The Reduced-Basis (RB)
method for Uncertainty
Quantification (UQ)
Sébastien Boyaval1,2
1
(collaborations with C. Le Bris, T. Lelièvre, Y. Maday, N. C. Nguyen and A. T. Patera )
Univ. Paris Est, Laboratoire d’hydraulique Saint-Venant (EDF R&D - Ecole des Ponts Paristech - CETMEF), Chatou, France
2
INRIA, MICMAC team–project, Rocquencourt, France
ICMS, Edinburgh May 2010
Outline
1
Introduction: motivation for UQ with RB
2
Methodology: principles of standard RB
3
Application: RB implementation in a Robin BVP
4
Extension: standard RB method for parameter estimation
5
Advanced: combination with RB-like variance reduction
6
Summary and bibliography
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
2 / 17
UQ: BVP with random parameter
uncertainty propagation – (Bayesian) parameter estimation
For PDEs with stochastic coefficients, e.g. diffusion A
− div(A∇u) = f
(1)
how to compute expensive (high-dim.) solutions to (1) ?
1
Sample u (+ outputs: moments. . . ) by Monte-Carlo (MC)
2
Accelerates the “many-query” problem by RB method
Notice that:
RB method can also reduce other discretizations (e.g.
quad. formula invoking indep. parametrized problems)
naive MC is often less favourable than deterministic quad.
(in small dim.), but OK with efficient variance reduction
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
3 / 17
Outline
1
Introduction: motivation for UQ with RB
2
Methodology: principles of standard RB
3
Application: RB implementation in a Robin BVP
4
Extension: standard RB method for parameter estimation
5
Advanced: combination with RB-like variance reduction
6
Summary and bibliography
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
3 / 17
Problem: compute many expensive input-output relationships
For many input parameters µ, compute an expensive u(µ) ∈ X .
RB = project u(µ) on best linear space spanned by “snapshots”
1 identify “offline”: XN = Span u(µN
n ), n = 1, . . . , N
2
compute fast at any µ: uN (µ) ∈ arginfku(µ) − v kµ,X
v ∈XN
Remarks:
“Online” computations can be certified supku(µ) − uN (µ)kµ,X ≤ ε
µ
RB (Patera-Madayr) ought to be a good strategy if. . .
S
{u(µ)} is close to a small-dimensional vector space
µ
(offline+online) effort = less expensive at fixed accuracy ε
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
4 / 17
Example: µ-elliptic variational pb. a(u(µ), v ; µ) = l(v ) , ∀v ∈ X
Given µ, find u(µ) ∈ X solution to − div(A(µ)∇u(µ)) = f + BC
uN best approx. in energy (Galerkin): k · kµ,X =
p
a(·, ·; µ)
XN N-linear space minimizing L∞ -width: supku(µ) − uN (µ)kµ,X
µ
goal-oriented cases (like homogenization) → RB also for adjoint eq.
This is a “best N-linear space” approximation problem:
computing minimizers of inf
supku(µ) − uN (µ)kµ,X hard !
µ1 ,...,µN
µ
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
5 / 17
Practical approach
Assume a good discretization XN of X (d.o.f. N 1)
1
2
a posteriori estimators ∆N ,N (µ) ≥ kuN (µ) − uN (µ)kµ,X ,
Greedy algorithms selecting iteratively (n = 1, . . .) µN
n = µn
within a training sample of µ while sup ∆N ,N (µ) ≤ ε
µ∈sample
!
µ1 = rand(); µn+1 ∈
sup ∆N ,n (µ)
, n = 1, . . . , N − 1
µ∈sample
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
6 / 17
Outline
1
Introduction: motivation for UQ with RB
2
Methodology: principles of standard RB
3
Application: RB implementation in a Robin BVP
4
Extension: standard RB method for parameter estimation
5
Advanced: combination with RB-like variance reduction
6
Summary and bibliography
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
6 / 17
A simple uncertainty propagation problem
−div (A(x)∇u(x, ω)) = 0 in D
(2)
n(x) A(x)∇u(x, ω) + B(x, ω) u(x, ω) = g(x) on ∂D
(3)
Z
s(ω) =
gu(·, ω)
B = B1ΓB > 0 g = g1ΓR ∈ L2 (∂D)
ΓR
k (x)
0
A(x) =
0
k (x)
(
k1 in D1
ΓB D1
k=
k2 in D2
T
∂D = ΓR ∪ ΓN ∪ ΓB
ΓN
D2
(ΓR ∩ ΓN ∩ ΓB = ∅)
ΓR
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
7 / 17
RB technicalities
parametrically-affine formulation
Sample output distribution: MC with B m (·, ω), m = 1 . . . M
Given B m (·, ω), accurate approx.: Finite-Element uN ∈ XN
a(uN , vN ; k1 , k2 , B(·, ω)) = l(vN ; g)
|
{z
}
∀vN ∈ XN ⊂ X := H 1 (D)
µ
Issue: many realizations (N 1 d.o.f.) → RB(µ)
Efficient reduction with RB to XN ⊂ XN needs
a fast projector in XN (Galerkin)
a fast a posteriori error estimator for kuN − uN kX
B(·, ω) = infinitely many coefficients ! ⇒ e.g.
