view presentation

Transcription

view presentation

Stochastic Galerkin and collocation methods for
PDEs with random coefficients:
a numerical comparison
Fabio Nobile
MOX, Department of Mathematics, Politecnico di Milano
Acknowledgements: J. Bäck, L. Tamellini, R. Tempone
Uncertainty Quantification Workshop, Edinburgh May 24-28, 2010
Fabio Nobile (MOX, Politecnico di Milano)
Galerkin and Collocation comparison
UQ Workshop Edinburgh, May 24-28, 2010
Outline
1
Elliptic PDE with random coefficients
2
Stochastic Polynomial approximation / Galerkin versus Collocation
3
Numerical comparison
Let (Ω, F, P) be a complete probability space. Consider the problem
(
− div(a(ω, x)∇u(ω, x)) = f (x) for a.e. x ∈ D, ω ∈ Ω,
u(ω, x) = 0
for a.e. x ∈ ∂D, ω ∈ Ω
a(ω, x) is a random field parametrized with N independent
random variables (finite dimensional noise assumption)
a(ω, x) = a(Y1 (ω), . . . , YN (ω), x)
Then u(ω, x) = u(Y1 (ω), . . . , YN (ω), x) is a deterministic
function of the random vector Y.
Remark: u is in general analytic w.r.t. Y.
We assume
QN that Y has Na joint +probability density function
ρ(y) = n=1 ρn (yn ) : Γ → R
(
u(ω, x) = 0
a(ω, x) = a(Y1 (ω), . . . , YN (ω), x)
We assume
ρ(y) = n=1 ρn (yn ) : Γ → R
(
u(ω, x) = 0
a(ω, x) = a(Y1 (ω), . . . , YN (ω), x)
We assume
ρ(y) = n=1 ρn (yn ) : Γ → R
(
u(ω, x) = 0
a(ω, x) = a(Y1 (ω), . . . , YN (ω), x)
We assume
ρ(y) = n=1 ρn (yn ) : Γ → R
(
u(ω, x) = 0
a(ω, x) = a(Y1 (ω), . . . , YN (ω), x)
We assume
ρ(y) = n=1 ρn (yn ) : Γ → R
Examples
Material with inclusions of random conductivity
a(ω, x) = a0 +
1
PN
n=8 Yn (ω) Ωn (x)
with Yn ∼ uniform, lognormal, ...
Random, spatially correlated, material properties
a(ω, x) is ∞-dimensional random field
(e.g. lognormal), suitably truncated
a(ω, x) ≈ amin + e
PN
n=1
Yn (ω)bn (x)
with Yn ∼ N(0, 1), i.i.d.
Examples
Material with inclusions of random conductivity
a(ω, x) = a0 +
1
PN
n=8 Yn (ω) Ωn (x)
with Yn ∼ uniform, lognormal, ...
Random, spatially correlated, material properties
a(ω, x) is ∞-dimensional random field
(e.g. lognormal), suitably truncated
random field with Lc=1/4
1
0.9
2.5
0.8
2
0.7
1.5
0.6
a(ω, x) ≈ amin + e
PN
n=1
Yn (ω)bn (x)
0.5
1
0.4
0.5
0.3
0
0.2
−0.5
0.1
0
with Yn ∼ N(0, 1), i.i.d.
0
0.2
0.4
0.6
0.8
1
−1
Stochastic polynomial approximation
Idea: approximate the response function u(y, ·) by global multivariate
polynomials. Since u(y, ·) is analytic, we expect fast convergence.
Polynomial space
(
PΛw (ΓN ) = span
N
Y
)
ynpn ,
with p ∈ Λw ,
with Λw ⊂ NN index set
n=1
Approximation: uw (y, x) =
P
p∈Λwup (x)ψp (y),
with {ψp } basis of PΛw
Polynomial space
(
PΛw (ΓN ) = span
N
Y
)
ynpn ,
with p ∈ Λw ,
n=1
P
