Optimal polynomial approximation of PDEs with stochastic

Transcription

Optimal polynomial approximation of PDEs with
stochastic coefficients by Galerkin and
collocation methods
Fabio Nobile
MOX, Department of Mathematics, Politecnico di Milano
Seminaire du Laboratoire Jacques-Louis Lions
Université Pierre et Marie Curie (Paris VI), May 27, 2011
Fabio Nobile (MOX, Politecnico di Milano)
Polynomial approx. of stochastic PDEs
LJLL, Paris, May 27, 2011
1
Acknowledgments
Collaborators: I. Babuška, L. Tamellini, M. Mohammad,
R. Tempone, J. Bäck
Funds:
Italian project FIRB-IDEAS (’09) Advanced Numerical Techniques for Uncertainty Quantification in Engineering and Life Science Problems
2
Outline
1
Problem setting: PDEs with random coefficients
2
Techniques for computing polynomial approximations
Galerkin projection
Collocation on sparse grids
numerical comparison
3
Optimization of polynomial spaces
4
Numerical examples
5
Hyperbolic problems with random coefficients
numerical results
6
Conclusions
3
Problem setting
Consider a differential problem
find u :
L(y)(u) = F
where the operator L(y) depends on a vector of N parameters:
y = (y1 , . . . , yN ).
The parameters are not perfectly known, or they have a lot of
variability in repeated experiments
Treat them as random variables with a given probability density
fuction (estimated from experimentsRor from prior knowledge /
expert opinion): ρ(y) : Γ → R+ ,
ρ(y)dy = 1
Γ
We assume that for each y ∈ Γ the corresponding solution
u = u(y) belongs to a given functional space V
=⇒
u = u(y) : Γ → V
5
Problem setting
Goal: compute statistics of the solution u or some quantity of
interest J(u). E.g.
Z
J̄ = E[J(u)] = J(u(y))ρ(y)dy,
mean value
Γ
2
Var [J] = E[J(u) ] − E[J(u)]2
Z
P[J(u) > Jcr ] = 1{J(u(y))>Jcr } ρ(y)dy
variance
exceedance prob.
Γ
6
Multivariate polynomial approximation
Computations of statistical quantities typically imply lots of
evaluations of u(y) ; lots of problems to solve
Idea: build a reduced model uΛ (y) ≈ u(y) that is cheap to
evaluate and use it to compute statistical moments. E.g.
J̄ ≈ J̄Λ = E[J(uΛ )]
In this talk we consider multivariate polynomial reduced models.
Let Λ ⊂ NN be an index set of cardinality |Λ| = M, and consider
the multivariate polynomial space
nQ
o
N
N
pn
PΛ (Γ ) = span
with p = (p1 , . . . , pN ) ∈ Λ
n=1 yn ,
find M particular solutions up ∈ V , ∀p ∈ Λ and build
X
uΛ (y) =
up y1p1 y2p2 · · · yNpN
p∈Λ
7
Examples of pol. spaces:
N = 2,
pn ≤ w
Total Degree (TD)
16
16
14
14
12
12
10
10
p2
Tensor
product:
p2
Tensor Product (TP)
8
6
4
4
2
Total
P
n
8
6
0
Q
(pn + 1)
≤w +1
n
2
4
6
8
p1
10
12
14
0
16
16
14
14
12
12
10
10
p2
p2
pn ≤ w
0
2
4
6
8
p1
10
12
14
16
Smolyak (SM)
16
8
6
4
4
2
Smolyak:
P
n
8
6
0
degree:
2
0
Hyperbolic Cross (HC)
Hyperbolic
cross:
p = 16
f (p) =
2
0
2
4
6
8
p1
10
12
14
16
0
0
2
4
6
8
p1
10
12
14
f8(pn ) ≤ f (w )
>
<0,
1,
>
:dlog (p)e,
2
p=0
p=1
p≥2
16
9
Anisotropic spaces:
N = 2,
Aniso Total Degree (ATD)
16
16
14
14
12
12
10
10
p2
Tensor
product:
αn pn ≤ w
p2
Aniso Tensor Product (ATP)
8
6
4
4
2
Total
degree:
P
α
p
≤
w
n
n
n
8
6
0
2
0
2
4
6
8
p1
10
12
14
0
16
0
2
16
14
14
12
12
10
10
8
6
4
4
2
2
0
2
4
6
8
p1
10
12
14
6
8
p1
10
12
14
16
16
Smolyak:
X
αn f (pn ) ≤ f (w )
8
6
0
4
Aniso Smolyak (ASM)
16
p2
p2
Aniso Hyperbolic Cross (AHC)
Hyperbolic
cross:
Q
αn
n (pn + 1)
≤w +1
pmax = 16
0
n
f (p) =
0
2
4
6
8
p1
10
12
14
16
8
>
<0,
1,
>
:dlog (p)e,
2
p=0
p=1
p≥2
10
Accuracy requirements
We look at mean square error control (strong convergence)
Z
err2,ρ = ku(y) − uΛ (y)k2V ρ(y)dy small
Γ
An easy implication:
[N.-Tempone, IJNME 09]
Assume ku(y)kV and kuΛ (y)kV uniformly bounded in Γ
Let J : V → R a locally Lipschitz functional, with J(0) = 0
Then
1
E[J(u)q − J(uΛ )q ] ≤ C (q)E[ku − uΛ k2V ] 2
1
i.e. convergence in E[ku − uΛ k2V ] 2 implies convergence of all
moments of the funciontal J.
