Multilevel Markov Chain Monte Carlo with Applications in Subsurface Flow Robert Scheichl Department of Mathematical Sciences Collaborators: AL Teckentrup (Warwick) & C Ketelsen (Boulder) Thanks also to my Bath colleagues F. Lindgren (Stats) & R. Jack (Physics) Workshop on “Stochastic and Multiscale Inverse Problems” October 2nd-3rd 2014, Ecole des Ponts Paristech, Paris R. Scheichl (Bath, UK) Multilevel MCMC & UQ in Subsurface Flow Ecole des Ponts, Oct ’14 1 / 35
Introduction Many problems involve PDEs with spatially varying data which is subject to uncertainty . Example: groundwater flow in rock underground. Uncertainty enters PDE via its coefficients (random fields) . The quantity of interest : is a random number or field derived from the PDE solution. Examples: effective permeability or breakthrough time of a pollution plume Typical Computational Goal: expected value of quantity of interest . ( Uncertainty quantification ) R. Scheichl (Bath, UK) Multilevel MCMC & UQ in Subsurface Flow Ecole des Ponts, Oct ’14 2 / 35
Uncertainty Propagation The Forward Problem R. Scheichl (Bath, UK) Multilevel MCMC & UQ in Subsurface Flow Ecole des Ponts, Oct ’14 3 / 35
Example: Uncertainty in Subsurface Flow (eg. risk analysis of radwaste disposal or optimisation of oil recovery) EDZ CROWN SPACE WASTE VAULTS FAULTED GRANITE GRANITE DEEP SKIDDAW N-S SKIDDAW DEEP LATTERBARROW N-S LATTERBARROW FAULTED TOP M-F BVG TOP M-F BVG FAULTED BLEAWATH BVG � q + k ∇ p = Darcy’s Law: f BLEAWATH BVG FAULTED F-H BVG F-H BVG FAULTED UNDIFF BVG Incompressibility: ∇ · � q = 0 UNDIFF BVG FAULTED N-S BVG N-S BVG FAULTED CARB LST CARB LST + Boundary Conditions FAULTED COLLYHURST COLLYHURST FAULTED BROCKRAM BROCKRAM SHALES + EVAP FAULTED BNHM BOTTOM NHM FAULTED DEEP ST BEES DEEP ST BEES FAULTED N-S ST BEES N-S ST BEES FAULTED VN-S ST BEES VN-S ST BEES FAULTED DEEP CALDER DEEP CALDER FAULTED N-S CALDER N-S CALDER FAULTED VN-S CALDER VN-S CALDER MERCIA MUDSTONE QUATERNARY Rock strata at Sellafield (potential UK radwaste site in 90s) c � NIREX UK Ltd R. Scheichl (Bath, UK) Multilevel MCMC & UQ in Subsurface Flow Ecole des Ponts, Oct ’14 4 / 35
Example: Uncertainty in Subsurface Flow (eg. risk analysis of radwaste disposal or optimisation of oil recovery) EDZ CROWN SPACE WASTE VAULTS FAULTED GRANITE GRANITE DEEP SKIDDAW N-S SKIDDAW DEEP LATTERBARROW N-S LATTERBARROW FAULTED TOP M-F BVG TOP M-F BVG FAULTED BLEAWATH BVG � q + k ∇ p = Darcy’s Law: f BLEAWATH BVG FAULTED F-H BVG F-H BVG FAULTED UNDIFF BVG → → uncertain k Incompressibility: ∇ · � q = 0 uncertain p , � UNDIFF BVG q FAULTED N-S BVG N-S BVG FAULTED CARB LST CARB LST + Boundary Conditions FAULTED COLLYHURST COLLYHURST FAULTED BROCKRAM BROCKRAM