Stochastic Polynomial approximation of PDEs with random coefficients Fabio Nobile CSQI-MATHICSE, EPFL, Switzerland and 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 Life Science Problems Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 1
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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
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]).
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]).
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]).
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])
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])
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])
Stochastic multivariate polynomial approximation Collocation on sparse grids
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
Stochastic multivariate polynomial approximation Collocation on sparse grids
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
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!
(1)
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 :
∑n
gnpn − log|p|!p!≤ w
In practice, we estimate numerically the rates gn by inexpensive 1Danalyses.
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!
(1)
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 :
∑n
gnpn − log|p|!p!≤ w
In practice, we estimate numerically the rates gn by inexpensive 1Danalyses.
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!
(1)
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 :
∑n
gnpn − log|p|!p!≤ w
In practice, we estimate numerically the rates gn by inexpensive 1Danalyses.
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.
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.
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.
Polynomial approximation by discrete projection on random points
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)
1
M
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
Polynomial approximation by discrete projection on random points
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)
1
M
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
Polynomial approximation by discrete projection on random points
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)
1
M
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
Polynomial approximation by discrete projection on random points
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
M
∑Mi=1 φ(yi )
2, with yi ∼ ρ(y)dy, i.i.d.
random discrete least square projection: ΠΛ,ωM φ ∈ PΛ(Γ),
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.
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