Stochastic Polynomial approximation of PDEs with random …€¦ · Stochastic Polynomial...

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Stochastic Polynomial approximation of PDEs

with random coefficients

Fabio Nobile

CSQI-MATHICSE, EPFL, Switzerlandand MOX, Politecnico di Milano, Italy

Joint work with: R. Tempone, E. von Schwerin (KAUST)

L. Tamellini, G. Migliorati (MOX, Politecnico Milano), J. Beck (UCL)

WS: “Numer. Anal. of Multiscale Problems & Stochastic Modelling”RICAM, Linz, December 12-16, 2011

Italian project FIRB-IDEAS (’09) Advanced Numerical Tech-niques for Uncertainty Quantification in Engineering and LifeScience Problems

Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 1

Outline

1 Elliptic PDE with random coefficients

2 Stochastic multivariate polynomial approximationGalerkin projectionCollocation on sparse gridsOptimized algorithmsNumerical results

3 Polynomial approximation by discrete projection on random points

4 Conclusions

Elliptic PDE with random coefficients

Outline

4 Conclusions

Let (Ω,F ,P) be a complete probability space. Consider

L(u) = F ⇔

− div(a(ω, x)∇u(ω, x)) = f (x) x ∈ D, ω ∈ Ω,

u(ω, x) = 0 x ∈ ∂D, ω ∈ Ω

wherea : Ω× D → R is a second order random field such thata ≥ amin > 0 a.e. in Ω× Df ∈ L2(D) deterministic (could be stochastic as well,f ∈ L2

P (Ω)⊗ L2(D))

By Lax-Milgram lemma, for a.e. ω ∈ Ω, there exists a unique solution

u(ω, ·) ∈ V ≡ H10 (D), and ‖u‖V⊗L2

P≤ CP

amin‖f ‖L2(D)

The uniform coercivity assumption can be relaxed toE[amin(ω)−p] <∞ for all p > 0. Useful for lognormal random fields(see e.g. [Babuska-N.-Tempone ’07, Garvis-Sarkis ’09, Gittelson ’10, Charrier ’11]).

L(u) = F ⇔

u(ω, x) = 0 x ∈ ∂D, ω ∈ Ω

P (Ω)⊗ L2(D))

u(ω, ·) ∈ V ≡ H10 (D), and ‖u‖V⊗L2

P≤ CP

amin‖f ‖L2(D)

L(u) = F ⇔

u(ω, x) = 0 x ∈ ∂D, ω ∈ Ω

P (Ω)⊗ L2(D))

u(ω, ·) ∈ V ≡ H10 (D), and ‖u‖V⊗L2

P≤ CP

amin‖f ‖L2(D)

Assumption – random field parametrization

The stochastic coefficient a(ω, x) can be parametrized by afinite number of independent random variables

a(ω, x) = a(y1(ω), . . . , yN(ω), x)

We assume that y has a joint probability density functionρ(y) =

∏Nn=1 ρn(yn) : ΓN → R+

Then u(ω, x) = u(y1(ω), . . . , yN(ω), x) is a deterministic function ofthe random vector y.

Extensions to the case of infinitely many (countable) randomvariables is possible provided the solution u(y1, . . . , yn, . . . , x)has suitable decay properties w.r.t. n.(see e.g. [Cohen-Devore-Schwab, ’09,’10])

a(ω, x) = a(y1(ω), . . . , yN(ω), x)

Examples

Material with inclusions of random conductivity

a(ω, x) = a0 +∑N

n=N yn(ω)1Ωn (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∑N

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 +∑N

n=N yn(ω)1Ωn (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∑N

n=1 yn(ω)bn(x)

with yn ∼ N(0, 1), i.i.d.

random field with Lc=1/4

0 0.2 0.4 0.6 0.8 10

−0.5

Analytic regularityTheorem [Back-N.-Tamellini-Tempone ’11, Babuska-N.-Tempone ’05, Cohen-DeVore-Schwab ’09/’10]

Assume y bounded (e.g. y ∈ ΓN ≡ [−1, 1]N)

Let i = (i1, . . . , iN) ∈ NN and r = (r1, . . . , rN) > 0. Setri =

∏n r in

assume ‖1a∂ ia∂yi‖L∞(D) ≤ ri uniformly in y

‖∂ iu∂yi‖V ≤ C |i|!( 1

log 2r)i uniformly in y

u : ΓN → V is analytic and can be extended analyticially to

z ∈ CN :

N∑n=1

rn|zn − yn| < log 2 for some y ∈ ΓN

Better estimates on analyticity region can be obtained by complex analysis.

