Parameter Estimation and Uncertainty Quantifjcation for Flow and Transport Processes Ole Klein WIAM 2016, Hamburg 02.09.2016 Ole Klein PE and UQ for Flow and Transport 02.09.2016 1 / 26
Introduction Richards equation 02.09.2016 PE and UQ for Flow and Transport Ole Klein observations Motivation Parameter estimation should: Groundwater fmow equation, transport equation, Here: observations PDE-based modeling often leads to: 2 / 26 • High-dimensional discretized parameterization • Sparse (but not necessarily low-dimensional) state • Produce estimates of parameters and uncertainty • Be applicable for high-resolution parameter fjelds • Remain feasible for large numbers of state
Introduction P : Parameter vectors 02.09.2016 PE and UQ for Flow and Transport Ole Klein Z : Measurement vectors U : System states Forward and Inverse Problem S : Parameter fjelds P Z U S 3 / 26 F I O G I : Input, Interpretation F : Forward model (continuous) O : Output, Observation (sparse) G : Forward model (discrete) Task: construct mapping Z → P to estimate parameters
Introduction Example: Groundwater Flow Groundwater fmow equation: Convection-difgusion equation: Parameters P : log-storage term Z s , log-conductivity Y , initial conditions Parameters are underdetermined, problem is ill-posed and requires regularization Ole Klein PE and UQ for Flow and Transport 02.09.2016 4 / 26 F φ ( Y , Z s , φ 0 ; φ ) := S s ( Z s ) ∂ t φ + ∇ · j θ w ( Y , φ ) − q θ w = 0 j θ w ( Y , φ ) := − K ( Y ) ∇ φ = − exp ( Y ) ∇ φ F c ( Y , Z s , φ 0 , c 0 , φ ; c ) := θ∂ t c + ∇ · j C ( Y , φ, c ) − q C = 0 j C ( Y , φ, c ) := − D ( φ ) ∇ c + cj θ w Measurements Z : hydraulic head φ ( x i , t i ) , tracer concentration c ( x j , t j )
Bayesian Inversion . . . . Prior Distribution . . Q p n P p n P . . . . n P Ole Klein PE and UQ for Flow and Transport 02.09.2016 . . 5 / 26 In more detail: . . . Treat parameters as random variables: p n P P ∼ N ( Q PP , P ∗ ) · · · p ∗ p 1 1 Q p 1 p 1 Q p 1 p 2 Q p 1 p n P p ∗ · · · p 2 2 Q p 2 p 1 Q p 2 p 2 Q p 2 p n P ∼ N , p ∗ · · · Q p n P p 1 Q p n P p 2 • Stationary random fjelds → constituent matrices are block Toeplitz • Multiplication with Q PP using circulant embedding • Multiplication with Q − 1 PP is problematic
Bayesian Inversion Maximum A Posteriori Apply Bayes’ theorem and drop constant term: Ole Klein PE and UQ for Flow and Transport 02.09.2016 6 / 26 Assume that measurement noise ǫ is Gaussian: ǫ = [ Z − G ( P )] ∼ N ( Q ZZ , 0 ) Z | P ∼ N ( Q ZZ , G ( P )) f Z · f P | Z = f P · f Z | P f P | Z ∝ f P · f Z | P Use maximizer P map of f P | Z as parameter estimate
Bayesian Inversion ZZ 02.09.2016 PE and UQ for Flow and Transport Ole Klein with gradient ZZ Objective Function 7 / 26 For Gaussian parameters and Gaussian noise, we have: ( − 1 ]) [ � P − P ∗ � 2 PP + � Z − G ( P ) � 2 f P | Z ∝ exp Q − 1 Q − 1 2 Finding P map is equivalent to minimizing objective function L ( P ) := 1 PP + 1 2 � P − P ∗ � 2 2 � Z − G ( P ) � 2 Q − 1 Q − 1 ∇ L = Q − 1 PP [ P − P ∗ ] − H T ZP Q − 1 ZZ [ Z − G ( P )] ( H ZP : linearization of G around current estimate)
Preconditioned Conjugate Gradients Prior Preconditioned Conjugate Gradients 02.09.2016 PE and UQ for Flow and Transport Ole Klein mesh width) 8 / 26 Apply Conjugate Gradients to objective function expressed in Q − 1/2 PP P Equivalent to using Q − 1 PP as preconditioner: δ P = − Q PP ∇ L | P i − 1 [+ conj. contrib. ] = − [ P i − 1 − P ∗ ] + Q PP H T ZP Q − 1 ZZ [ Z − G ( P )] • Allows using combined adjoint (full assembly of H ZP not needed) • Completely eliminates Q − 1 PP from algorithm (negative preconditioner cost) • Low-rank perturbation of identity (resulting convergence rate independent of
Preconditioned Conjugate Gradients Conjugate Gradients 02.09.2016 PE and UQ for Flow and Transport Ole Klein ) Convergence behavior for 2D mesh of size Preconditioned Conjugate Gradients Normalized Objective Function Number of Iterations Prior Preconditioned Conjugate Gradients 9 / 26 Normalized Objective Function Number of Iterations 10 0 10 0 10 − 1 10 − 1 10 − 2 10 − 2 0 20 40 60 80 100 0 20 40 60 80 100 64 × 64 ( ), 128 × 128 ( ), 256 × 256 ( ) and 512 × 512 (
Preconditioned Conjugate Gradients Prior Preconditioned Conjugate Gradients 02.09.2016 PE and UQ for Flow and Transport Ole Klein ) ) and unpreconditioned CG ( Gauss-Newton reference ( ), Time to solution for prior preconditioned CG ( number of measurements n Z 10 / 26 number of cells n P number of parameters n P = 256 × 256 number of measurements n Z = 16 10 4 1 , 000 t total [ s ] t total [ s ] 500 10 2 0 0 500 10 4 10 5
Preconditioned Conjugate Gradients Example: 2D Groundwater Flow Synthetic Reference Ole Klein PE and UQ for Flow and Transport 02.09.2016 11 / 26 Estimate for n φ = 25 Estimate for n φ = 900
Uncertainty Quantifjcation and Analysis PP 02.09.2016 PE and UQ for Flow and Transport Ole Klein PP PP PP PP Q post Uncertainty Quantifjcation PP PP H T 12 / 26 Q post ZZ H ZP Q post Provide linearized uncertainty quantifjcation through posterior covariance matrix: ] − 1 Q − 1 ZP Q − 1 [ PP := PP + H T Use transformation P → Q − 1/2 PP P again, equivalent to split PP = Q 1/2 PP [ I + M like ] − 1 Q 1/2 M like := Q 1/2 ZZ H ZP Q 1/2 ZP Q − 1 M like has low rank, use spectral decomposition M like = V Λ V T : I + V Λ V T ] − 1 Q 1/2 PP = Q 1/2 [ = Q 1/2 Q 1/2 [ I − V Υ V T ] ( with Υ = diag ( λ i /( λ i + 1))) = Q PP − Q 1/2 PP V Υ V T Q 1/2 PP , Approximating Q 1/2 PP through circulant embedding introduces systematic error!