SHALES + EVAP FAULTED BNHM BOTTOM NHM FAULTED DEEP ST BEES DEEP ST BEES FAULTED N-S ST BEES N-S ST BEES FAULTED VN-S ST BEES VN-S ST BEES FAULTED DEEP CALDER DEEP CALDER FAULTED N-S CALDER N-S CALDER FAULTED VN-S CALDER VN-S CALDER MERCIA MUDSTONE QUATERNARY Rock strata at Sellafield (potential UK radwaste site in 90s) c � NIREX UK Ltd R. Scheichl (Bath, UK) Multilevel MCMC & UQ in Subsurface Flow Ecole des Ponts, Oct ’14 4 / 35
Stochastic Modelling of Uncertainty: Model uncertain conductivity tensor k as a lognormal random field Typical simplified model (prior): log k ( x , ω ) isotropic, scalar, Gaussian e.g. meanfree with exponential covariance R ( x , y ) := σ 2 exp ( −� x − y � /λ ) e.g. truncated Karhunen-Lo` eve expansion typical realisation 64 , σ 2 = 8) 1 ( λ = s � √ µ j φ j ( x ) Z j ( ω ) , Z j ( ω ) iid N (0 , σ 2 ) log k ( x , ω ) ≈ j =1 R. Scheichl (Bath, UK) Multilevel MCMC & UQ in Subsurface Flow Ecole des Ponts, Oct ’14 5 / 35
Stochastic Modelling of Uncertainty: Model uncertain conductivity tensor k as a lognormal random field Typical simplified model (prior): log k ( x , ω ) isotropic, scalar, Gaussian e.g. meanfree with exponential covariance R ( x , y ) := σ 2 exp ( −� x − y � /λ ) e.g. truncated Karhunen-Lo` eve expansion typical realisation 64 , σ 2 = 8) 1 ( λ = s � √ µ j φ j ( x ) Z j ( ω ) , Z j ( ω ) iid N (0 , σ 2 ) log k ( x , ω ) ≈ j =1 1 Typical quantities of interest: 0.9 0.8 0.7 0.6 p ( x ∗ ), � q ( x ∗ ), travel time, water cut,. . . 0.5 0.4 0.3 � 0.2 0.1 outflow through Γ out : Q out = Γ out � q · d � n 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R. Scheichl (Bath, UK) Multilevel MCMC & UQ in Subsurface Flow Ecole des Ponts, Oct ’14 5 / 35
Why is this problem so challenging? R. Scheichl (Bath, UK) Multilevel MCMC & UQ in Subsurface Flow Ecole des Ponts, Oct ’14 6 / 35
Why is this problem so challenging? 0 10 λ =0.01 λ =0.1 λ =1 −2 10 eigenvalue −4 10 −6 10 0 1 2 3 10 10 10 10 n KL-eigenvalues in 1D Convergence of q | x =1 w.r.t. s R. Scheichl (Bath, UK) Multilevel MCMC & UQ in Subsurface Flow Ecole des Ponts, Oct ’14 6 / 35
Why is this problem so challenging? 0 10 λ =0.01 λ =0.1 λ =1 −2 10 eigenvalue −4 10 −6 10 0 1 2 3 10 10 10 10 n KL-eigenvalues in 1D Convergence of q | x =1 w.r.t. s Small correlation length λ = ⇒ high dimension s ≫ 10 and fine mesh h ≪ 1 Large σ 2 & exponential k max k min > 10 6 = ⇒ large heterogeneity R. Scheichl (Bath, UK) Multilevel MCMC & UQ in Subsurface Flow Ecole des Ponts, Oct ’14 6 / 35
Monte Carlo for large scale problems (plain vanilla) Model( h ) Output Z s ( ω ) ∈ R s X h ( ω ) ∈ R M h − → − → Q h , s ( ω ) ∈ R random input state vector quantity of interest e.g. Z s multivariate Gaussian; X h numerical solution of PDE; Q h , s a (non)linear functional of X h R. Scheichl (Bath, UK) Multilevel MCMC & UQ in Subsurface Flow Ecole des Ponts, Oct ’14 7 / 35