parametrically-affine decomposition with Karhunen–Loève
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
8 / 17
Karhunen–Loève expansion of random input
B(x, ω) = B̄
!
K p
X
λk Φk (x)Zk (ω)
G(x) + Υ
k =1
R
B̄, Υ = positive amplitude parameters ( ∂D G = 1)
K = rank (possibly ∞) of covariance operator for B(x, ω). . .
R
. . . with eigenpairs B̄Υ2 λk , Φk (x) k ( ∂D Var(B) = B̄Υ2 )
(Zk (ω))1≤k ≤K = L2P (Ω)-orthonormal random variables
X p
λk Zk (ω) bk (u, v )
⇒ a(u, v ; µ) ≡ k1 a1 (u, v )+k2 a2 (u, v )+
{z
}
k |
µk (ω)
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
9 / 17
Mathematical technicalities
some limitations (at least for simple, rigorous error estimators)
RB method applies to truncated problems
with KL at finite order K ≤ K → ∆N,K ((µk )k ≤K )
Only for random fields B ∈ L∞
(⇒ well-posedness + existence of KL expansion)
with uniformly converging KL expansions
P√
(⇐ |Zk |, kΦk k uniformly bounded,
λk < ∞)
Only when B and KL expansions BK > B− uniformly
(⇐ Υ ≤ Υmax )
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
10 / 17
Example of reduced basis
H 1 orthonormalized for well-conditioned reduced problems
Greedy algorithm : solutions generated by gaussian covariance kernels exp(−|x − y |2 /δ 2 )
k1 = 1, k2 ∈ (.1, 10), B̄ = (.1, 1), G(x) ≡ 1, δ = .5, K = 25, Υ = .058, Zk ∼ U (−1, 1), ∀k ≤ K = 25
[3] S. B. C. Le Bris Y. Maday N.C. Nguyen A.T. Patera, A Reduced Basis Approach for Variational Problems with
Stochastic Parameters: Application to Heat Conduction with Variable Robin Coefficient,CMAME 198,2009.
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
11 / 17
Output: response surfaces
m → s m → E [s
sample (Zkm )k ≤K → uN,K
M N,K ] =
N,K
1
M
PM
m
m=1 sN,K
−3
3.702
4.6
3.701
4.55
x 10
4.5
3.7
20
4.45
3.699
4.4
15
4.35
10
3.698
3.697
4.3
3.696
4.25
3.695
4.2
3.694
0
5
0.8
0.6
2000
4000
M
6000
8000
10000
0
2000
4000
M
6000
8000
10000
Bi
0.4
0.2
2
4
6
8
10
κ
Figure: Mean EM [sN,K ] and variance VM [sN,K ] w.r.t. M at
k2 = 2.0, B̄ = 0.5 and variations of mean with (k2 , B̄).
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
12 / 17
Error bounds (global = truncation + reduction)
1
1
10
10
0
10
0
0
−1
10
−2
−3
2
4
8
N
10
12
14
0
2
4
−1
K =5
K = 10
K = 15
K = 20
−6
6
8
N
10
12
14
10
2
4
6
8
10
12
14
8
10
12
14
N
0
10
10
−1
10
−2
10
−2
10
−3
10
−2
10
ΔsV [sN,K ](2, 0.5)
ΔtV [sN,K ](2, 0.5)
10
ΔoV [sN,K ](2, 0.5)
10
0
10
−4
10
−3
10
K =5
K = 10
K = 15
K = 20
−4
10
K =5
K = 10
K = 15
K = 20
−3
6
−4
10
−2
10
K =5
K = 10
K = 15
K = 20
−2
10
−1
10
10
10
ΔsE [sN,K ](2, 0.5)
10
ΔtE [sN,K ](2, 0.5)
ΔoE [sN,K ](2, 0.5)
10
2
4
−4
6
8
N
10
12
14
10
2
−6
K =5
K = 10
K = 15
K = 20
4
10
6
8
N
10
12
14
2
K =5
K = 10
K = 15
K = 20
4
6
N
Figure: Error bounds for (a) E(S) and (b) V (S) w.r.t. N, K .
