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
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 3
Elliptic PDE with random coefficients
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
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 4
Elliptic PDE with random coefficients
Elliptic PDE with random coefficients
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]).
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 5
Elliptic PDE with random coefficients
Elliptic PDE with random coefficients
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]).
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 5
Elliptic PDE with random coefficients
Elliptic PDE with random coefficients
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]).
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 5
Elliptic PDE with random coefficients
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])
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 6
Elliptic PDE with random coefficients
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])
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 6
Elliptic PDE with random coefficients
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])
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 6
Elliptic PDE with random coefficients
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.
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 7
Elliptic PDE with random coefficients
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
0.1
0.2
0.3
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0.5
0.6
0.7
0.8
0.9
1
−1
−0.5
0
0.5
1
1.5
2
2.5
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 7
Elliptic PDE with random coefficients
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
n .
assume ‖1a∂ ia∂yi‖L∞(D) ≤ ri uniformly in y
Then
‖∂ iu∂yi‖V ≤ C |i|!( 1
log 2r)i uniformly in y
u : ΓN → V is analytic and can be extended analyticially to
Σ =
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.
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 8
Elliptic PDE with random coefficients
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
n .
assume ‖1a∂ ia∂yi‖L∞(D) ≤ ri uniformly in y
Then
‖∂ iu∂yi‖V ≤ C |i|!( 1
log 2r)i uniformly in y
u : ΓN → V is analytic and can be extended analyticially to
Σ =
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.
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 8
Stochastic multivariate polynomial approximation
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
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 10
Stochastic multivariate polynomial approximation
Stochastic multivariate polynomial approximation
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
N
Compute statistics of functionals J(u), J ∈ H−1(D) of thesolution as E[J(u)] ≈ E[J(uΛ)]
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 11
Stochastic multivariate polynomial approximation
Stochastic multivariate polynomial approximation
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
N
Compute statistics of functionals J(u), J ∈ H−1(D) of thesolution as E[J(u)] ≈ E[J(uΛ)]
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 11
Stochastic multivariate polynomial approximation
Stochastic multivariate polynomial approximation
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
N
Compute statistics of functionals J(u), J ∈ H−1(D) of thesolution as E[J(u)] ≈ E[J(uΛ)]
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 11
Stochastic multivariate polynomial approximation
Examples of pol. spaces: N = 2, p = 16
Tensor
product:pn ≤ w
0 2 4 6 8 10 12 14 160
2
4
6
8
10
12
14
16
p1
p2
Tensor Product (TP)
0 2 4 6 8 10 12 14 160
2
4
6
8
10
12
14
16
p1
p2
Total Degree (TD)
Total degree:∑n pn ≤ w
Hyperbolic
cross:∏n (pn + 1)≤ w + 1
0 2 4 6 8 10 12 14 160
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4
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10
12
14
16
p1
p2
Hyperbolic Cross (HC)
0 2 4 6 8 10 12 14 160
2
4
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12
14
16
p1
p2
Smolyak (SM)
Smolyak:∑n f (pn) ≤ f (w)
f (p) =
0, p = 0
1, p = 1
dlog2(p)e, p ≥ 2
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 12
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.
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 13
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)
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 15
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)
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 15
Stochastic multivariate polynomial approximation Collocation on sparse grids
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]
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 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
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 17
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
−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
0
0.2
0.4
0.6
0.8
1
y1
y2
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
0
0.2
0.4
0.6
0.8
1
y1
y2
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
0
0.2
0.4
0.6
0.8
1
y1
y2
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
0
0.2
0.4
0.6
0.8
1
y1
y2
Smolyak Gauss (SM)
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−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
y1
y2
Smolyak CC (SM)
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 17
Stochastic multivariate polynomial approximation Collocation on sparse grids
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
−3
−2
−1
0
1
2
3
x
log
− c
ondu
ctiv
ity
Exponential cov.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−3
−2
−1
0
1
2
3
x
log
− c
ondu
ctiv
ity
Gaussian cov.Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 18
Stochastic multivariate polynomial approximation Collocation on sparse grids
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.
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 19
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?
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 20
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?
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 20
Stochastic multivariate polynomial approximation Optimized algorithms
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
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 21
Stochastic multivariate polynomial approximation Optimized algorithms
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
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 21
Stochastic multivariate polynomial approximation Optimized algorithms
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
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 21
Stochastic multivariate polynomial approximation Optimized algorithms
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
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 21
Stochastic multivariate polynomial approximation Optimized algorithms
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!
(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.
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 24
Stochastic multivariate polynomial approximation Optimized algorithms
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!
(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.
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 24
Stochastic multivariate polynomial approximation Optimized algorithms
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!
(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.
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 24
Stochastic multivariate polynomial approximation Optimized algorithms
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 :
∑n
gnpn ≤ w
.
(Aniso) TD-FC space:
Λ(w) =
p ∈ NN :
N∑n=1
gnpn − log|p|!p!≤ w
.
The rates gn have been estimated numerically by inexpensive 1Danalyses.
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 26
Stochastic multivariate polynomial approximation Optimized algorithms
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 :
∑n
gnpn ≤ w
.
(Aniso) TD-FC space:
Λ(w) =
p ∈ NN :
N∑n=1
gnpn − log|p|!p!≤ w
.
The rates gn have been estimated numerically by inexpensive 1Danalyses.