Analytic regularityTheorem [Back-N.-Tamellini-Tempone ’11, Babuska-N.-Tempone ’05, Cohen-DeVore-Schwab ’09/’10]

Assume y bounded (e.g. y ∈ ΓN ≡ [−1, 1]N)

Let i = (i1, . . . , iN) ∈ NN and r = (r1, . . . , rN) > 0. Setri =

∏n r in

assume ‖1a∂ ia∂yi‖L∞(D) ≤ ri uniformly in y

‖∂ iu∂yi‖V ≤ C |i|!( 1

log 2r)i uniformly in y

u : ΓN → V is analytic and can be extended analyticially to

z ∈ CN :

N∑n=1

rn|zn − yn| < log 2 for some y ∈ ΓN

Better estimates on analyticity region can be obtained by complex analysis.

Stochastic multivariate polynomial approximation

Outline

4 Conclusions

Idea: approximate the response function u(y, ·) by global multivariatepolynomials. Since u(y, ·) is analytic, we expect fast convergence.

Let Λ ⊂ NN be an index set of cardinality |Λ| = M , and consider themultivariate polynomial space

PΛ(ΓN) = span∏N

n=1 y pnn , with p = (p1, . . . , pN) ∈ Λ

Polynomial approximation

find M particular solutions up ∈ V , ∀p ∈ Λ and build

uΛ(y, x) =∑p∈Λ

up(x)y p11 y p2

2 · · · ypN

Compute statistics of functionals J(u), J ∈ H−1(D) of thesolution as E[J(u)] ≈ E[J(uΛ)]

PΛ(ΓN) = span∏N

n=1 y pnn , with p = (p1, . . . , pN) ∈ Λ

up(x)y p11 y p2

2 · · · ypN

PΛ(ΓN) = span∏N

n=1 y pnn , with p = (p1, . . . , pN) ∈ Λ

up(x)y p11 y p2

2 · · · ypN

Examples of pol. spaces: N = 2, p = 16

Tensor

product:pn ≤ w

0 2 4 6 8 10 12 14 160

Tensor Product (TP)

0 2 4 6 8 10 12 14 160

Total Degree (TD)

Total degree:∑n pn ≤ w

Hyperbolic

cross:∏n (pn + 1)≤ w + 1

0 2 4 6 8 10 12 14 160

Hyperbolic Cross (HC)

0 2 4 6 8 10 12 14 160

Smolyak (SM)

Smolyak:∑n f (pn) ≤ f (w)

f (p) =

0, p = 0

1, p = 1

dlog2(p)e, p ≥ 2

Stochastic multivariate polynomial approximation Galerkin projection

Galerkin projection[Ghanem-Spanos, Karniadakis et al, Matthies-Keese, Schwab-Todor et al., Knio-Le Maıtre et

al,Babuska et al.,. . . ]

Project the equation L(y)(u) = F onto the subspace V ⊗ PΛ(Γ)

Let ψjMj=1 be an orthonormal basis w.r.t. the probability density

ρ(y). Expand uΛ(y) on the basis: uΛ(y) =∑M

j=1 ujψj (y)

Galerkin formulation

Find uj ∈ V , j = 1, . . .M s.t.

E[L(y)(

M∑j=1

ujψj )ψi

]= E[Fψi ], i = 1, . . . ,M

This approach leads to solving M coupled deterministicproblems; difficult to assemble and need good preconditioners.

Stochastic multivariate polynomial approximation Collocation on sparse grids

Collocation approach[Smolyak ’63, Griebel et al ’98-’03-’04, Barthelmann-Novak-Ritter ’00, Hesthaven-Xiu ’05,

N.-Tempone-Webster ’08, Zabaras et al ’07]

1 Choose a set of points y(j) ∈ Γ, j = 1, . . . , M

2 Compute the solutions uj ∈ V : L(y(j))(uj ) = F3 Interpolate the obtained values: uΛ(y) =

∑Mj=1 ujφj (y).

φj ∈ PΛ(Γ): suitable combinations of Lagrange polynomials

Always leads to solving M uncoupled deterministic problems

The number M of points needed is larger than the dimension Mof the polynomial space (Except for tensor product spaces).