Weak convergence could (should) be considered as well.
11
Example I: Thermal conduction with inclusions of
random conductivity (baked cookies problem)
(
− div(a∇u) = f ,
u=0
in D
on ∂D
Each circular inclusion Ci , i = 1, . . . 8 has a random conductivity
coefficient yi .
a(y1 , . . . , yN , x) = a0 +
N
X
(yn − a0 )1Cn (x),
x ∈ D, y ∈ Γ
n=1
8-dimensional parametric problem
12
Example II: Darcy flow in a medium with random
permeability (groundwater flow problem)
(
u = −a∇p
div u = f
a = a(ω, x): random permeability field (with a > 0 a.s.)
Each realization of the stochastic process gives a spatially
varying permeability field.
The random field can be conditioned to
available measurements (e.g. by Kriging
techniques)
Infinite-dimensional parametric problem!
13
Approximation of an ∞-dimensional random field
Let {bn (x)} be a complete orthonormal basis in L2 (D) (trigon., wavelet,
Karhunen-Loève, ...) and a(ω, x) a ∞-dimensional random field with finite
second moments. Then, a can be expanded as
a(ω, x) = E[a](x) +
∞
X
yn (ω)bn (x)
n=1
Z
(a(ω, x) − E[a](x))bn (x) dx
with yn (ω) =
D
If the basis {bn } has spectral approx. properties and the realizations of a
n→∞
are smooth, then Var [yn ] → 0 suff. fast and we can truncate the series
a(ω, x) ≈ aN (ω, x) = E[a](x) +
N
X
yn (ω)bn (x)
n=1
WARNING: the truncated expansion might not be positive
almost surely!
PN
Possible remedy: a(ω, x) ≈ aN (ω, x) = amin + e b0 (x)+ n=1 yn (ω)bn (x)
14
Galerkin projection
Galerkin projection
[Ghanem-Spanos, Karniadakis et al, Matthies-Keese, Schwab-Todor et al., Knio-Le Maı̂tre et
al,Babuska et al.,. . . ]
Project the equation onto the subspace PΛ (Γ)
Suitable for stochastic problems
Let {ψj }M
j=1 be an orthonormal basis w.r.t. the probability density
P
ρ(y). Expand uΛ (y) on the basis: uΛ (y) = M
j=1 uj ψj (y)
Galerkin formulation
Find uj ∈ V , j = 1, . . . M s.t.
M
h
i
X
E L(y)(
uj ψj )ψi = E[Fψi ],
i = 1, . . . , M
j=1
This approach leads to solving M coupled deterministic
problems; difficult to assemble and need good preconditioners.
18
[Smolyak ’63, Griebel et al ’98-’03-’04, Barthelmann-Novak-Ritter ’00, Hesthaven-Xiu ’05,
N.-Tempone-Webster ’08, Zabaras et al ’07]
1
2
3
e
Choose a set of points y(j) ∈ Γ, j = 1, . . . , M
Compute the solutions uj ∈ V : L(y(j) )(uj ) = F
Pe
Interpolate the obtained values: uΛ (y) = M
j=1 uj φj (y).
