STOCHASTIC GALERKIN FINITE ELEMENT METHODS FOR …web.mat.bham.ac.uk/A.Bespalov/talks/naspde12_talk.pdf · STOCHASTIC GALERKIN FINITE ELEMENT METHODS FOR SADDLE POINT PROBLEMS WITH

STOCHASTIC GALERKIN FINITE ELEMENT METHODS FOR

SADDLE POINT PROBLEMS WITH RANDOM DATA

Alex Bespalov, Catherine Powell, David Silvester

School of Mathematics, University of Manchester,

Manchester, United Kingdom

Workshop “Numerical Analysis of Stochastic PDEs”

Mathematics Institute, University of Warwick

11 – 12 June, 2012

A. Bespalov ∗ sGFEM for saddle point problems with random data 1/22

What is this talk about...

∗ Saddle point problems with random data

∗ Stochastic Galerkin mixed finite element method

∗ Inf-sup stability of discrete problem, solution regularity, error analysis


Saddle point problems

Find (u, p) ∈ V ×W such that

a(u, v) + b(v, p) = f(v) ∀ v ∈ V,

b(u, q) = g(q) ∀ q ∈ W.

Here, V and W represent Hilbert spaces;

a : V × V → IR is a symmetric bounded bilinear form,

b : V ×W → IR is a bounded bilinear form and

f : V → IR and g : W → IR are linear functionals.


Saddle point problems with random data

Random coefficient(s): find (u, p) ∈ V ×W such that

a(u, v) + b(v, p) = f(v) ∀ v ∈ V,

b(u, q) = g(q) ∀ q ∈ W.

Examples: groundwater flow modelling, steady state Navier-Stokes flow




a(u, v) + b(v, p) = f(v) ∀ v ∈ V,

b(u, q) = g(q) ∀ q ∈ W.

Random domain: find (u, p) ∈ V ×W such that

a(u, v) + b(v, p) = f(v) ∀ v ∈ V,

b(u, q) = g(q) ∀ q ∈ W.

Fictitious domain approach for elliptic PDEs in random domains:

[Canuto and Kozubek ’07].




a(u, v) + b(v, p) = f(v) ∀ v ∈ V,

b(u, q) = g(q) ∀ q ∈ W.

Random domain: find (u, p) ∈ V ×W such that

a(u, v) + b(v, p) = f(v) ∀ v ∈ V,

b(u, q) = g(q) ∀ q ∈ W.

Random forces and/or boundary conditions: find (u, p) ∈ V ×W such that

a(u, v) + b(v, p) = f(v) ∀ v ∈ V,

b(u, q) = g(q) ∀ q ∈ W.


Example: steady flow over a step with data uncertainty

Model problem:

−ν∇2u + u · ∇u +∇p = f in D,

∇ · u = 0 in D,

u = g on ∂DDir,

ν∇u · n− n p = 0 on ∂DNeu.

Figure 1. The backward-facing step domain.



Model problem:

−ν∇2u + u · ∇u +∇p = f in D,

∇ · u = 0 in D,

u = g on ∂DDir,


We can model uncertainty in the viscosity as ν(ω) = ν0 + ν1ξ1(ω).

If ξ1 ∼ U(−√3,√

3), then ν is a uniform random variable with

E[ν(ω)] = ν0, Var[ν(ω)] = ν21 .



Model problem:

−ν∇2u + u · ∇u +∇p = f in D,

∇ · u = 0 in D,

u = g on ∂DDir,


We can model uncertainty in the viscosity as ν(ω) = ν0 + ν1ξ1(ω).

If ξ1 ∼ U(−√3,√

3), then ν is a uniform random variable with

E[ν(ω)] = ν0, Var[ν(ω)] = ν21 .

Then ν ∼ U(νmin, νmax) with νmin = ν0 − ν1

√3, νmax = ν0 + ν1

√3, and

Re(ω) =constν(ω)

, E[Re] = const E[ν−1] =const∗

ν1log

(νmax

νmin

).



Random viscosity: ν(ω) = ν0 + ν1ξ1(ω) with ν0 = 1/50 and ν1 = 1/500.