Polynomial space
(
PΛw (ΓN ) = span
)
N
Y
ynpn ,
with p ∈ Λw ,
n=1
P
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
0
1
2
3
4
5
6
7
8
9
Total Degree:
P
n pn ≤ w
0
1
2
3
4
5
6
7
8
9
Hyperbolic Cross:
Q
n (pn + 1) ≤ w + 1
Polynomial space
(
PΛw (ΓN ) = span
)
N
Y
ynpn ,
with p ∈ Λw ,
n=1
9
P
9
TD
TD−aniso
8
HC
HC−aniso
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
0
1
2
3
4
5
6
7
8
Aniso Total Degree:
P
n αn p n ≤ w
9
0
1
2
3
4
5
6
7
8
9
Aniso Hyperbolic Cross:
Q
αn ≤ w + 1
n (pn + 1)
Stochastic Galerkin approximation
Project the equation onto the subspace VΛw = PΛw (ΓN ) ⊗ H01 (D)
[Ghanem-Spanos, Karniadakis et al, Matthies-Keese, Schwab-Todor et al., Knio-Le Maı̂tre et
al,Babuska et al.,. . . ]
Find uwSG ∈ VΛw such that ∀v ∈ VΛw
Z
Z
SG
f (x)v (y, x) dx
E
a(y, x)∇uw (y, x) · ∇v (y, x) dx = E
D
D
Leads to M coupled deterministic problems
with sparse matrix.
We solve it by PCG [Ghanem-Pellissetti,
Elman-Powell et al., ...] with mean-based
preconditioner (block diagonal)
Z
Z
SG
f (x)v (y, x) dx
E
a(y, x)∇uw (y, x) · ∇v (y, x) dx = E
D
D
with sparse matrix.
0
20
40
60
80
100
120
140
160
0
20
40
60
80
100
nz = 885
120
140
160
Z
Z
SG
f (x)v (y, x) dx
E
a(y, x)∇uw (y, x) · ∇v (y, x) dx = E
D
D
with sparse matrix.
0
20
40
60
80
100
120
140
160
0
20
40
60
80
100
nz = 885
120
140
160
Stochastic Collocation approximation
Collocate the equation in a set of points yk
([Tatang 95], [Mathelin-Hussaini ’03], [Hesthaven-Xiu ’05], [Babuška -N.-Tempone ’05],
[Zabaras et al ’06], [Karniadakis et al ’08] . . . )
Generalized sparse grid construction
m(i)
1
Define 1D polynomial interpolants Un
on m(i) Gauss knots;
2
3
m(i)
m(i)
m(i−1)
Define ∆n = Un
− Un
;
N
Given an increasing function
X g : N → N, define
uwSC = Swm,g (u) =
∆1 m(i1 ) ⊗ · · · ⊗ ∆N m(iN ) (u)
i∈N N : g (i)≤w
Theorem [Back-N.-Tamellini-Tempone ’10]. Given a Polynomial space PΛ
there exist functions m and g , such that Swm,g (u) ∈ PΛ and
S m,g is exact in PΛ .
m(i)
1
2
3
m(i)
m(i)
m(i−1)
Define ∆n = Un
− Un
;
N
uwSC = Swm,g (u) =
∆1 m(i1 ) ⊗ · · · ⊗ ∆N m(iN ) (u)
i∈N N : g (i)≤w
m(i)
1
2
3
m(i)
m(i)
m(i−1)
Define ∆n = Un
− Un
;
N
uwSC = Swm,g (u) =
∆1 m(i1 ) ⊗ · · · ⊗ ∆N m(iN ) (u)
i∈N N : g (i)≤w
Equivalence table
SC:
Tensor Product
(TP)
Total Degree
(TD)
Hyperbolic Cross
(HC)
Smolyak (SM)
m, rule (g )
m(i) = i
g (i) = maxn (in − 1) ≤ w
m(i) = i
P
g (i) = n (in − 1) ≤ w
m(i) = i
Q
g (i) =( n (in ) ≤ w + 1
2i−1 + 1, i > 1
m(i) =
1, i = 1
g (i) =
P
n (in
− 1) ≤ f (w )
SG:
polynomial space
{p ∈ NN : maxn pn ≤ w }
{p ∈ NN :
{p ∈ NN :
Q
P
n (pn
n
pn ≤ w }
+ 1) ≤ w + 1}
P
{p ∈ NN :
n f (pn ) ≤ f (w )}
(
0, for p = 0; 1, for p = 1
f (p) =
dlog2 (p)e p ≥ 2
Equivalence table