φj ∈ PΛ (Γ): suitable combinations of Lagrange polynomials
e uncoupled deterministic problems
Always leads to solving M
e of points needed is larger than the dimension M
The number M
of the polynomial space (Except for tensor product spaces).
20
(Generalized) Sparse Grid approximation
1
Choose 1D abscissae. E.g.
Clenshaw-Curtis (extrema on Chebyshev polynomials)
Gauss points w.r.t. the weight
Q ρn , assuming that the probability
density factorizes as ρ(y) = N
n=1 ρn (yn )
2
3
Take a sequence of 1D polynomial interpolant operators
m(i)
Un
: C 0 (Γn ) → Pm(i)−1 (Γn ) with increasing number of points
i
i
, . . . , yn,m
.
The i-th interpolant uses m(i) abscissae ϑin = yn,1
i
Take differences of consecutive operators:
∆m(i)
= Unm(i) − Unm(i−1) ,
n
4
Unm(0) = 0.
Multidimensional Smolyak approx.: let i = [i1 , . . . , iN ] ∈ NN+ , and
Λ ⊂ NN an index set
X m(i )
m(i )
uΛSC =
∆1 1 ⊗ · · · ⊗ ∆N N (u)
i∈Λ
21
By choosing properly the function m and the set Λ one can obtain a
polynomial approximation in any given multivariate polynomial space
([Back-N.-Tamellini-Tempone LNCSE vol. 76, 2010])
Examples of sparse grids:
N = 2,
max. polynomial degree p = 16
Total Degree (TD)
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
y2
y2
Tensor Product (TP)
1
0
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
y1
0.2
0.4
0.6
0.8
Hyperbolic Cross (HC)
−1
−1
1
−0.8
−0.6
−0.4
−0.2
0
0.2
y1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
y2
1
0
−0.2
−0.2
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.8
−0.8
−0.6
−0.4
−0.2
0
y1
0.2
0.4
0.6
0.8
1
−1
−1
0.8
1
0
−0.2
−0.8
0.6
Smolyak CC (SM)
1
−1
−1
0.4
Smolyak Gauss (SM)
y2
y2
0
−0.2
−0.8
−0.8
−0.6
−0.4
−0.2
0
y1
0.2
0.4
0.6
0.8
1
−1
−1
−0.8
−0.6
−0.4
−0.2
0
y1
0.2
0.4
0.6
0.8
1
22
Thermal conduction with random inclusions
Conductivity coefficient: matrix k=1
circular inclusions: k|Ωi ∼ U(0.01, 0.8)
→ 8 iid uniform random variables
forcing term f = 1001F
zero boundary conditions
R
quantity of interest ψ(u) = F u
mean
std
29
Convergence plot for E[ψ(u)]
−2
−2
10
10
−4
−4
10
−6
E(ψ1,p) − E(ψ1,ovk)
E(ψ1,p) − E(ψ1,ovk)
10
10
−8
10
−10
10
MC
SG−TP
SG−TD
SG−HC
SG−SM
−12
10
−14
10
0
10
1
10
−6
10
−8
10
SG−TD
MC
SC−TP
SC−TD
SC−HC
SC−SM−G
SC−SM−CC
−10
10
−12
10
−14
2
10
3
10
4
10
5
10
SG: dim−stoc * iter−CG / MC: sample size
Galerkin
6
10
10
0
10
1
2
10
10
3
10
4
10
5
10
6
10
SC: num−pts / SG: dim−stoc * iter−CG / MC: sample size
Collocation
error versus estimated cost
(see [Back-N.-Tamellini-Tempone, LNCSE ’10])
30
Thermal conduction with random inclusions –
anisotropic version
Conductivity coefficient: matrix k=1
circular inclusions: k|Ωi ∼ γi U(0.01, 0.8)
→ 4 indep. uniform random variables
forcing term f = 100
zero boundary conditions
R
quantity of interest ψ(u) = F u
mean
std
31
Convergence plot for E[ψ(u)]
Anisotropic Total degree spaces with different anisotropy weights αn
0
0
10
10
−1
−1
10
10
−2
−2
10
E(ψ1)
E(ψ1)
10
−3
10
−4
−4
10
10
MC
SG−isospaces
SG−1−7−11−15
SG−1−2−3−4
SG−1−2.5−4−5.5(exp)
SG−1−3.5−5.5−7.5(th)
−5
10
−6
10
−3
10
0
10
2
MC
SC−isospaces
SC−1−7−11−15
SC−1−2−3−4
SC−1−2.5−4−5.5(exp)
SC−1−3.5−5.5−7.5(th)
−5
10
−6
4
10
10
SG: dim−stoc * iter−CG / MC: sample size
Galerkin
6
10
10
0
10
2
4
6
10
10
10
SC: num−pts / SG: dim−stoc * iter−CG / MC: sample size
Collocation
Error versus estimated cost
(see [Back-N.-Tamellini-Tempone, LNCSE ’10])
32
Consider the diffusion problem
(
− div(a(x, y1 , . . . , yN )∇u) = f , in D
u = 0,
on ∂D
assume a(y) ≥ α > 0, ∀y ∈ Γ (uniform coerciveness)
Analyticity result [Back-N.-Tamellini-Tempone ’11, Babuska-N.-Tempone ’05, Cohen-DeVore-Schwab ’09/’10]
let i = (i1 , . . . , iN ) ∈ NN and r = (r1 , . . . , rN ) > 0. Set ri =
Q
in
n rn .
