Discrete least squares polynomial approximation with random evaluations – application to PDEs with random parameters F. Nobile 1 , G. Migliorati 1 , R. Tempone 2 1 CSQI-MATHICSE, EPFL, Switzerland 2 AMCS and SRI-UQ Center, KAUST, Saudi Arabia Acknowlegments : A. Cohen, A. Chkifa (UPMC - Paris VI), E. von Schwerin (KTH), Advances in UQ Methods, Algorthms and Applications KAUST, January 6-9, 2015 F. Nobile (EPFL) Discrete least squares for random PDEs UQAW 2015, KAUST 1
Outline Introduction – PDEs with random parameters 1 Stochastic polynomial approximation 2 Discrete projection using random evaluations 3 Stability Convergence results in expectation and probability The case of noisy observations Conclusions 4 F. Nobile (EPFL) Discrete least squares for random PDEs UQAW 2015, KAUST 2
Introduction – PDEs with random parameters Outline Introduction – PDEs with random parameters 1 Stochastic polynomial approximation 2 Discrete projection using random evaluations 3 Stability Convergence results in expectation and probability The case of noisy observations Conclusions 4 F. Nobile (EPFL) Discrete least squares for random PDEs UQAW 2015, KAUST 3
Introduction – PDEs with random parameters UQ for deterministic PDE models Consider a deterministic PDE model in D ⊂ R d find u : L ( y )( u ) = F (1) with suitable boundary / initial conditions. The operator L ( y ) depends on a vector of N parameters: y = ( y 1 , . . . , y N ) ∈ R N ( N = ∞ when dealing with distributed fields). Often in applications the parameters y are not perfectly known or are intrinsically variable. Examples are: subsurface modeling: porous media flows; seismic waves; basin evolutions; ... modeling of living tissues: mechanical response; growth models; material science: properties of composite materials Probabilistic approach: y is a random vector with probability density function ρ : Γ → R + . F. Nobile (EPFL) Discrete least squares for random PDEs UQAW 2015, KAUST 4
Introduction – PDEs with random parameters UQ for deterministic PDE models Consider a deterministic PDE model in D ⊂ R d find u : L ( y )( u ) = F (1) with suitable boundary / initial conditions. The operator L ( y ) depends on a vector of N parameters: y = ( y 1 , . . . , y N ) ∈ R N ( N = ∞ when dealing with distributed fields). Often in applications the parameters y are not perfectly known or are intrinsically variable. Examples are: subsurface modeling: porous media flows; seismic waves; basin evolutions; ... modeling of living tissues: mechanical response; growth models; material science: properties of composite materials Probabilistic approach: y is a random vector with probability density function ρ : Γ → R + . F. Nobile (EPFL) Discrete least squares for random PDEs UQAW 2015, KAUST 4
Introduction – PDEs with random parameters UQ for deterministic PDE models Consider a deterministic PDE model in D ⊂ R d find u : L ( y )( u ) = F (1) with suitable boundary / initial conditions. The operator L ( y ) depends on a vector of N parameters: y = ( y 1 , . . . , y N ) ∈ R N ( N = ∞ when dealing with distributed fields). Often in applications the parameters y are not perfectly known or are intrinsically variable. Examples are: subsurface modeling: porous media flows; seismic waves; basin evolutions; ... modeling of living tissues: mechanical response; growth models; material science: properties of composite materials Probabilistic approach: y is a random vector with probability density function ρ : Γ → R + . F. Nobile (EPFL) Discrete least squares for random PDEs UQAW 2015, KAUST 4
Introduction – PDEs with random parameters UQ for deterministic PDE models Consider a deterministic PDE model in D ⊂ R d find u : L ( y )( u ) = F (1) with suitable boundary / initial conditions. The operator L ( y ) depends on a vector of N parameters: y = ( y 1 , . . . , y N ) ∈ R N ( N = ∞ when dealing with distributed fields). Often in applications the parameters y are not perfectly known or are intrinsically variable. Examples are: subsurface modeling: porous media flows; seismic waves; basin evolutions; ... modeling of living tissues: mechanical response; growth models; material science: properties of composite materials Probabilistic approach: y is a random vector with probability density function ρ : Γ → R + . F. Nobile (EPFL) Discrete least squares for random PDEs UQAW 2015, KAUST 4