Figure 2. Streamlines of the mean flow field (top) and contours of the variance of

the magnitude of flow field (bottom).



Random viscosity: ν(ω) = ν0 + ν1ξ1(ω) with ν0 = 1/50 and ν1 = 1/500.

mean pressure field

variance of the pressure field

Figure 3. The mean (top) and the variance (bottom) of the pressure field.



More details on this problem (including stochastic Galerkin mixed finite element

scheme, properties of saddle point linear systems, and analysis of precondition-

ing strategies):

D. Silvester, A. B. and C. Powell, A framework for the development of implicit

solvers for incompressible flow problems, Discrete and Continuous Dynamical

Systems - Series S, 2012 (to appear).

C. Powell and D. Silvester, Preconditioning steady-state Navier-Stokes equa-

tions with random data, MIMS EPrint 2012.35, The University of Manchester,

2012 (submitted).


Model problem

D ⊂ Rd (d = 2, 3) – spatial domain;

(Ω,F ,P) – complete probability space;

A−1(x, ω) : D × Ω → R – second-order correlated random field.

Model problem:

find random fields p(x, ω) and u(x, ω) such that P-almost everywhere in Ω

A−1 (x, ω)u (x, ω)−∇p (x, ω) = 0 x ∈ D,

∇ · u (x, ω) = 0 x ∈ D,

p (x, ω) = g(x) x ∈ ∂DDir,

u (x, ω) · n = 0 x ∈ ∂DNeu.


Useful references

• Primal formulations, stochastic Galerkin FEM, error analysis

[Babuska, Tempone and Zouraris ’04],

[Frauenfelder, Schwab, Todor ’05].

• Stochastic collocation FEM, mixed formulation,

log-normal distribution of random data

[Ganis, Klie, Wheeler, Wildey, Yotov, and Zhang ’08].

• Stochastic Galerkin FEM, mixed formulation,

linear algebra and fast solvers

[Ernst, Powell, Silvester, and Ullmann ’09],

[Elman, Furnival, and Powell ’10].


Weak formulation

X(D) – a Banach space of real-valued functions on D with norm ‖ · ‖X(D).

Vector spaces of random fields

L2P(Ω, X(D)) :=

v (x, ω) ; v : D × Ω → R,

‖v‖L2P(Ω,X(D)) :=

(E

[‖v‖2X(D)

])1/2

< ∞

;

V := L2P(Ω,H0(div, D)) and W := L2

P(Ω, L2(D)).


Weak formulation

X(D) – a Banach space of real-valued functions on D with norm ‖ · ‖X(D).

Vector spaces of random fields

L2P(Ω, X(D)) :=

v (x, ω) ; v : D × Ω → R,

‖v‖L2P(Ω,X(D)) :=

(E

[‖v‖2X(D)

])1/2

< ∞

;

V := L2P(Ω,H0(div, D)) and W := L2

P(Ω, L2(D)).

Weak formulation:

find u(x, ω) ∈ V and p(x, ω) ∈ W such that

E[(

A−1(x, ω)u,v)]

+ E [(p,∇ · v)] = E[(g,v · n)∂DDir

],

E [(w,∇ · u)] = 0

for all v(x, ω) ∈ V and w(x, ω) ∈ W.


Discretisation strategy

Discretisation method: stochastic Galerkin mixed finite elements.

Three levels of approximation

• Approximation of random data, A−1 (x, ω) ≈ A−1M (x, ξ(ω)):

e.g., using the truncated Karhunen-Loeve expansion of A−1 (x, ω);

• Spatial discretisation on D:

e.g., by the lowest-order mixed FEM with mesh-size h;

• Discretisation on Γ = ξ(Ω) ⊂ RM :

e.g., global polynomial approximation of total degree ≤ k.


Discretisation strategy

Discretisation method: stochastic Galerkin mixed finite elements.

Three levels of approximation

• Approximation of random data, A−1 (x, ω) ≈ A−1M (x, ξ(ω)):

e.g., using the truncated Karhunen-Loeve expansion of A−1 (x, ω);

• Spatial discretisation on D:

e.g., by the lowest-order mixed FEM with mesh-size h;

• Discretisation on Γ = ξ(Ω) ⊂ RM :

e.g., global polynomial approximation of total degree ≤ k.