Tensor grids are impractical in high dimension (curse ofdimensionality)

Collocation approach[Smolyak ’63, Griebel et al ’98-’03-’04, Barthelmann-Novak-Ritter ’00, Hesthaven-Xiu ’05,

N.-Tempone-Webster ’08, Zabaras et al ’07]

1 Choose a set of points y(j) ∈ Γ, j = 1, . . . , M

2 Compute the solutions uj ∈ V : L(y(j))(uj ) = F3 Interpolate the obtained values: uΛ(y) =

∑Mj=1 ujφj (y).

φj ∈ PΛ(Γ): suitable combinations of Lagrange polynomials

Always leads to solving M uncoupled deterministic problems

The number M of points needed is larger than the dimension Mof the polynomial space (Except for tensor product spaces).

Tensor grids are impractical in high dimension (curse ofdimensionality)

Collocation on a (generalized) Sparse Grid

Let i = [i1, . . . , iN ] ∈ NN+ and m(i) : N+ → N+ an increasing function

1 1D polynomial interpolant operators: U m(in)n on m(in) abscissas.

We use either

Clenshaw-Curtis (extrema on Chebyshev polynomials)Gauss points w.r.t. the weight ρn, assuming that the probabilitydensity factorizes as ρ(y) =

∏Nn=1 ρn(yn)

2 Detail operator: ∆m(in)n = U m(in)

n −U m(in−1)n , U m(0)

n = 0.

3 Sparse grid approximation: on an index set Λ ⊂ NN

uΛ =∑i∈Λ

N⊗n=1

∆m(in)n [u]

By choosing properly the function m and the set Λ one can obtain apolynomial approximation in any given multivariate polynomialspace ([Back-N.-Tamellini-Tempone, 2010])

Examples of sparse grids: N = 2, max. polynomial degree p = 16

By choosing properly the function m and the set Λ one can obtain apolynomial approximation in any given multivariate polynomialspace ([Back-N.-Tamellini-Tempone, 2010])

Examples of sparse grids: N = 2, max. polynomial degree p = 16

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

Tensor Product (TP)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

Total Degree (TD)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

Hyperbolic Cross (HC)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

Smolyak Gauss (SM)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

Smolyak CC (SM)

A simple numerical test

1D problem:

−(a(x , ω)u(x , ω)′)′ = sin(πx), x ∈ (0, 1),

u(0, ω) = u(1, ω) = 0

Diffusion coefficient a(x , ω) = eγ(x ,ω): lognormal with eitherexponential or Gaussian covariance

γ(x , ω): stationary Gaussian random field, mean=0; std=1;correlation length: 0.3Karhunen-Loeve expansion

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Exponential cov.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−3

Gaussian cov.Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 18

A simple numerical test

Stochastic Collocation approximation:

Gauss-Hermite nodes

isotropic TD grid (m(i) = i; Λ ≡ i : |i|1 ≤ w)measured error ‖E[u]− E[uΛ]‖L2ρ

comparison with Monte Carlo

Exponential cov. Gaussian cov.

Stochastic multivariate polynomial approximation Optimized algorithms

Optimization of polynomial spaces

Galerkin

uSGΛ =

∑p∈Λ

up(x)ψp(y)

find uSGΛ by Galerkin projection

of the equation onPΛ = spanψp, p ∈ Λ.

Collocation

uSCΛ =

∑i∈Λ

⊗n=1,...,N

∆m(in)n [u].

Compute uSCΛ by collocation on

the corresponding sparse grid

Question: What is the best index set Λ in both cases?

Optimization of polynomial spaces

Galerkin

uSGΛ =

∑p∈Λ

up(x)ψp(y)

find uSGΛ by Galerkin projection

of the equation onPΛ = spanψp, p ∈ Λ.

Collocation

uSCΛ =

∑i∈Λ

⊗n=1,...,N

∆m(in)n [u].

Compute uSCΛ by collocation on

the corresponding sparse grid

Question: What is the best index set Λ in both cases?