SC:
Tensor Product
(TP)
Total Degree
(TD)
Hyperbolic Cross
(HC)
m, rule (g )
Smolyak (SM)
g (i) =
P
n (in
polynomial space
{p ∈ NN :
{p ∈ NN :
Q
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0
−0.2
−0.2
−0.4
−0.4
−0.4
−0.6
−0.6
−0.6
−0.8
−0.8
−0.4
−0.2
0
y1
0.2
0.4
0.6
0.8
1
−1
−1
+ 1) ≤ w + 1}
0
−0.2
−0.6
pn ≤ w }
0.2
y2
y2
1
0.8
0
n
Smolyak Gauss (SM)
1
0.8
−0.8
n (pn
Hyperbolic Cross (HC)
1
0.8
−1
−1
P
P
{p ∈ NN :
n f (pn ) ≤ f (w )}
(
0, for p = 0; 1, for p = 1
f (p) =
dlog2 (p)e p ≥ 2
− 1) ≤ f (w )
Total Degree (TD)
y2
SG:
m(i) = i
g (i) = maxn (in − 1) ≤ w
m(i) = i
P
g (i) = n (in − 1) ≤ w
m(i) = i
Q
g (i) =( n (in ) ≤ w + 1
2i−1 + 1, i > 1
m(i) =
1, i = 1
−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
Equivalence table
SC:
Tensor Product
(TP)
Total Degree
(TD)
Hyperbolic Cross
(HC)
m, rule (g )
SG:
Smolyak (SM)
g (i) =
P
n (in
{p ∈ NN :
Q
P
n (pn
1
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
y2
0
0
0
−0.2
−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
+ 1) ≤ w + 1}
Aniso Smolyak Gauss (SM)
1
0.8
−0.8
pn ≤ w }
Aniso Hyperbolic Cross (HC)
1
−1
−1
n
P
{p ∈ NN :
n f (pn ) ≤ f (w )}
(
0, for p = 0; 1, for p = 1
f (p) =
dlog2 (p)e p ≥ 2
0.8
y2
2
{p ∈ NN :
− 1) ≤ f (w )
Aniso Total Degree (TD)
y
polynomial space
m(i) = i
g (i) = maxn (in − 1) ≤ w
m(i) = i
P
g (i) = n (in − 1) ≤ w
m(i) = i
Q
g (i) =( n (in ) ≤ w + 1
2i−1 + 1, i > 1
m(i) =
1, i = 1
0.8
1
−1
−1
−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
Random inclusions – uniform
Conductivity coefficient: matrix k=1
8 circular inclusions: k|Ωi ∼ U(0.01, 0.8), i.i.d.
forcing term f = 1001F
zero boundary conditions
quantity of interest ψ(u) =
R
F
u
std
mean
Computational cost: ] calls to deterministic solver
1
Galerkin : ]stocDoF × ]iterPCG
2
Collocation : ]collocation points
Random inclusions – uniform
8 circular inclusions: k|Ωi ∼ U(0.01, 0.8), i.i.d.
forcing term f = 1001F
R
F
u
std
mean
Computational cost: ] calls to deterministic solver
1
Galerkin : ]stocDoF × ]iterPCG
2
Collocation : ]collocation points
Error on E[ψ(u)] versus Computational Cost
−2
10
−4
E(ψ1,p) − E(ψ1,ovk)
10
−6
10
−8
10
Collocation
−10
10
MC
SG−TP
SG−TD
SG−HC
SG−SM
−12
10
−14
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
SG: dim−stoc * iter−CG / MC: sample size
Galerkin
−2
−2
10
10
−4
−4
10
−6
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
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 on E[ψ(u)] versus Dim. Polynomial Space
−2
10
SG−TD
SG−SM
SC−TD
SC−SM
−4
10
−6
10
−8
10
−10
10
−12
10
−14
10
0
10
1
10
2
3
10
10
4
10
5
10
dim−stoc
On optimal spaces for Stochastic Galerkin
Why TD space is the best for Stochastic Galerkin?