Introduction – PDEs with random parameters UQ for deterministic PDE models Assumption: ∀ y ∈ Γ the problem admits a unique solution u ∈ V in a Hilbert space V . Moreover, � u ( y ) � V ≤ C ( y ) �F� V ′ Then, the PDE (1) induces a map u = u ( y ) : Γ → V . if C ( y ) ∈ L p ρ (Γ), then u ∈ L p ρ (Γ , V ). Goals: Compute statistics of the solution pointwise Expected value : ¯ u ( x ) = E [ u ( x , · )], x ∈ D u ( x )) 2 ]( x ) pointwise Variance : Var [ u ]( x ) = E [( u ( x , · ) − ¯ two points corr. : Cov u ( x 1 , x 2 ) = E [( u ( x 1 , · ) − ¯ u ( x 1 ))( u ( x 2 , · ) − ¯ u ( x 2 ))] or of specific Quantities of Interest Q ( u ) : V → R . Then ϕ ( y ) = Q ( u ( y )) is a real-valued function of the random vector y and we would like to approximate its law. F. Nobile (EPFL) Discrete least squares for random PDEs UQAW 2015, KAUST 5
Introduction – PDEs with random parameters UQ for deterministic PDE models Assumption: ∀ y ∈ Γ the problem admits a unique solution u ∈ V in a Hilbert space V . Moreover, � u ( y ) � V ≤ C ( y ) �F� V ′ Then, the PDE (1) induces a map u = u ( y ) : Γ → V . if C ( y ) ∈ L p ρ (Γ), then u ∈ L p ρ (Γ , V ). Goals: Compute statistics of the solution pointwise Expected value : ¯ u ( x ) = E [ u ( x , · )], x ∈ D u ( x )) 2 ]( x ) pointwise Variance : Var [ u ]( x ) = E [( u ( x , · ) − ¯ two points corr. : Cov u ( x 1 , x 2 ) = E [( u ( x 1 , · ) − ¯ u ( x 1 ))( u ( x 2 , · ) − ¯ u ( x 2 ))] or of specific Quantities of Interest Q ( u ) : V → R . Then ϕ ( y ) = Q ( u ( y )) is a real-valued function of the random vector y and we would like to approximate its law. F. Nobile (EPFL) Discrete least squares for random PDEs UQAW 2015, KAUST 5
Introduction – PDEs with random parameters UQ for deterministic PDE models Assumption: ∀ y ∈ Γ the problem admits a unique solution u ∈ V in a Hilbert space V . Moreover, � u ( y ) � V ≤ C ( y ) �F� V ′ Then, the PDE (1) induces a map u = u ( y ) : Γ → V . if C ( y ) ∈ L p ρ (Γ), then u ∈ L p ρ (Γ , V ). Goals: Compute statistics of the solution pointwise Expected value : ¯ u ( x ) = E [ u ( x , · )], x ∈ D u ( x )) 2 ]( x ) pointwise Variance : Var [ u ]( x ) = E [( u ( x , · ) − ¯ two points corr. : Cov u ( x 1 , x 2 ) = E [( u ( x 1 , · ) − ¯ u ( x 1 ))( u ( x 2 , · ) − ¯ u ( x 2 ))] or of specific Quantities of Interest Q ( u ) : V → R . Then ϕ ( y ) = Q ( u ( y )) is a real-valued function of the random vector y and we would like to approximate its law. F. Nobile (EPFL) Discrete least squares for random PDEs UQAW 2015, KAUST 5
Introduction – PDEs with random parameters Example: Elliptic PDE with random coefficients � − div( a ( y , x ) ∇ u ( y , x )) = f ( x ) x ∈ D , y ∈ Γ , u ( y , x ) = 0 x ∈ ∂ D , y ∈ Γ with a min ( y ) = inf x ∈ D a ( y , x ) > 0 for all y ∈ Γ and f ∈ L 2 ( D ). Then C P u ( y ) ∈ V ≡ H 1 ∀ y ∈ Γ , 0 ( D ) , and � u ( y ) � V ≤ a min ( y ) � f � L 2 ( D ) . Inclusions problem Random fields problem y describes the a ( y , x ) is a random field, random field with L c =1/4 conductivity in each 1 e.g. lognormal: 0.9 2.5 inclusion a ( y , x ) = e γ ( y , x ) with γ 0.8 2 0.7 1.5 0.6 1 expanded e.g. in 0.5 N 0.4 0.5 Karhunen-Lo` eve series 0.3 0 � a ( y , x ) = a 0 + y n ✶ D n ( x ) 0.2 −0.5 0.1 −1 0 n = N 0 0.2 0.4 0.6 0.8 1 ∞ � � γ ( y , x ) = λ n y n b n ( x ) , y n ∼ N (0 , 1) i . i . d . n =1 F. Nobile (EPFL) Discrete least squares for random PDEs UQAW 2015, KAUST 6
Stochastic polynomial approximation Outline Introduction – PDEs with random parameters 1 Stochastic polynomial approximation 2 Discrete projection using random evaluations 3 Stability Convergence results in expectation and probability The case of noisy observations Conclusions 4 F. Nobile (EPFL) Discrete least squares for random PDEs UQAW 2015, KAUST 7
Stochastic polynomial approximation Stochastic multivariate polynomial approximation The parameter-to-solution map u ( y ) : Γ → V is often smooth (even analytic for the elliptic diffusion model). It is therefore sound to approximate it by global multivariate polynomials. Let Λ ⊂ N N be an index set of cardinality | Λ | = M , and consider the multivariate polynomial space �� N � n =1 y p n P Λ (Γ) = span n , with p = ( p 1 , . . . , p N ) ∈ Λ We seek an approximation P Λ u ∈ P Λ (Γ) ⊗ V . The optimal choice of Λ depends heavily on the problem at hand and the “structure” of the map u ( y ). Definition . An index set Λ is downward closed if p ∈ Λ and q ≤ p = ⇒ q ∈ Λ F. Nobile (EPFL) Discrete least squares for random PDEs UQAW 2015, KAUST 8
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