Galerkin projection – best M term approximation

Galerkin optimality:

‖u − uSGΛ ‖V⊗L2

ρ(Γ) ≤ C infvΛ∈V⊗PΛ

‖u − vΛ‖V⊗L2ρ(Γ)

Let ψp, p ∈ NN be the orthonormal basis of multivariate

polynomials w.r.t. the denisty ρ(y) =∏N

n=1 ρn(yn) and vΛ thetruncated expansion of u

vΛ =∑p∈Λ

upψp, up = E[uψp]

Parseval’s identity: ‖u − vΛ‖2V⊗L2

ρ(Γ) =∑

p/∈Λ ‖up‖2V

Best M terms approximation

The optimal index set Λ of cardinality M is the one that contains theM largest Legendre coefficients ‖up‖V

vΛ =∑p∈Λ

upψp, up = E[uψp]

ρ(Γ) =∑

p/∈Λ ‖up‖2V

vΛ =∑p∈Λ

upψp, up = E[uψp]

ρ(Γ) =∑

p/∈Λ ‖up‖2V

vΛ =∑p∈Λ

upψp, up = E[uψp]

ρ(Γ) =∑

p/∈Λ ‖up‖2V

Estimate of Fourier coefficients

For the diffusion problem with uniform random variables, the solutionu(y) is analytic in Γ = [−1, 1]N and the following estimate of theLegendre coefficients holds [Cohen-DeVore-Schwab ’10, Back-N.-Tamellini-Tempone ’11]

‖up‖V ≤ C0e−∑

n gnpn|p|!p!

for some gn > 0, with |p| =∑

n pn, p! =∏

n pn!.

Then the optimal index set of level w is (TD-FC)

Λ(w) =p ∈ NN :

gnpn − log|p|!p!≤ w

In practice, we estimate numerically the rates gn by inexpensive 1Danalyses.

‖up‖V ≤ C0e−∑

n gnpn|p|!p!

n pn, p! =∏

n pn!.

Λ(w) =p ∈ NN :

‖up‖V ≤ C0e−∑

n gnpn|p|!p!

n pn, p! =∏

n pn!.

Λ(w) =p ∈ NN :

Numerical tests

We consider the 1D problem−(a(x , y)u(x , y)′)′ = 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 :

gnpn ≤ w

(Aniso) TD-FC space:

Λ(w) =

p ∈ NN :

N∑n=1

The rates gn have been estimated numerically by inexpensive 1Danalyses.

Numerical tests

We consider the 1D problem−(a(x , y)u(x , y)′)′ = 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 :

gnpn ≤ w

(Aniso) TD-FC space:

Λ(w) =

p ∈ NN :

N∑n=1

The rates gn have been estimated numerically by inexpensive 1Danalyses.

A numerical check: a(x, y) = 1 + 0.1xy1 + 0.5x2y2

0 20 40 60 80 10010

10−18

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

Legendre coeffTD apprTD−FC appr

Figure: Legendre coeffs of Θ(u)in lexicographic order, with TDand TD-FC estimates

0 20 40 60 8010

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

best M termTDiso−TDTD−FC

Figure: Convergence plot for‖Θ(u)−Θ(uM)‖2

L2ρ(Γ) w.r.t.

M = |Λ|

The Legendre coefficients have been computed with a sufficiently high

level sparse grids.Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 27

Optimization of sparse grids

uM = SmΛ [u] =

∑i∈Λ

N⊗n=1

∆m(in)n [u].

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 (errorcontribution)∆W (i): how much work, i.e. number of evaluations, increases ifi is added to Λ (work contribution)

Then estimate the profit of each i as

P(i) =∆E (i)

∆W (i)

and build the sparse grid using the set Λ of the M indices with thelargest profit.

uM = SmΛ [u] =

∑i∈Λ

N⊗n=1

∆m(in)n [u].

P(i) =∆E (i)

∆W (i)

uM = SmΛ [u] =

∑i∈Λ

N⊗n=1

∆m(in)n [u].

P(i) =∆E (i)

∆W (i)

Estimates for ∆W and ∆E

1 ∆W (i): for nested points (e.g. Clenshaw Curtis, Gauss-Patterson)

∆W (i) = nb. new pts. inN⊗

∆m(in)n =

N∏n=1

( m(in)−m(in − 1) )

2 ∆E (i): we use the heuristic argument: use expansion onorthnormal basis u =

∑p upψp

∆E (i) = ‖∆m(i)[u]‖V⊗L2ρ

= ‖∑p

up∆m(i)ψp‖V⊗L2ρ

≤∑

p≥m(i−1)

‖up‖V‖∆m(i)ψp‖L2ρ≈ ‖um(i−1)‖V‖∆m(i)ψm(i−1)‖L2

where ‖um(i−1)‖V : estimated with a-priori / a-posterioriL(m(i)) := ‖∆m(i)ψm(i−1)‖L2