i
assume k 1a ∂∂yai kL∞ (D) ≤ ri uniformly in y
Then
i
k ∂∂yui kV ≤ C |i|!( log 2 r)i
uniformly in y
u : Γn→ V is analytic and can be extended analyticially
to
o
PN
N
Σ= z∈C :
n=1 rn |zn
− ỹn | < log 2 for some ỹ ∈ Γ
34
Remark:
The assumption is satisfied both for linear and exponential expansions
of a random field:
P
linear expansion: a(y, x) = a0 + Nn=1 bn (x)yn ,
P
with amin = a0 − Nn=1 kbn kL∞ (D) > 0.
In this case rn = kbn kL∞ (D) /amin
P
N
exponential expansion: a(y, x) = a0 + exp
b
(x)y
n
n
n=1
with a0 > 0. In this case: rn = kbn kL∞ (D) .
Better estimates on analyticity region can be obtained by complex
analysis.
35
Collocation
Galerkin
uΛSG =
X
up (x)ψp (y)
p∈Λ
find uΛSG by Galerkin projection
of the equation on
PΛ = span{ψp , p ∈ Λ}.
uΛSC =
X O
n)
[u].
∆m(i
n
i∈Λ n=1,...,N
Compute uΛSC by collocation on
the corresponding sparse grid
Question: What is the best index set Λ in both cases?
36
Galerkin projection – best M term approximation
Galerkin optimality:
ku − uΛSG kV ⊗L2ρ (Γ) ≤ C
inf
vΛ ∈V ⊗PΛ
ku − vΛ kV ⊗L2ρ (Γ)
Let {ψp , p ∈ NN } be the orthonormal basis of Legendre
multivariate polynomials and vΛ the truncated Legendre
expansion of u
X
vΛ =
E[uψp ]ψp
p∈Λ
Parseval’s identity:
P
P
2
ku − vΛ k2V ⊗L2ρ (Γ) = ku − p∈Λ E[uψp ]ψp k2V ⊗L2ρ (Γ) = p∈Λ
/ kE[uψp ]kV
Best M terms approximation
The optimal index set Λ of cardinality M is the one that contains the
M largest Legendre coefficients kE[uψp ]kV
37
Abstract construction of quasi optimal spaces
Suppose we have an a priori estimate of the form
kE[uψp ]kV ≤ G (p)
(1)
Fix a threshold ∈ R+ , and define the index set Λ as
Λ() = p ∈ NN : G (p) ≥ or equivalently
Λ(w ) = p ∈ NN : − log G (p) ≤ w , w = d− log e
The sharper the estimate (1), the better Λ approximates the
“best M terms” index set.
38
Estimate of Legendre coefficients
For the diffusion problem with random coefficients, the solution u(y)
is analytic in Γ and the following estimate of the Legendre
coefficients holds [Cohen-DeVore-Schwab ’10, Back-N.-Tamellini-Tempone ’11]
P
|p|!
kE[uψp ]kV ≤ C0 e − n gn pn
p!
P
Q
for some gn > 0, with |p| = n pn , p! = n pn !.
Then the induced optimal index set is (TD-FC)
n
o
X
|p|!