Three levels of approximation =⇒ three discretisation parameters (M, h, k)

and three sources of error.


Approximation of random data

A−1(x, ω) ≈ A−1M (x, ω)

... leads to

Perturbed weak formulation:

find uM (x, ω) ∈ V and pM (x, ω) ∈ W such that

E[(

A−1M (x, ω)uM ,v

)]+ E [(pM ,∇ · v)] = E

[(g,v · n)∂DDir

],

E [(w,∇ · uM )] = 0

for all v(x, ω) ∈ V and w(x, ω) ∈ W.


Estimating the perturbation error

Lemma 1. Assume that

0 < Amin ≤ A−1(x, ω) ≤ Amax < ∞ a. e. in D × Ω,

0 < AMmin ≤ A−1

M (x, ω) ≤ AMmax < ∞ a. e. in D × Ω.

Then there exist unique solution pairs (u, p) ∈ V×W, (uM , pM ) ∈ V×W and

‖u− uM‖V + ‖p− pM‖W ≤ C ‖A−1 −A−1M ‖L∞(D×Ω). (1)





0 < AMmin ≤ A−1

M (x, ω) ≤ AMmax < ∞ a. e. in D × Ω.



Remark 1. The constant C in (1) depends on Amin, Amax, AMmin, AM

max, on

the inf-sup constant and the Dirichlet data...





0 < AMmin ≤ A−1

M (x, ω) ≤ AMmax < ∞ a. e. in D × Ω.



Remark 1. The constant C in (1) depends on Amin, Amax, AMmin, AM

max, on

the inf-sup constant and the Dirichlet data...

...but if ‖A−1 − A−1M ‖L∞(D×Ω) → 0 as M →∞ (see next Lemma), then,

for sufficiently large M , we can set

AMmin := 1

2Amin, AMmax := Amax + 1

2Amin.

Then, the constant C in (1) is independent of M .


Estimating the error in approximation of random data

Goal: upper bound for ‖A−1 −A−1M ‖L∞(D×Ω).




Representation of random data using Karhunen-Loeve (KL) expansion:

A−1(x, ω) = E[A−1](x) +∑∞

n=1

√λn ϕn(x) ξn(ω)

≈ E[A−1](x) +∑M

n=1

√λnϕn(x) ξn(ω) =: A−1

M (x, ω).

Lemma 2 [Frauenfelder, Schwab, Todor ’05]. Assume:

(i) ξn∞n=1 is uniformly bounded;

(ii) covariance function C[A−1](x,x′) is (piecewise) analytic on D×D. Then

‖A−1 −A−1M ‖L∞(D×Ω) ≤ Ce−cM1/d

.




Representation of random data using Karhunen-Loeve (KL) expansion:

A−1(x, ω) = E[A−1](x) +∑∞

n=1

√λn ϕn(x) ξn(ω)

≈ E[A−1](x) +∑M

n=1

√λnϕn(x) ξn(ω) =: A−1

M (x, ω).

Lemma 2 [Frauenfelder, Schwab, Todor ’05]. Assume:

(i) ξn∞n=1 is uniformly bounded;

(ii) covariance function C[A−1](x,x′) is (piecewise) analytic on D×D. Then

‖A−1 −A−1M ‖L∞(D×Ω) ≤ Ce−cM1/d

.

Lemma 1 + Lemma 2 =⇒Theorem 1. ‖u− uM‖V + ‖p− pM‖W = O

(e−cM1/d

).


More assumptions ...