ρ: estimated numerically

∆m(in)n =

N∏n=1

( m(in)−m(in − 1) )

∑p upψp

∆E (i) = ‖∆m(i)[u]‖V⊗L2ρ

= ‖∑p

≤∑

p≥m(i−1)

∆m(in)n =

N∏n=1

( m(in)−m(in − 1) )

∑p upψp

∆E (i) = ‖∆m(i)[u]‖V⊗L2ρ

= ‖∑p

≤∑

p≥m(i−1)

All the pieces together

Optimal index set

Λ(w) =

i ∈ NN

+ :N∑

m(in − 1)gn − log|m(i− 1)|!m(i− 1)!

N∑n=1

logL(m(in))

m(in)−m(in − 1)≤ w

(EW - Error Work grids)

Legendre coeff + L(m(i)) = error estimate

work estimate

A numerical check: a(x, y) = 1 + 0.1xy1 + 0.5x2y2

0 5 10 15 20 25 30

10−15

10−12

10−9

10−6

10−3

∆E(i)

Legm(i−1)[ψ(u)]

Legm(i−1)[ψ(u)] ⋅ Leb[m(i)]

Figure: Estimates of ∆E (i) forΘ(u)] in lexicographic order

0 20 40 60 80 100 120 140

10−9

10−7

10−5

10−3

10−1

iso SMEWadaptivebest M terms

Figure: Convergence plot for‖Θ(u)−Θ(uM)‖2

L2ρ(Γ) w.r.t. |Λ|

Nested Clenshaw-Curtis knots

The terms ∆E (i) have been computed with a sufficiently highlevel sparse grids.

Comparison with dimension adaptive algorithm [Gerstner-Griebel ’03, Klimke, PhD ’06]

Stochastic multivariate polynomial approximation Numerical results

Numerical test - 1D stationary lognormal fieldL = 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) = eγ(x,y)

µγ(x) = 0

Covγ(x, x′) = σ2e−|x1−x′1|

We approximate γ as

γ(y, x) ≈ µ(x) + σa0y0 + σ

K∑k=1

[y2k−1 cos

(πLkx1

)+ y2k sin

(πLkx1

)]with yi ∼ N (0, 1), i.i.d.

Given the Fourier series σ2e−|z|2

LC2 =∑∞

k=0 ck cos(πL kz), ak =

√ck .

Numerical test - 1D stationary lognormal field

Quantity of interest: effective permeability

E[Φ(u)], with Φ =

[∫ L

k(·, x)∂u(·, x)

∂xdx

Convergence: |E[Φ(uSG )]− E[Φ(u)]|

We compare Monte Carlo estimate with Knapsack grids based onGauss-Hermite-Patterson points (nested Gauss-Hermite)

Estimate of Hermite coefficients decay:

for the simpler problem ∇ · a(y)∇u(y, x) = f ,

a(y) = eb0+∑N

n=1 ynbn , we have ‖ui‖V = C binn√in!

Heuristic: use the same ansaz ‖ui‖V ≈ C∏N

n=1e−gnin√

in!but

estimate the rates gn numerically.

E[Φ(u)], with Φ =

[∫ L

k(·, x)∂u(·, x)

∂xdx

a(y) = eb0+∑N

n=1e−gnin√

in!but

E[Φ(u)], with Φ =

[∫ L

k(·, x)∂u(·, x)

∂xdx

a(y) = eb0+∑N

n=1e−gnin√

in!but

Correlation length: LC = 0.2Standard deviation: σ = 0.3 (c.o.v. ∼ 30%)

10−10

10−8

10−6

10−4

10−2

sparse grid

1/N0.5

1/N1.7

MC run1

MC run2

MC run3

MC run4

The optimal set construction automatically adds new variableswhen needed.

No need to truncate a-priori the random fieldFabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 34

How the sparse grid looks like

−5 0 5−5

Polynomial approximation by discrete projection on random points

Outline

4 Conclusions

Discrete L2 projection using random evaluations

(see poster 11 – G. Migliorati)

Another way to compute a polynomial approximation (besidesGalerkin and sparse grid collocation) consists in doing a discrete leastsquare approx. using random evaluations (see e.g. [Burkardt-Eldred

2009, Hosder-Walters et al. 2010, Eldred 2011, Blatman-Sudret 2008, ...])