Answer:
P
1
Expand u in Legendre series: u(·, y) = p∈NN cp (·)ψp (y) with
{ψp } multivariate orthonormal Legendre polynomials.
2
Since u is analytic, the Fourier coefficients decay exponentially:
!
N
X
Y
gn pn
kcp kH 1 (D) ∼
e −gn pn = exp −
n
n=1
3
In this example gn ≈ g for all n (isotropic problem)
Optimal index set Λ of cardinality M: the one corresponding to
the M largest Fourier coefficients.
X
kcp kH 1 (D) ≥ e −gw ⇒ Λ ≡ {p ∈ NN :
pn ≤ w } (TD space)
n
Answer:
P
1
2
!
N
X
Y
gn pn
kcp kH 1 (D) ∼
n
n=1
3
X
kcp kH 1 (D) ≥ e −gw ⇒ Λ ≡ {p ∈ NN :
n
Answer:
P
1
2
!
N
X
Y
gn pn
kcp kH 1 (D) ∼
n
n=1
3
X
kcp kH 1 (D) ≥ e −gw ⇒ Λ ≡ {p ∈ NN :
n
Answer:
P
1
2
!
N
X
Y
gn pn
kcp kH 1 (D) ∼
n
n=1
3
X
kcp kH 1 (D) ≥ e −gw ⇒ Λ ≡ {p ∈ NN :
n
Random inclusions – lognormal
3
( 1+ uniform) d.f.
(1 + l0 + lognormal) d.f.
Zi
k|Ωi = l0 + e ,
with Zi ∼ N(µ, σ 2 ), i.i.d.
l0 , µ, σ so as to have same mean,
var. and min. value as uniform test
use Hermite polynomials
2.5
2
1.5
1
0.5
0
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Optimal space for Stochastic Galerkin
In this case the coefficients of the Hermite expansion decay as
!
Y
X √
√
kcp kH 1 (D) ∼
gn pn
n
n
We introduce the new space square root (SQ)
X√
Λ ≡ {p ∈ NN :
pn ≤ w }
n
Random inclusions – lognormal
3
( 1+ uniform) d.f.
(1 + l0 + lognormal) d.f.
Zi
k|Ωi = l0 + e ,
with Zi ∼ N(µ, σ 2 ), i.i.d.
l0 , µ, σ so as to have same mean,
var. and min. value as uniform test
use Hermite polynomials
2.5
2
1.5
1
0.5
0
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Optimal space for Stochastic Galerkin
In this case the coefficients of the Hermite expansion decay as
!
Y
X √
√
kcp kH 1 (D) ∼
gn pn
n
n
We introduce the new space square root (SQ)
X√
Λ ≡ {p ∈ NN :
pn ≤ w }
n
Convergence plots
0
0
10
10
SG−TP−L
SG−TD−L
SG−HC−L
SG−SM−L
SG−SQ−L
10
−4
10
−6
10
−8
10
−10
10
10
−4
10
−6
10
−8
10
−10
10
−12
10
SG−TP−L
SG−TD−L
SG−HC−L
SG−SM−L
SG−SQ−L
SC−TP−L
SC−TD−L
SC−HC−L
SC−SM−L
SC−SQ−L
montecarlo
−2
( E(ψ1,p) − E(ψ1,ovk) )/ E(ψ1,ovk)
( E(ψ1,p) − E(ψ1,ovk) )/ E(ψ1,ovk)
−2
−12
0
10
2
4
6
8
10
10
10
10
SG: dim−stoc * iter−CG / SC: nb points/ MC: sample size
10
0
10
2
4
6
8
10
10
10
10
SG method
SC method
Remarks
TD underperforms in this case. SQ is the best space
Higher extra-cost between Galerkin and SC due to ill
conditioning of the Galerkin matrix.
Convergence plots
0
0
10
10
SG−TP−L
SG−TD−L
SG−HC−L
SG−SM−L
SG−SQ−L
10
−4
10
−6
10
−8
10
−10
10
10
−4
10
−6
10
−8
10
−10
10
−12
10
SG−TP−L
SG−TD−L
SG−HC−L
SG−SM−L
SG−SQ−L
SC−TP−L
SC−TD−L
SC−HC−L
SC−SM−L
SC−SQ−L
montecarlo
−2
( E(ψ1,p) − E(ψ1,ovk) )/ E(ψ1,ovk)
( E(ψ1,p) − E(ψ1,ovk) )/ E(ψ1,ovk)
−2
−12
0
10
2
4
6
8
10
10
10
10
10
0
10