We further assume that

• random variables ξn : Ω → R (n = 1, 2, . . .) are independent;

• images Γn = ξn(Ω) are bounded intervals in R;

• ∃ ρn : Γn → R+ – a density function of ξn (n = 1, . . . , M);







... then

• uM (x, ω) = uM (x, ξ1(ω), . . . , ξM (ω)),pM (x, ω) = pM (x, ξ1(ω), . . . , ξM (ω));







... then


• ρ(y) :=∏M

n=1 ρn – the joint probability density of (ξ1, . . . , ξM ), where

y = (y1, . . . ,yM ) ∈ Γ with yn = ξn(ω) (n = 1, . . . ,M), and

Γ = supp ρ = Γ1 × . . .× ΓM ⊂ RM ;







... then


• ρ(y) :=∏M

n=1 ρn – the joint probability density of (ξ1, . . . , ξM ), where

y = (y1, . . . ,yM ) ∈ Γ with yn = ξn(ω) (n = 1, . . . ,M), and

Γ = supp ρ = Γ1 × . . .× ΓM ⊂ RM ;

• (Ω,F ,P) can be replaced by(Γ,B(Γ), ρ(y)dy

);

• for any measurable ϕ = ϕ(ξ1, . . . , ξM ), one has E[ϕ] =∫Γ

ϕ(y) ρ(y)dy.


... and another weak formulation

Denote

V := L2ρ(Γ,H0(div; D)), W := L2

ρ(Γ, L2(D));

aM (u,v) =(A−1

M u,v), b(p,v) = (p,∇ · v); `(v) = (g,v · n)∂DDir

.

Parametric deterministic formulation:

find uM (x,y) ∈ V and pM (x,y) ∈ W such that

E [aM (uM ,v)] + E [b (pM ,v)] = E [`(v)] ,

E [b (w,uM )] = 0

for all v(x,y) ∈ V and w(x,y) ∈ W.

Remark 2. This problem is uniquely solvable under the assumptions in the

statement of Lemma 1.


Stochastic Galerkin mixed FEM

Discrete subspaces

(i) on the spatial domain D ⊂ Rd: Xdivh ⊂ H0(div; D), X0

h ⊂ L2(D);(ii) on the outcomes set Γ ⊂ RM : Sk ⊂ L2

ρ(Γ).



Discrete subspaces



ρ(Γ).

Discrete formulation (sGFEM):

find uhk(x,y) ∈ Xdivh ⊗ Sk and phk(x,y) ∈ X0

h ⊗ Sk satisfying

E [aM (uhk,v)] + E [b (phk,v)] = E [` (v)] ,

E [b (w,uhk)] = 0

for all v ∈ Xdivh ⊗ Sk and w ∈ X0

h ⊗ Sk.



Discrete subspaces



ρ(Γ).

Discrete formulation (sGFEM):

find uhk(x,y) ∈ Xdivh ⊗ Sk and phk(x,y) ∈ X0

h ⊗ Sk satisfying

E [aM (uhk,v)] + E [b (phk,v)] = E [` (v)] ,

E [b (w,uhk)] = 0

for all v ∈ Xdivh ⊗ Sk and w ∈ X0

h ⊗ Sk.

Theorem 2. Let Xdivh , X0

h be a deterministic inf-sup stable pairing with

discrete inf-sup constant β. Then, for any choice of Sk ⊂ L2ρ(Γ), the pairing(

Xdivh ⊗ Sk

),(X0

h ⊗ Sk

)for the sGFEM is inf-sup stable with the same discrete

inf-sup constant β, and the sGFEM converges quasi-optimally.


Estimating the stochastic Galerkin error

Total error of the stochastic Galerkin FEM:

Ehk := ‖uM − uhk‖V + ‖pM − phk‖W .



Total error of the stochastic Galerkin FEM:

Ehk := ‖uM − uhk‖V + ‖pM − phk‖W .

Decomposition of the error

Ehk ¹ infv∈Xdiv

h ⊗Sk

‖uM − v‖V + infw∈X0

h⊗Sk

‖pM − w‖W

¹ ‖uM −Πdivh uM‖V + ‖pM −Π0

h pM‖W

+ ‖uM −Π0,ρk uM‖V + ‖pM −Π0,ρ

k pM‖W

=‘spatial’ h-error

+

‘stochastic’ k-error

,

Πdivh is an H(div; D)-conforming interpolation operator (defined elementwise),

Π0h is L2(D)-projector onto X0

h ⊂ L2(D),

Π0,ρk is L2

ρ(Γ)-orthogonal projection onto Sk ⊂ L2ρ(Γ).


