ, ˜ujµ .
Numerical cost is O (r + 1)2 · d
µ=1 nµ .

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Computing QoI in low-rank tensor format
Now, we consider how to
ﬁnd ‘level sets’,
for instance, all entries of tensor u from interval [a, b].

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Definitions of characteristic and sign functions
1. To compute level sets and frequencies we need
characteristic function.
2. To compute characteristic function we need sign function.
The characteristic χI(u) ∈ T of u ∈ T in I ⊂ R is for every multi-
index i ∈ I pointwise defined as
(χI(u))i :=
1, ui ∈ I,
0, ui /∈ I.
Furthermore, the sign(u) ∈ T is for all i ∈ I pointwise defined
by
(sign(u))i :=



1, ui > 0;
−1, ui < 0;
0, ui = 0.
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sign(u) is needed for computing χI(u)
Lemma
Let u ∈ T , a, b ∈ R, and 1 = d
µ=1
˜1µ, where
˜1µ := (1, . . . , 1)t ∈ Rnµ .
(i) If I = R<b, then we have χI(u) = 1
2 (1 + sign(b1 − u)).
(ii) If I = R>a, then we have χI(u) = 1
2(1 − sign(a1 − u)).
(iii) If I = (a, b), then we have
χI(u) = 1
2 (sign(b1 − u) − sign(a1 − u)).
Computing sign(u), u ∈ Rr , via hybrid Newton-Schulz iteration
with rank truncation after each iteration.
Quantiﬁcation
ation Logo Lock-up
35

4*
Level Set, Frequency
Definition (Level Set, Frequency)
Let I ⊂ R and u ∈ T . The level set LI(u) ∈ T of u respect to I is
pointwise defined by
(LI(u))i :=
ui, ui ∈ I ;
0, ui /∈ I ,
for all i ∈ I.
The frequency FI(u) ∈ N of u respect to I is defined as
FI(u) := # supp χI(u).
Quantification
ation Logo Lock-up
36

4*
Computation of level sets and frequency
Proposition
Let I ⊂ R, u ∈ T , and χI(u) its characteristic. We have
LI(u) = χI(u) u
and rank(LI(u)) ≤ rank(χI(u)) rank(u).
The frequency FI(u) ∈ N of u respect to I is
FI(u) = χI(u), 1 ,
where 1 = d
µ=1
˜1µ, ˜1µ := (1, . . . , 1)T ∈ Rnµ .
Quantiﬁcation
ation Logo Lock-up
37

4*
Definition of tensor of order d
Tensor of order d is a multidimensional array over a d-tuple
index set I = I1 × · · · × Id ,
A = [ai1...id
: i ∈ I ] ∈ RI
, I = {1, ..., n }, = 1, .., d.
A is an element of the linear space
Vn =
d
=1
V , V = RI
equipped with the Euclidean scalar product ·, · : Vn × Vn → R,
defined as
A, B :=
(i1...id )∈I
ai1...id
bi1...id
, for A, B ∈ Vn.
Quantification
ation Logo Lock-up
38

4*
Examples of rank-1 and rank-2 tensors
Rank-1:
f(x1, ..., xd ) = exp(f1(x1) + ... + fd (xd )) = d
j=1 exp(fj(xj))
Rank-2: f(x1, ..., xd ) = sin( d
j=1 xj), since
2i · sin( d
j=1 xj) = ei d
j=1 xj
− e−i d
j=1 xj
Rank-d function f(x1, ..., xd ) = x1 + x2 + ... + xd can be
approximated by rank-2: with any prescribed accuracy:
f ≈
d
j=1(1 + εxj)
ε
−
d
j=1 1
ε
+ O(ε), as ε → 0
Quantiﬁcation
ation Logo Lock-up
39

4*
Conditional probability and expectation
Classically, Bayes’s theorem gives conditional probability
P(Iq|Mz) =
P(Mz|Iq)
P(Mz)
P(Iq) (or πq(q|z) =
p(z|q)
Zs
pq(q));
Expectation with this posterior measure is conditional
expectation.
Kolmogorov starts from conditional expectation E (·|Mz),
from this conditional probability via P(Iq|Mz) = E χIq
|Mz .
Quantiﬁcation
ation Logo Lock-up
40

4*
Conditional expectation
The conditional expectation is deﬁned as
orthogonal projection onto the closed subspace L2(Ω, P, σ(z)):
E(q|σ(z)) := PQ∞ q = argmin˜q∈L2(Ω,P,σ(z)) q − ˜q 2
L2
The subspace Q∞ := L2(Ω, P, σ(z)) represents the available
information.
The update, also called the assimilated value
qa(ω) := PQ∞ q = E(q|σ(z)), is a Q-valued RV
and represents new state of knowledge after the measurement.
Doob-Dynkin: Q∞ = {ϕ ∈ Q : ϕ = φ ◦ z, φ measurable}.
Quantiﬁcation
ation Logo Lock-up
41

Developing fast low-rank tensor methods for solving PDEs with uncertain coefficients

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