Λ(w ) = p ∈ NN :
gn pn − log
≤w
p!
n
(2)
The factorial term |p|!
accounts for the interaction between the
p!
random variables and is purely isotropic
P
Estimate (2) is meaningful only if n e −gn < 1!
40
Numerical tests
We consider the 1D problem
(
−(a(x, y)u(x, y)0 )0 = 1 x ∈ D = (0, 1), y ∈ Γ
u(0, y) = u(1, y) = 0, y ∈ Γ
with several choices of a(x, y) and compute Θ(u) = u( 12 ).
We compare:
(Aniso) TD space:
(
Λ(w ) =
)
p ∈ NN :
X
gn pn ≤ w
.
n
(Aniso) TD-FC space:
(
Λ(w ) =
N
X
|p|!
p ∈ NN :
gn pn − log
≤w
p!
n=1
)
.
41
Test 1: a(x, y) = 1 + 0.1y1 + 0.5y2
0
10
Legendre coeff
TD appr
TD−FC appr
−5
10
best M term
TD
iso−TD
TD−FC
−3
10
−6
10
−10
10
−9
10
−15
10
−12
10
−20
10
−15
10
0
50
100
150
200
Figure: Legendre coeffs of Θ(u)
in lexicographic order, with TD
and TD-FC estimates
0
10
20
30
40
50
Figure: Convergence plot for
kΘ(u) − Θ(uM )k2L2ρ (Γ) w.r.t.
M = |Λ|
The Legendre coefficients have been computed with a sufficiently high
level sparse grids.
42
12
12
10
10
8
8
6
6
4
4
2
2
0
0
2
4
6
8
10
12
0
“true” Legendre coeffs.
12
10
10
8
8
6
6
4
4
2
2
0
2
4
6
8
10
12
aniso TD estimate
2
4
6
8
10
12
iso-TD estimate.
12
0
0
0
0
2
4
6
8
10
12
TD-FC estimate.
43
Test 2
log a(x, y) = y1 + 0.2 sin(πx)y2 + 0.04 sin(2πx)y3 + 0.008 sin(3πx)y4
best M term
TD
iso−TD
TD−FC
−3
10
−6
10
−9
10
−12
10
0
10
20
Convergence plot for kΘ(u) −
30
40
Θ(uM )k2L2ρ (Γ)
w.r.t. M
44
Optimization of sparse grids
uM =
SΛm [u]
=
N
XO
n)
∆m(i
[u].
n
i∈Λ n=1
We use a knapsack problem-approach [Griebel-Knapek ’09, Gerstner-Griebel ’03,
Bungartz-Griebel ’04]: for each multiindex i estimate
∆E (i): how much error decreases if i is added to Λ (error
contribution)
∆W (i): how much work, i.e. number of evaluations, increases if
i is added to Λ (work contribution)
Then estimate the profit of each i as
∆E (i)
P(i) =
∆W (i)
and build the sparse grid using the set Λ of the M indices with the
largest profit.
45
Estimate for ∆W
Suppose we use nested abscissae, e.g. Clenshaw Curtis. The number
of points added to the grid by i is then
∆W (i) = nb. new pts. in
N
O
n=1
n)
=
∆m(i
n
N
Y
( m(in ) − m(in − 1) )
n=1
Recall that for Clenshaw-Curtis


0 if i = 0
m(i) = 1 if i = 1

 i−1
2 + 1, if i > 1,
and the Lebesgue constant is
L(m(i)) =
2
log(m(in ) + 1) + 1
π
46
Estimate for ∆E
1
rewrite ∆E (i) as
∆E (i) = (u − SΛ [u]) − u − S{Λ∪i} [u] V ⊗L2 (Γ) =
ρ
X
X
m(j)
m(j)
k
∆ [u] −
∆ [u] kV ⊗L2ρ (Γ) = ∆m(i) [u]V ⊗L2 (Γ) .
ρ
j∈{Λ∪i}
2
j∈{Λ}
use the following estimate (numerically validated)
N
Y
∆E (i)[u] ≈ um(i−1) V
1 + Lm(in −1)
n=1
3
where um(i−1) is the corresponding Legendre coefficient.
estimate um(i−1) as in the Galerkin case
P
um(i−1) ≤ C0 e − n gn m(in −1) |m(i − 1)|!
V
m(i − 1)!