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
13 / 17
Outline
1
Introduction: motivation for UQ with RB
2
Methodology: principles of standard RB
3
Application: RB implementation in a Robin BVP
4
Extension: standard RB method for parameter estimation
5
Advanced: combination with RB-like variance reduction
6
Summary and bibliography
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
13 / 17
Classical RB in parameter estimation
One approach to parameter estimation with Bayesian inferences
−div (A(x)∇u(x, ω)) = 0 in D
(4)
n(x) A(x)∇u(x, ω) + B u(x, ω) = g(x) on ∂D
(5)
Z
s(ω) =
gu(·, ω)
B = B1ΓB > 0 g = g1ΓR ∈ L2 (∂D)
ΓR
k (x)
0
A(x) =
0
k (x)
(
k1
in D1
ΓB D1
k=
k2 (ω) in D2
T
∂D = ΓR ∪ ΓN ∪ ΓB
ΓN
D2
(ΓR ∩ ΓN ∩ ΓB = ∅)
ΓR
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
14 / 17
Parameter estimation from Bayesian inference
The MMSE case
Standard RB useful to MMSE e.g.
(exactly like for statistics of uncertainty propagation in PDEs !)
k̂2 (s ) = E(k2 |s + ξ = s ) =
?
?
Z
k2 π(k2 |s(u( k2 )) + ξ = s? )dk2
|{z}
µ
where observations s + ξ = l ? are spoiled by ξ ∼ f
and a prior k2 ∼ π gives a Bayes formula
π(k2 |s + ξ = s? ) =
f (s? − s|k2 )π(k2 )
P(s + ξ = s? )
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
15 / 17
RB → OK with deterministic quadrature for MMSE.
See [4] (parametrized parabolic PDEs).
Nguyen, N.C., Rozza, G., Huynh, D.B.P., Patera, A.T. Reduced basis approximation and a posteriori error estimation
for parametrized parabolic pdes; application to real-time bayesian parameter estimation. Computational Methods for
Large Scale Inverse Problems and Uncertainty Quantification, John Wiley & Sons, UK (2009).
On-going with Monte-Carlo (higher dimensions)
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
16 / 17
Outline
1
Introduction: motivation for UQ with RB
2
Methodology: principles of standard RB
3
Application: RB implementation in a Robin BVP
4
Extension: standard RB method for parameter estimation
5
Advanced: combination with RB-like variance reduction
6
Summary and bibliography
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
16 / 17
MC acceleration
reduce statistical error
Response surfaces, parameter estimation: many MC samples
→ opportunity for RB ideas in variance reduction too !
First applied to parametrized SDEs (rheology, finance) [2]
S. B., T. Lelièvre A variance reduction method for parametrized stochastic differential equations using the reduced
basis paradigm CMS 8 spec. iss. (P. Zhang ed.), 2010 (arXiv:0906.3600).
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
17 / 17
Outline
1
Introduction: motivation for UQ with RB
2
Methodology: principles of standard RB
3
Application: RB implementation in a Robin BVP
4
Extension: standard RB method for parameter estimation
5
Advanced: combination with RB-like variance reduction
6
Summary and bibliography
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
17 / 17
Computational reductions and RB ideas
A general philosophy able to adapt
UQ: many reduction opportunities
Efficient certified reduction possible with the
(now standard) RB method [Patera-Madayr]
Combination with fast MC approach possible with
control-variates (on-going)
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
18 / 17
For Further Reading I
S.B., C. Le Bris, T. Lelièvre, Y. Maday, N.C. Nguyen and
A.T. Patera
Reduced basis techniques for stochastic problems
ArCME special issue (E. Cueto, F. Chinesta, P. Ladeveze
and A. Nouy ed.), 2010 (arXiv:1004.0357).