2
4
6
8
10
10
10
10
SG method
SC method
Remarks
TD underperforms in this case. SQ is the best space
Higher extra-cost between Galerkin and SC due to ill
conditioning of the Galerkin matrix.
Performance of PCG iterations
80
80
70
70
60
60
50
50
iter CG
iter CG
The mean-based preconditioner is less effective for logn
variables. PCG requires more iterations than for uniform random
variables (for the same variance) [Powell-Ullmann 2009, MIMS
tech.report]
Conversely, no additional cost for Collocation (]coll.points is the
same)
40
30
uniform TD
lognormal TD
uniform SM
lognormal SM
40
30
20
20
10
10
uniform test
lognormal test
0
0
10
1
10
2
10
3
10
4
10
5
10
0
0
10
log(dim−stoc)
2
10
3
10
4
10
5
10
log(dim−stoc)
PCG iterations for TD
1
10
PCG iterations for SM
Thermal conduction – anisotropic version
4 circular inclusions: k|Ωi ∼ γi U(0.01, 0.8)
forcing term f = 100
mean
R
F
u
std
Approximation in Anisotropic Total Degree polynomial spaces
Weights αn estimated theoretically or numerically
Galerkin
Galerkin versus Collocation
0
10
−1
10
−2
E(ψ1)
10
−3
10
−4
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
0
10
2
4
10
10
6
10
Galerkin
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
SG−1−2.5−4−5.5(exp)
SG−1−3.5−5.5−7.5(th)
SC−1−2.5−4−5.5(exp)
SC−1−3.5−5.5−7.5(th)
−5
10
−6
4
10
10
Galerkin
6
10
10
0
10
1
2
3
4
5
10
10
10
10
10
SC: num−pts / SG: dim−stoc * iter−CG / MC: sample size
Conclusions
It is possible to set Collocation and Galerkin in the same polynomial space
Polynomial spaces / sparse grids can be tuned to the problem if we have
information on the 1D convergence and weights
Galerkin is better if we compare the accuracy of the methods w.r.t. the
dimension of the underlying polynomial space (L2 optimality of Galerkin)
Good preconditioners are crucial for Galerkin Collocation cost is less
affected by the choice of rand. var.
Collocation seems to be better in terms of error versus computational cost.
However, a better preconditioner for Galerkin may lead to opposite
conclusions
The picture may change considerably for non-linear problems
References
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, to appear in Proc. of ICOSAHOM09. LNCSE, Springer.
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

view presentation

Transcription

Similar documents

Optimal polynomial approximation of PDEs with stochastic

are oltre la risonanza magnetica nucleare - E

LFR fuel overview and perspectives

Sara Moccia – - nearlab

brochure - Dipartimento di Meccanica

Presentation slides

POLI.design Welcome Kit

Il Fatto Quotidiano_Cristina Sivieri Tagliabue

I Liguriani

Scarica la brochure

Benvenuti ad... 2016/17