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 26
Stochastic multivariate polynomial approximation Optimized algorithms
A numerical check: a(x, y) = 1 + 0.1xy1 + 0.5x2y2
0 20 40 60 80 10010
−20
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
−9
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
Stochastic multivariate polynomial approximation Optimized algorithms
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.
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 28
Stochastic multivariate polynomial approximation Optimized algorithms
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.
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 28
Stochastic multivariate polynomial approximation Optimized algorithms
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.
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 28
Stochastic multivariate polynomial approximation Optimized algorithms
Estimates for ∆W and ∆E
1 ∆W (i): for nested points (e.g. Clenshaw Curtis, Gauss-Patterson)
∆W (i) = nb. new pts. inN⊗
n=1
∆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
ρ
where ‖um(i−1)‖V : estimated with a-priori / a-posterioriL(m(i)) := ‖∆m(i)ψm(i−1)‖L2
ρ: estimated numerically
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 29
Stochastic multivariate polynomial approximation Optimized algorithms
Estimates for ∆W and ∆E
1 ∆W (i): for nested points (e.g. Clenshaw Curtis, Gauss-Patterson)
∆W (i) = nb. new pts. inN⊗
n=1
∆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
ρ
where ‖um(i−1)‖V : estimated with a-priori / a-posterioriL(m(i)) := ‖∆m(i)ψm(i−1)‖L2
ρ: estimated numerically
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 29
Stochastic multivariate polynomial approximation Optimized algorithms
Estimates for ∆W and ∆E
1 ∆W (i): for nested points (e.g. Clenshaw Curtis, Gauss-Patterson)
∆W (i) = nb. new pts. inN⊗
n=1
∆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
ρ
where ‖um(i−1)‖V : estimated with a-priori / a-posterioriL(m(i)) := ‖∆m(i)ψm(i−1)‖L2
ρ: estimated numerically
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 29
Stochastic multivariate polynomial approximation Optimized algorithms
All the pieces together
Optimal index set
Λ(w) =
i ∈ NN
+ :N∑
i=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)
where
Legendre coeff + L(m(i)) = error estimate
work estimate
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 30
Stochastic multivariate polynomial approximation Optimized algorithms
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
100
∆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]
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 31
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|
2
LC2
We approximate γ as
γ(y, x) ≈ µ(x) + σa0y0 + σ
K∑k=1
ak
[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 .
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 32
Stochastic multivariate polynomial approximation Numerical results
Numerical test - 1D stationary lognormal field
Quantity of interest: effective permeability
E[Φ(u)], with Φ =
[∫ L
0
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.
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 33
Stochastic multivariate polynomial approximation Numerical results
Numerical test - 1D stationary lognormal field
Quantity of interest: effective permeability
E[Φ(u)], with Φ =
[∫ L
0
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.
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 33
Stochastic multivariate polynomial approximation Numerical results
Numerical test - 1D stationary lognormal field
Quantity of interest: effective permeability
E[Φ(u)], with Φ =
[∫ L
0
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.
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 33
Stochastic multivariate polynomial approximation Numerical results
Numerical test - 1D stationary lognormal field
Correlation length: LC = 0.2Standard deviation: σ = 0.3 (c.o.v. ∼ 30%)
100
101
102
103
104
105
10−10
10−8
10−6
10−4
10−2
100
4
1114
17
19
2124
29
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
Stochastic multivariate polynomial approximation Numerical results
How the sparse grid looks like
−5 0 5−5
0
5
−5 0 5−5
0
5
−5 0 5−5
0
5
−5 0 5−5
0
5
−5 0 5−5
0
5
−5 0 5−5
0
5
−5 0 5−5
0
5
−5 0 5−5
0
5
−5 0 5−5
0
5
−5 0 5−5
0
5
−5 0 5−5
0
5
−5 0 5−5
0
5
−5 0 5−5
0
5
−5 0 5−5
0
5
−5 0 5−5
0
5
−5 0 5−5
0
5
−5 0 5−5
0
5
−5 0 5−5
0
5
−5 0 5−5
0
5
−5 0 5−5
0
5
−5 0 5−5
0
5
η4
η3
η1
η2
η5
η6
η2
η3
η4
η5
η6
η7
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 35
Polynomial approximation by discrete projection on random points
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
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 38
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),
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∞
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 40
Polynomial approximation by discrete projection on random points
Some theoretical results [Migliorati-N.-von Schwerin-Tempone ’11]
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√
3w2
Then, it holds
P
(‖φ−Πw,ω
M φ‖L2ρ≤
(1 + 2
√3 log
M + 1
α
)inf
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 ?
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 41
Polynomial approximation by discrete projection on random points
Some theoretical results [Migliorati-N.-von Schwerin-Tempone ’11]
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√
3w2
Then, it holds
P
(‖φ−Πw,ω
M φ‖L2ρ≤
(1 + 2
√3 log
M + 1
α
)inf
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 ?
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 41
Polynomial approximation by discrete projection on random points
Some theoretical results [Migliorati-N.-von Schwerin-Tempone ’11]
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√
3w2
Then, it holds
P
(‖φ−Πw,ω
M φ‖L2ρ≤
(1 + 2
√3 log
M + 1
α
)inf
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 ?
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 41
Conclusions
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
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 48
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.
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 49
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
Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 50