Regularity of the solution

Parameterised coefficient:

A−1M (x,y) = E[A−1](x) +

∑Mn=1

√λn ϕn(x) yn.

Spatial regularity

If E[A−1] ∈ C1(D) and C[A−1] is smooth on D ×D, then ∃ r > 0 such that(uM , pM

) ∈ L2ρ(Γ;Hr(div, D))× L2

ρ(Γ; Hr(D)).




A−1M (x,y) = E[A−1](x) +

∑Mn=1

√λn ϕn(x) yn.

Spatial regularity



ρ(Γ; Hr(D)).

Regularity with respect to y1, . . . , yM

Lemma 4. If Γn (n ∈ 1, 2, . . . ,M) is a bounded interval in R then the

functions uM and pM , as functions of variable yn, can be analytically extended

to the same region of the complex plane:




A−1M (x,y) = E[A−1](x) +

∑Mn=1

√λn ϕn(x) yn.

Spatial regularity



ρ(Γ; Hr(D)).

Regularity with respect to y1, . . . , yM

Lemma 4. If Γn (n ∈ 1, 2, . . . ,M) is a bounded interval in R then the

functions uM and pM , as functions of variable yn, can be analytically extended

to the same region of the complex plane:

Σn :=

z ∈ C; |z − y0

n| <infx∈D

(A−1

M (x, y0n,y∗n)

)√

λn ‖ϕn‖L∞(D)

∀ y0n ∈ Γj

,

where y∗n = (y1, . . . , yn−1, yn+1, . . . , yM ) for any n ∈ 1, 2, . . . , M.



Theorem 3. Assume:

(i) KL-expansion of A−1 with uniformly distributed random variables ξn;

(ii) given k = (k1, . . . , kM ) ∈ NM0 , Sk := Sk1(Γ1)⊗ . . .⊗ SkM (ΓM );

(iii) technical coercivity assumption. Then there holds

‖uM − uhk‖V + ‖pM − phk‖W ≤ C

(hmin r,1 +

M∑n=1

ηnkn+1

),

where ηn =(χn +

√χ2

n − 1)−1

∈ (0, 1) with χn = 1 + const√λn ‖ϕn‖L∞(D)

for n = 1, . . . ,M .


References

More details of this work:

A. B., C. Powell and D. Silvester, A priori error analysis of stochastic Galerkin

mixed approximations of elliptic PDEs with random data, SIAM J. Numer.

Anal., 2012 (to appear).

Useful references

[1] I. Babuska, R. Tempone and G. E. Zouraris, SINUM, 42 (2004).

[2] H.C. Elman, D.G. Furnival and C.E. Powell, Math. Comp., 79 (2010).

[3] O.G. Ernst, C.E. Powell, D.J. Silvester and E. Ullmann, SISC, 31 (2009).

[4] P. Frauenfelder, C. Schwab and R.A.Todor, CMAME, 194 (2005).

[5] B. Ganis, H. Klie, M. Wheeler, T. Wildey, I. Yotov and D. Zhang, CMAME,

197 (2008).


References

More details of this work:

A. B., C. Powell and D. Silvester, A priori error analysis of stochastic Galerkin

mixed approximations of elliptic PDEs with random data, SIAM J. Numer.

Anal., 2012 (to appear).

Useful references

[1] I. Babuska, R. Tempone and G. E. Zouraris, SINUM, 42 (2004).

[2] H.C. Elman, D.G. Furnival and C.E. Powell, Math. Comp., 79 (2010).

[3] O.G. Ernst, C.E. Powell, D.J. Silvester and E. Ullmann, SISC, 31 (2009).

[4] P. Frauenfelder, C. Schwab and R.A.Todor, CMAME, 194 (2005).

[5] B. Ganis, H. Klie, M. Wheeler, T. Wildey, I. Yotov and D. Zhang, CMAME,

197 (2008).

Thank you for your attention!


Documents

STOCHASTIC GALERKIN FINITE ELEMENT METHODS FOR …web.mat.bham.ac.uk/A.Bespalov/talks/naspde12_talk.pdf · STOCHASTIC GALERKIN FINITE ELEMENT METHODS FOR SADDLE POINT PROBLEMS WITH