47
Example: Comparison ∆E vs. estimate:
Let y1 , y2 ∼ U(−1, 1).
(
−∇ · [(1 + c1 y1 + c2 y2 )∇u(x, y1 , y2 ) ] = f (x) x ∈ D
u(x) = 0
x ∈ ∂D
u(x, y1 , y2 ) =
∆−1 f (x)
1+c1 y1 +c2 y2
admits a Legendre expansion.
Nested knots: Clenshaw-Curtis: m(i) = 2i+1 − 1, y k = cos
∆m(i)
0
10
−2
10
−4
m(i−1)
Leg
fm(i−1) ⋅ Leb(m(i))
−2
10
−4
10
10
−6
−6
10
10
−8
−8
10
10
−10
10
kπ
m(i)
∆m(i)
0
10
m(i−1)
Leg
fm(i−1) ⋅ Leb(m(i))
0
−10
5
10
15
20
25
c1 = 0.3, c2 = 0.3
30
10
0
5
10
15
20
25
30
c1 = 0.1, c2 = 0.5
48
All the pieces together
The set {i ∈ NN : P(i) ≥ } is then equivalent to
(
i∈
NN
+:
N
X
m(in − 1)gn − log
i=n
|m(i − 1)|!
−
m(i − 1)!
N
X
n=1
log
2
π
log(m(in − 1) + 1) + 2
≤w
m(in ) − m(in − 1)
)
(EW - Error Work grids)
where
Legendre coeff + Lebesgue constant = error estimate
work estimate
49
Numerical examples
Numerical test 1 - Uniform case
(
−(a(x, y)u(x, y)0 )0 = 1 x ∈ D = (0, 1),
u(0, y) = u(1, y) = 0
• y ∈ Γ = [−1, 1]N , N = 2, 4
• different choices of diffusion coefficient a(x, y).
• We focus on a linear functional ψ : V → R, ψ(v ) = v ( 12 );
• Convergence: kψ(uSG ) − ψ(u)kL2ρ (Γ) vs. nb of points in sparse grid
• We compare
standard isotropic Smolyak Sp. Grid, I = {i ∈ NN :
PN
n=1 (in
− 1) ≤ w }
the Knapsack grid derived
“best M terms”: knapsack grid, with computed profits P(i)
dimension adaptive algorithm [Gerstner-Griebel ’03, Klimke, PhD ’06],
”www.ians.uni-stuttgart.de/spinterp”
52
Numerical examples
−1
−1
10
iso SM
EW
adaptive
best M terms
−2
10
−3
10
10
iso SM
adaptive
best M terms
EW
−2
10
−3
10
−4
10
−4
10
−5
10
−5
10
−6
10
−6
−7
10
10
−8
−7
10
0
20
40
60
80
100
120
a = 1 + 0.3y1 + 0.3y2
140
10
0
20
40
60
80
100
120
140
a = 1 + 0.1y1 + 0.5y2
53
Numerical examples
−1
−2
10
10
iso SM
EW
adaptive
best M terms
−3
10
−4
10
10
−6
−5
10
10
−7
−6
10
10
−8
−7
10
10
−9
0
−3
10
−4
−5
10
10
iso SM
EW
adaptive
best M terms
−2
10
−8
20
40
60
80
100
120
140
a(x, y) = 4 + y1 + 0.2 sin(πx)y2 +
0.04 sin(2πx)y3 + 0.008 sin(3πx)y4
10
0
20
40
60
80
100
120
140
log a(x, y) = y1 + 0.2 sin(πx)y2 +
0.04 sin(2πx)y3 + 0.008 sin(3πx)y4
54
Numerical examples
Numerical test 2 - 1D lognormal field
L = 1, D = [0, L]2 .


−∇ · a(y, x)∇u(y, x) = 0
u = 1 on x = 0, h = 0 on x = 1


no flux otherwise
a(x, y) = e γ(x,y)
µγ (x) = 0
Cov γ (x, x0 ) = σ 2 e −
|x1 −x01 |2
LC 2
We approximate γ as
γ(y, x) ≈ µ(x) + σa0 Y0 + σ
K
X
k=1
h
π
π
i
ak Y2k−1 cos
kx1 + Y2k sin
kx1
L
L
with Yi ∼ N (0, 1), i.i.d.
|z|2
Given the Fourier series σ 2 e − LC 2 =
P∞
k=0 ck
cos
π
L kz
, ak =
√
ck .
