Stochastic Polynomial approximation of PDEs with random coefficients - PowerPoint PPT Presentation

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

Outline Elliptic PDE with random coefficients 1 Stochastic multivariate polynomial approximation 2 Galerkin projection Collocation on sparse grids Optimized algorithms Numerical results Polynomial approximation by discrete projection on random points 3 Conclusions 4 Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 3

Elliptic PDE with random coefficients Outline Elliptic PDE with random coefficients 1 Stochastic multivariate polynomial approximation 2 Galerkin projection Collocation on sparse grids Optimized algorithms Numerical results Polynomial approximation by discrete projection on random points 3 Conclusions 4 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 � − div( a ( ω, x ) ∇ u ( ω, x )) = f ( x ) x ∈ D , ω ∈ Ω , L ( u ) = F ⇔ u ( ω, x ) = 0 x ∈ ∂ D , ω ∈ Ω where a : Ω × D → R is a second order random field such that a ≥ a min > 0 a.e. in Ω × D f ∈ L 2 ( D ) deterministic (could be stochastic as well, f ∈ L 2 P (Ω) ⊗ L 2 ( D )) By Lax-Milgram lemma, for a.e. ω ∈ Ω, there exists a unique solution P ≤ C P u ( ω, · ) ∈ V ≡ H 1 0 ( D ) , and � u � V ⊗ L 2 � f � L 2 ( D ) a min The uniform coercivity assumption can be relaxed to E [ a min ( 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 � − div( a ( ω, x ) ∇ u ( ω, x )) = f ( x ) x ∈ D , ω ∈ Ω , L ( u ) = F ⇔ u ( ω, x ) = 0 x ∈ ∂ D , ω ∈ Ω where a : Ω × D → R is a second order random field such that a ≥ a min > 0 a.e. in Ω × D f ∈ L 2 ( D ) deterministic (could be stochastic as well, f ∈ L 2 P (Ω) ⊗ L 2 ( D )) By Lax-Milgram lemma, for a.e. ω ∈ Ω, there exists a unique solution P ≤ C P u ( ω, · ) ∈ V ≡ H 1 0 ( D ) , and � u � V ⊗ L 2 � f � L 2 ( D ) a min The uniform coercivity assumption can be relaxed to E [ a min ( 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 � − div( a ( ω, x ) ∇ u ( ω, x )) = f ( x ) x ∈ D , ω ∈ Ω , L ( u ) = F ⇔ u ( ω, x ) = 0 x ∈ ∂ D , ω ∈ Ω where a : Ω × D → R is a second order random field such that a ≥ a min > 0 a.e. in Ω × D f ∈ L 2 ( D ) deterministic (could be stochastic as well, f ∈ L 2 P (Ω) ⊗ L 2 ( D )) By Lax-Milgram lemma, for a.e. ω ∈ Ω, there exists a unique solution P ≤ C P u ( ω, · ) ∈ V ≡ H 1 0 ( D ) , and � u � V ⊗ L 2 � f � L 2 ( D ) a min The uniform coercivity assumption can be relaxed to E [ a min ( 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 a finite number of independent random variables a ( ω, x ) = a ( y 1 ( ω ) , . . . , y N ( ω ) , x ) We assume that y has a joint probability density function ρ ( y ) = � N n =1 ρ n ( y n ) : Γ N → R + Then u ( ω, x ) = u ( y 1 ( ω ) , . . . , y N ( ω ) , x ) is a deterministic function of the random vector y . Extensions to the case of infinitely many (countable) random variables is possible provided the solution u ( y 1 , . . . , y n , . . . , 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 a finite number of independent random variables a ( ω, x ) = a ( y 1 ( ω ) , . . . , y N ( ω ) , x ) We assume that y has a joint probability density function ρ ( y ) = � N n =1 ρ n ( y n ) : Γ N → R + Then u ( ω, x ) = u ( y 1 ( ω ) , . . . , y N ( ω ) , x ) is a deterministic function of the random vector y . Extensions to the case of infinitely many (countable) random variables is possible provided the solution u ( y 1 , . . . , y n , . . . , 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 a finite number of independent random variables a ( ω, x ) = a ( y 1 ( ω ) , . . . , y N ( ω ) , x ) We assume that y has a joint probability density function ρ ( y ) = � N n =1 ρ n ( y n ) : Γ N → R + Then u ( ω, x ) = u ( y 1 ( ω ) , . . . , y N ( ω ) , x ) is a deterministic function of the random vector y . Extensions to the case of infinitely many (countable) random variables is possible provided the solution u ( y 1 , . . . , y n , . . . , 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 ) = a 0 + � N n = N y n ( ω ) ✶ Ω n ( x ) with y n ∼ uniform, lognormal, ... Random, spatially correlated, material properties a ( ω, x ) is ∞ -dimensional random field (e.g. lognormal), suitably truncated � N n =1 y n ( ω ) b n ( x ) a ( ω, x ) ≈ a min + e with y n ∼ 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 ) = a 0 + � N n = N y n ( ω ) ✶ Ω n ( x ) with y n ∼ uniform, lognormal, ... Random, spatially correlated, material properties random field with L c =1/4 a ( ω, x ) is ∞ -dimensional random field 1 0.9 2.5 (e.g. lognormal), suitably truncated 0.8 2 0.7 1.5 0.6 � N 1 0.5 n =1 y n ( ω ) b n ( x ) a ( ω, x ) ≈ a min + e 0.4 0.5 0.3 0 0.2 −0.5 0.1 with y n ∼ N (0 , 1), i.i.d. −1 0 0 0.2 0.4 0.6 0.8 1 Fabio Nobile (EPFL & PoliMi) Polynomial approximation of SPDEs RICAM, Linz, 12-16/12/2011 7

Elliptic PDE with random coefficients Analytic regularity Theorem [Back-N.-Tamellini-Tempone ’11, Babuska-N.-Tempone ’05, Cohen-DeVore-Schwab ’09/’10] Assume y bounded (e.g. y ∈ Γ N ≡ [ − 1 , 1] N ) Let i = ( i 1 , . . . , i N ) ∈ N N and r = ( r 1 , . . . , r N ) > 0. Set r i = � n r i n n . ∂ y i � L ∞ ( D ) ≤ r i uniformly in y ∂ i a assume � 1 a Then � ∂ i u log 2 r ) i 1 ∂ y i � V ≤ C | i | !( uniformly in y u : Γ N → V is analytic and can be extended analyticially to � � N � z ∈ C N : y ∈ Γ N Σ = r n | z n − ˜ y n | < log 2 for some ˜ n =1 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 regularity Theorem [Back-N.-Tamellini-Tempone ’11, Babuska-N.-Tempone ’05, Cohen-DeVore-Schwab ’09/’10] Assume y bounded (e.g. y ∈ Γ N ≡ [ − 1 , 1] N ) Let i = ( i 1 , . . . , i N ) ∈ N N and r = ( r 1 , . . . , r N ) > 0. Set r i = � n r i n n . ∂ y i � L ∞ ( D ) ≤ r i uniformly in y ∂ i a assume � 1 a Then � ∂ i u log 2 r ) i 1 ∂ y i � V ≤ C | i | !( uniformly in y u : Γ N → V is analytic and can be extended analyticially to � � N � z ∈ C N : y ∈ Γ N Σ = r n | z n − ˜ y n | < log 2 for some ˜ n =1 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