Uncertainty Quantifjcation and Analysis Example: 2D Groundwater Flow Ole Klein PE and UQ for Flow and Transport 02.09.2016 13 / 26 Uncertainty for n φ = 25 Uncertainty for n φ = 900
Uncertainty Quantifjcation and Analysis Q post 02.09.2016 PE and UQ for Flow and Transport Ole Klein Quality Assurance / Sample Generation 14 / 26 Assumption: posterior distribution can be modeled as approx. PP Results of previous steps: P map and Q post ( ) P | Z ∼ N PP , P map • P map is mode of posterior distribution, not its mean • Q post PP is linearization of true posterior variance • Higher-order terms are neglected • Construct has to be checked for consistency with data set • Realistic applications require samples from posterior distribution
Uncertainty Quantifjcation and Analysis measurement residuals, and generate conditional samples: PP Q post ZZ Quality Assurance / Sample Generation Z Z ZZ This can be used to check distribution of estimation error (if known) or Q post P P P Q post Z Z Ole Klein PE and UQ for Flow and Transport 02.09.2016 P 15 / 26 PP like Q post PP Q post PP ZZ Instead of spectral decomposition M like = V Λ V T , use ] − 1 Q 1/2 PP = Q 1/2 [ I + L like L T L like := Q 1/2 PP H ZP Q − 1/2 Performing singular value decomposition (SVD) L like = V P Λ 1/2 V T Z leads to PP = Q PP − Q 1/2 P Q 1/2 = Q 1/2 ] T Q 1/2 PP V + I − Υ + ] [ V + [ PP V P Υ V T ZZ = Q ZZ − Q 1/2 Z Q 1/2 = Q 1/2 ] T Q 1/2 ZZ V + I − Υ + ] [ V + [ ZZ V Z Υ V T I − Υ + ] 1/2 [ P with L P := Q 1/2 PP V + V + [ ] T PP = L P L T I − Υ + ] 1/2 [ Z with L Z := Q 1/2 ZZ V + V + [ ] T ZZ = L Z L T
Examples Example: 2D Groundwater Flow 02.09.2016 PE and UQ for Flow and Transport Ole Klein Optimization stopped early, only normalized residuals refmect this ) measurements ), 900 ( ), 625 ( for 400 ( Normalized residuals and errors Count Value 16 / 26 Count Value Residual of PCG, tol = 10 − 4 Error of PCG, tol = 10 − 4 100 4 , 000 50 2 , 000 0 0 − 20 − 10 − 4 − 2 0 10 20 0 2 4
Examples Example: 2D Groundwater Flow 02.09.2016 PE and UQ for Flow and Transport Ole Klein Optimization has converged, both measures indicate linearization is appropriate ) measurements ), 900 ( ), 625 ( for 400 ( Normalized residuals and errors Count Value 17 / 26 Value Count Residual of PCG, tol = 10 − 5 Error of PCG, tol = 10 − 5 80 4 , 000 60 40 2 , 000 20 0 0 − 4 − 2 − 4 − 2 0 2 4 0 2 4
Examples Example: 3D Groundwater Flow Synthetic Reference (Interior) Estimation of 3D log-conductivity parameter fjeld Ole Klein PE and UQ for Flow and Transport 02.09.2016 18 / 26 Estimate for n φ = 225 (Interior) 128 × 128 × 16 = 2 . 65 · 10 5 parameters, 225 measurements Moments of normalized residual: ( − 0 . 195 , 0 . 960 , 0 . 029 , 2 . 99)
Examples Example: 3D Groundwater Flow Synthetic Reference (Interior) Estimation of 3D log-conductivity parameter fjeld Ole Klein PE and UQ for Flow and Transport 02.09.2016 19 / 26 Estimate for n φ = 225 (Interior) 128 × 128 × 16 = 2 . 65 · 10 5 parameters, 225 measurements Moments of normalized residual: ( − 0 . 195 , 0 . 960 , 0 . 029 , 2 . 99)
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