S. B., T. Lelièvre
A variance reduction method for parametrized stochastic
differential equations using the reduced basis paradigm
CMS 8 spec. iss. (P. Zhang ed.), 2010 (arXiv:0906.3600).
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
18 / 17
For Further Reading II
S. B., C. Le Bris, Y. Maday, N.C. Nguyen and A.T. Patera
A Reduced Basis Approach for Variational Problems with
Stochastic Parameters: Application to Heat Conduction
with Variable Robin Coefficient
CMAME 198(41–44):3187–3206, 2009.
N.C.Nguyen, G. Rozza, D.B.P. Huynh, A.T. Patera
Reduced basis approximation and a posteriori error
estimation for parametrized parabolic pdes; application to
real-time bayesian parameter estimation.
Computational Methods for Large Scale Inverse Problems
and Uncertainty Quantification, John Wiley & Sons, UK
(2009)
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
19 / 17
Outline
7
Appendix
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
19 / 17
Parametrized r.v. Z µ ∈ L2 (Ω) in Monte-Carlo
Goal: compute expectation E(Z µ ) for many µ.
Monte-Carlo with confidence intervals (CLT+Slutsky) ∀a > 0
EM (Z µ ) :=
M
1 X µ P−a.s.
Zm −−−−→ E(Z µ )
M
M→∞
VarM (Z µ ) = . . .
m=1
r
P
|EM (Z ) − E(Z )| ≤ a
µ
µ
VarM (Z µ )
M
!
−→
M→∞
Z
a
−a
2
e−x /2
√
dx
2π
Faster MC with variance reduced by control variates Y µ :
Compute E(Z µ ) = E(Z µ − Y µ ) + E(Y µ ) where
E(Y µ ) is known (here E(Y µ ) = 0)
Var (Z µ ) ≥ Var (Z µ − Y µ ).
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
18 / 17
Control variates: practical variance reduction
Ideally Y µ = Z µ − E(Z µ ) ⇒ Var (Z µ − Y µ ) = 0, but in practice:
Compute Ỹ µ ≈ Y µ minimizing
2
2
Var (Z µ − Ỹ µ ) = E((Z µ − E(Z µ )) − Ỹ µ ) = E(Y µ − Ỹ µ )
Faster MC with RB approach :
µ
Ỹ :=
N
X
αn (µ)Y
µn
=
n=1
N
X
αn (µ) (Z µn − E(Z µn ))
n=1
where the αn (µ) minimize Var (Z µ − Ỹ µ )
EMsmall Z µ − Ỹ µ =
1
Msmall
M
small
X
P−a.s.
(Zmµ − Ỹmµ ) −−−−−−→ E(Z µ ).
m=1
Msmall →∞
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
18 / 17
Control variates: practical variance reduction
Ideally Y µ = Z µ − E(Z µ ) ⇒ Var (Z µ − Y µ ) = 0, but in practice:
Compute Ỹ µ ≈ Y µ minimizing
2
2
Var (Z µ − Ỹ µ ) = E((Z µ − E(Z µ )) − Ỹ µ ) = E(Y µ − Ỹ µ )
Faster MC with RB approach :
µ
Ỹ :=
N
X
αn (µ)Y
µn
≈
N
X
αn (µ) Z µn − EMlarge (Z µn )
n=1
n=1
where the αn (µ) minimize VarMsmall (Z µ − Ỹ µ )
EMsmall Z µ − Ỹ µ =
1
Msmall
M
small
X
P−a.s.
(Zmµ − Ỹmµ ) −−−−−−→ E(Z µ ).
m=1
Msmall →∞
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
18 / 17
Effective RB control variates method in practice
Effective numerical variance minimizations:
For all µ, solve the least-square problem by usual methods
!
N
X
inf
VarMsmall Z µ −
αn (µ) (Z µn − E(Z µn ))
{α1 (µ),...,αN (µ)}
n=1
For
or QR for the normal equations (i = 1, . . . , N)
PN instance, SVD
µ
µi , Z µj ) αµ = Cov
µi
Cov
(Z
Msmall (Z , Z )
M
small
j=1
j
Computational gains:
Only in the many-query limit = many µ !
(Offline: N expensive computations EMlarge (Z µn ) + greedy)
⇒ OK for many observations s? , or a large response surface !
S. Boyaval
The Reduced-Basis (RB) method for Uncertainty Quantification (UQ)
18 / 17