55
Numerical examples
Quantity of interest:
"Z
E[Φ(u)],
with Φ =
0
L
∂u(·, x)
k(·, x)
dx
∂x
#
Convergence: |E[Φ(uSG )] − E[Φ(u)]|
We compare Monte Carlo estimate with Knapsack grids
Gauss-Hermite-Patterson points (nested Gauss-Hermite)
Estimate of Hermite coefficients decay (heuristic)
kui kV ≤
N
Y
e −gn in
√
in !
n=1
m(in )
Estimate of Lebesgue constant (heuristic) Ln
'1
56
Numerical examples
Here LC = 0.2, σ = 0.3.
K = 6 → N = 13 r.v., and 99% of total variability of e γ .
K = 10 → N = 21 r.v., and 99.99% of total variability of e γ .
mean
0
10
−2
10
−4
10
−6
10
MC
MC
MC
MC
MC
MC−slope
sparse grid 21 var
sparse grid 13 var
−8
10
−10
10
0
10
1
10
2
10
3
10
4
10
5
10
57
Numerical examples
L = 1, D = [0, L]2 .
(
−∇ · a(x, y)∇h(y, x) = 0
B.C. : see figure
a(x, y) = e γ(y,x)
µγ (x) = 0
Cov γ (x, x0 ) = σ 2 e −
|x−x0 |2
LC 2
We approximate γ as
γ(x, y) ≈ µ(x) + σ
X
ak [Yk,1 cos (πk1 x1 ) cos (πk2 x2 ) + Yk,2 cos (πk1 x1 ) sin (πk2 x2 ) +
k∈K
Yk,3 sin (πk1 x1 ) cos (πk2 x2 ) + Yk,4 sin (πk1 x1 ) sin (πk2 x2 )]
with Yi ∼ N (0, 1), i.i.d.
|z|2
Given the Fourier series σ 2 e − LC 2 =
P∞
k=0 ck
cos
π
L kz
, ak =
√
ck1 ck2 .
58
Numerical examples
Here LC = 0.4, σ = 0.3.
N = 21 r.v., 92% of total variability of e γ .
mean
0
10
−1
10
−2
10
−3
10
−4
10
−5
MC
MC
MC
MC
MC
MC−slope
sparse grid
10
−6
10
−7
10
0
10
1
10
2
10
3
10
4
10
59
Hyperbolic problems
 2
∂ u
2

 ∂t 2 − div(a (x, y1 (ω), . . . , yN (ω))∇u) = f , in D, t > 0
u = 0,
on ∂D, t > 0


∂u
u|t=0 = u0 ,
|
= v0 ,
in D
∂t t=0
assume a(x, y(ω)) ≤ amax < ∞, ∀y ∈ Γ, ∀x ∈ D
boundedness)
(uniform
The solution is in general not smooth
61
Example: 1D problem
( 2
2
∂ u
− y 2 ∂∂xu2 = 0,
∂t 2
u(x, 0) = u0 (x),
∂u
(x, 0)
∂t
in R, t > 0
= 0, in R
D’Alambert formula: u(x, t) = 21 u0 (x − y t) + 12 u0 (x + y t)
=⇒
t ∂u0 ∂u
t ∂u0 +
(x, t) = −
∂y
2 ∂ξ ξ=x−yt 2 ∂ξ ξ=x+yt
Remarks:
u0 (·) ∈ C k (R) −→ u(y )|(x,t) ∈ C k (Γ)
In particular, a discontinuous initial datum implies discontinuous
dependendence on the parameter (waveR speed)
However, consider a functional J(y ) = R u(y ; x, T )g (x)dx with
g ∈ C m (R) with compact support. Then, J(y )|(x,t) ∈ C k+m (Γ).
Linear functionals of the solution can be much smoother than
the solution itself. It might still be good to approximate J(y ) by
polynomials (not u itself).
62
Layered random medium
Assume that the wave speed has the form
a(x, y(ω)) =
N
X
yn (ω)ρn (x)1Dn (x)
n=1
where 1Dn are characteristic functions corresponding to a
non-overlapping partition of the domain: D = ∪Nn=1 Dn with smooth
interfaces, ρn are smooth funtions and yn ∈ Γn are random variables.