1 Generate M random i.i.d. samples yk ∈ Γ, k = 1, . . . ,M2 Compute the corresponding solutions uk = u(y(k))3 Find the discrete least square approximation ΠΛ,ω

M u ∈ V ⊗ PΛ(Γ)

ΠΛ,ωM u = argmin

v∈V⊗PΛ(Γ)

M∑k=1

‖uk − v(y(k))‖2V

Two relevant questionsFor a given set Λ, how many samples should one use?What is the accuracy of the random discrete least squareapprox.?Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 39

ΠΛ,ωM u = argmin

v∈V⊗PΛ(Γ)

M∑k=1

‖uk − v(y(k))‖2V

ΠΛ,ωM u = argmin

v∈V⊗PΛ(Γ)

M∑k=1

‖uk − v(y(k))‖2V

Some theoretical results [Migliorati-N.-von Schwerin-Tempone ’11]

For functions φ : Γ ⊂ RN → R, define

continuous norm: ‖φ‖2L2ρ

Γv 2(y)ρ(y)dy

discrete norm: ‖φ‖2M,ω = 1

∑Mi=1 φ(yi )

2, with yi ∼ ρ(y)dy, i.i.d.

random discrete least square projection: ΠΛ,ωM φ ∈ PΛ(Γ),

ΠΛ,ωM φ = argmin

v∈PΛ(Γ)

‖φ− v‖2M,ω.

Theorem 1

Let Cω(M ,Λ) := supv∈PΛ(Γ)

‖v‖2L2ρ

‖v‖2M,ω

. Then

1 Cω(M ,Λ)→ 1 almost surely when M →∞2 ‖φ− ΠΛ,ω

M φ‖L2ρ≤ (1 +

√Cω(M ,Λ)) infv∈PΛ(Γ) ‖φ− v‖L∞

So far we have only a result for 1D functions, Γ = [−1, 1], uniformdistribution and approx. in the polynomial space Pw of degree w.

Theorem 2

For any α ∈ (0, 1), let M be such that M3 log((M+1)/α)

= 4√

Then, it holds

(‖φ−Πw,ω

M φ‖L2ρ≤

(1 + 2

√3 log

v∈Pw

‖φ− v‖L∞

)≥ 1− α.

Notice that log((M + 1)/α) ≈ log(Cw2/α)

These results are confirmed numerically; the high dimentionalcase seems more foregiving w.r.t. the constraint M ∼ #Λ2

To achieve an approximation in PΛ, does this technique need lesssampling points than the corresponding sparse grid ?

Theorem 2

= 4√

Then, it holds

(‖φ−Πw,ω

M φ‖L2ρ≤

(1 + 2

√3 log

v∈Pw

‖φ− v‖L∞

)≥ 1− α.

Theorem 2

= 4√

Then, it holds

(‖φ−Πw,ω

M φ‖L2ρ≤

(1 + 2

√3 log

v∈Pw

‖φ− v‖L∞

)≥ 1− α.

Conclusions

Outline

4 Conclusions

Conclusions

Solutions to elliptic equations with random coefficients typicallyfeature analytic dependence on the parameters. Polynomialapproximations are very effective.

Sharp a-priori / a-posteriori analysis of the decay of the expansion ofthe solution in polynomial chaos allows to construct optimizedpolynomial spaces / sparse grids that provide effectiveapproximations also in the infinite dimensional case.

Discrete least square projection using random evaluations is apossible alternative to Galerkin or Collocation approaches. However,a better understanding is needed on the stability of the projectionand the correct number of samples to use.

Conclusions

References

G. Migliorati and F. Nobile and E. von Schwrin and R. Tempone

Analysis of the discrete L2 projection on polynomial spaces with random evaluations,submitted. Available as MOX-Report.

J. Back and F. Nobile and L. Tamellini and R. TemponeOn the optimal polynomial approximation of stochastic PDEs by Galerkin and Collocationmethods, MOX Report 23/2011, to appear in M3AS.

I. Babuska, 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. TemponeAnalysis and implementation issues for the numerical approximation of parabolic equationswith random coefficients, IJNME, 80:979–1006, 2009

F. Nobile, R. Tempone and C. WebsterAn anisotropic sparse grid stochastic collocation method for PDEs with random inputdata, SIAM J. Numer. Anal., 46(5):2411–2442, 2008

Stochastic Polynomial approximation of PDEs with random …€¦ · Stochastic Polynomial...

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