Monte Carlo for large scale problems (plain vanilla) Model( h ) Output Z s ( ω ) ∈ R s X h ( ω ) ∈ R M h − → − → Q h , s ( ω ) ∈ R random input state vector quantity of interest e.g. Z s multivariate Gaussian; X h numerical solution of PDE; Q h , s a (non)linear functional of X h h → 0 , s →∞ Q ( ω ) inaccessible random variable s.t. E [ Q h , s ] − → E [ Q ] and | E [ Q h , s − Q ] | = O ( h α ) + O ( s − α ′ ) Standard Monte Carlo estimator for E [ Q ]: N � Q MC := 1 Q ( i ) ˆ h , s N i =1 where { Q ( i ) h , s } N i =1 are i.i.d. samples computed with Model( h ) R. Scheichl (Bath, UK) Multilevel MCMC & UQ in Subsurface Flow Ecole des Ponts, Oct ’14 7 / 35
Convergence of plain vanilla MC (mean square error) : � � 2 �� ˆ � 2 � V [ Q h , s ] Q MC − E [ Q ] = + E [ Q h , s − Q ] E � �� � N � �� � � �� � =: MSE sampling error model error (“bias”) Typical (2D) : α = 1 ⇒ MSE = O ( N − 1 ) + O ( M − 1 h ) = O ( ε 2 ) R. Scheichl (Bath, UK) Multilevel MCMC & UQ in Subsurface Flow Ecole des Ponts, Oct ’14 8 / 35
Convergence of plain vanilla MC (mean square error) : � � 2 �� ˆ � 2 � V [ Q h , s ] Q MC − E [ Q ] = + E [ Q h , s − Q ] E � �� � N � �� � � �� � =: MSE sampling error model error (“bias”) Typical (2D) : α = 1 ⇒ MSE = O ( N − 1 ) + O ( M − 1 h ) = O ( ε 2 ) Thus M h ∼ N ∼ ε − 2 and Cost = O ( NM h ) = O ( ε − 4 ) (w. MG solver) (e.g. for ε = 10 − 3 we get M h ∼ N ∼ 10 6 and Cost = O (10 12 ) !!) Quickly becomes prohibitively expensive ! Complexity Theorem for (plain vanilla) Monte Carlo Assume that E [ Q h , s ] → E [ Q ] with O ( h α ) and cost per sample is O ( h − γ ). Then � α � ε − 2 − γ Cost( ˆ Q MC ) = O to obtain MSE = O ( ε 2 ) . R. Scheichl (Bath, UK) Multilevel MCMC & UQ in Subsurface Flow Ecole des Ponts, Oct ’14 8 / 35
Numerical Example (Standard Monte Carlo) � � D = (0 , 1) 2 , covariance R ( x , y ) := σ 2 exp − � x − y � 2 and Q = � − k ∂ p ∂ x 1 � L 1 ( D ) λ using mixed FEs and the AMG solver amg1r5 [Ruge, St¨ uben, 1992] R. Scheichl (Bath, UK) Multilevel MCMC & UQ in Subsurface Flow Ecole des Ponts, Oct ’14 9 / 35
Numerical Example (Standard Monte Carlo) � � D = (0 , 1) 2 , covariance R ( x , y ) := σ 2 exp − � x − y � 2 and Q = � − k ∂ p ∂ x 1 � L 1 ( D ) λ using mixed FEs and the AMG solver amg1r5 [Ruge, St¨ uben, 1992] Numerically observed FE-error: ≈ O ( h 3 / 4 ) = ⇒ α ≈ 3 / 4. Numerically observed cost/sample: ≈ O ( M h ) = O ( h − 2 ) = ⇒ γ ≈ 2. R. Scheichl (Bath, UK) Multilevel MCMC & UQ in Subsurface Flow Ecole des Ponts, Oct ’14 9 / 35
Recommend
More recommend