Result 1 [Motamed-N.-Tempone ’11]
Given f ∈ L2 (0, T ; H01 (D)), u0 ∈ H01 (D), v0 ∈ L2 (D),
the solution has in general only one bounded derivative
∂yn u ∈ L2 (0, T ; L2 (D)).
The solution might be smoother for smoother data not
intersecting any interface between the layers.
63
Layered random medium
Result 2 [Motamed-N.-Tempone ’11]
Consider a finite dimensional approximation of the equation in space
with discretization parameter h.
The discrete solution uh (y) is always analytic with respect to y
for all y ∈ RN . However, if we replace y with
z ∈ Σ ≡ {z ∈ CN : dist(Γ, z) ≤ τ }, and study the problem in
the complex domain
maxz∈CN kuh (z)kL2 (0,T ;H01 ) ≤ C τh exp{γT τh }
Consider a tensor product polynomial approximation if y of
degree p. We have two regimes
for hp CT ; exponential convergence in p
for hp CT ; algebraic slow convergence in p due to small
regularity of u w.r.t. y.
64
numerical results
Wave equation in two layered random medium
layer 1
y1~ U(0.2 , 1)
layer 2
y2~ U(0.3 , 1.5)
initial solution
wave equation in two-layered medium
random wave speed in each layer
smooth initial deformation across the
interface
E[u](x, t = 1)
std[u](x, t = 1)
65
numerical results
Isotropic Smolyak grid approximation on Gauss-Legendre
abscissae
Finite difference approximation in space + leapfrog in time
Maximum error in the expected value at t=1 versus # of
e
collocation points M.
0
10
−5
e2
10
εh, ∆ x = 0.1
εh, ∆ x = 0.05
−10
10
εh, ∆ x = 0.025
εh, ∆ x = 0.0125
η−1/2
−15
10
0
10
1
10
2
10
3
10
e1
4
10
5
10
6
10
The solution has low regularity ; slow convergence (the convergence
e denpends on the mesh size ∆x)
rate in M
66
numerical results
Smooth case:
The initial displacement is smooth and
confined in the second layer
Maximum error in the expected value at
e
t=1 versus # of collocation points M.
y1~ U(0.2 , 1)
layer 1
layer 2
y2~ U(0.3 , 1.5)
0
10
−5
e2
10
εh, ∆ x = 0.1
−10
10
ε , ∆ x = 0.05
h
ε , ∆ x = 0.025
h
ε , ∆ x = 0.0125
h
−15
10
0
10
1
10
2
3
10
10
4
10
5
10
e1
67
numerical results
Conclusions
1
Solution may depend on a high number of parameters /
random variables. An accurate choice of the approximation
space is needed, to avoid unaffordable computational costs
2
Under regularity assumptions it makes sense to look for global
polynomial approximations, either modal (galerkin procedure) or
nodal (collocation procedure)
3
We propose a general procedure based on estimates of
Legendre coefficients / profits of grids to build optimal
polynomial approximations
4
There is no “one for all” recipe: the structure of the problem
(hence of the solution) leads to the appropriate choice of
approximation
68
Conclusions
References
J. Bäck and F. Nobile and L. Tamellini and R. Tempone
On the optimal polynomial approximation of stochastic PDEs by Galerkin and Collocation
methods, MOX Report 2011, submitted.
J. Bäck and F. Nobile and L. Tamellini and R. Tempone
Stochastic Spectral Galerkin and collocation methods for PDEs with random coefficients:
a numerical comparison, LNCSE Springer, Vol. 76, 2011.
I. Babuška, F. Nobile and R. Tempone.
A stochastic collocation method for elliptic PDEs with random input data, SIAM Review,
52(2):317–355, 2010
F. Nobile and R. Tempone
Analysis and implementation issues for the numerical approximation of parabolic equations
with random coefficients, IJNME, 80:979–1006, 2009
F. Nobile, R. Tempone and C. Webster
An anisotropic sparse grid stochastic collocation method for PDEs with random input
data, SIAM J. Numer. Anal., 46(5):2411–2442, 2008
F. Nobile, R. Tempone and C. Webster
A sparse grid stochastic collocation method for PDEs with random input data, SIAM J.
Numer. Anal., 46(5), 2309–2345, 2008
71

Optimal polynomial approximation of PDEs with stochastic

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