Inverse Modeling with the aid of Surrogate Models Dongxiao Zhang, - PowerPoint PPT Presentation

Inverse Modeling with the aid of Surrogate Models Dongxiao Zhang, Qinzhuo Liao, Haibin Chang College of Engineering Peking (Beijing) University dxz@pku.edu.cn The 2017 EnKF Workshop

Inverse modeling • Estimate parameters from physical models and observations model: f input parameter: θ output observation: s s = f( θ ) + e Input Model Output  conductivity  groundwater supply  hydraulic head  porosity  contaminant control  velocity/flux  boundary condition  oil and gas production  phase saturation  source & sink  CO 2 sequestration  solute concentration 2

Stochastic approach • Bayesian inference  ( ): prior density p    ( ) s f e  ( | ): posterior density p s   ( ) ( | ) p p s    ( | ) p s ( | ):likelihood function p s ( ) p s ( ): normalization factor p s • Markov chain Monte Carlo − Use Monte Carlo simulations to construct a Markov chain − Computationally expensive: repeated evaluations of the forward model • Surrogate model − Can generate a large number of samples at low cost − Posterior error depends on forward solution error 3

Forwar ward d Stochastic ochastic Formulation ulation • SPDE:    ξ ξ ξ ( ; , ) ( , ), , , L u x g x P x D     ξ ) T where ( , ,..., 1 2 N which has a finite (random) dimensionality. • Weak form solution:            ˆ ( ; , ) ( ) ( ) ( , ) ( ) ( ) L u x w p d g x w p d P P where    ˆ ( , ) , where trial function space u x V V    ( ) , where test (weighting) function space w W W p    ξ ( ) probability density function of ( )

Forwar ward d Stochastic ochastic Metho ethods ds • Galerkin polynomial chaos expansion (PCE) [e.g., Ghanem and Spanos, 1991]:        M    M ( ) , ( ) V span W span   i i 1 1 i i     M  where ( ) orthogonal polynomials i  i 1 • Probabilistic collocation method (PCM) [ Tatang et al., 1997; Sarma et al ., 2005; Li and Zhang , 2007, 2009]:             M M ( ) , ( ) V span W span   i i i 1 i 1 • Stochastic collocation method (SCM) [Mathelin et al., 2005; Xiu and Hesthaven, 2005; Chang and Zhang, 2009]:            M M ( ) , ( ) V span L W span   i i 1 1 i i    M where { ( )} lagrange interpolation basis L 1 i i

Key y Components ponents fo for Stochastic ochastic Method ethods • Random dimensionality of underlying stochastic fields • How to effectively approximate the input random fields with finite dimensions • Karhunen-Loeve and other expansions may be used • Trial function space • How to approximate the dependent random fields • Perturbation series, polynomial chaos expansion, or Lagrange interpolation basis • Test (weighting) function space • How to evaluate the integration in random space? • Intrusive or non-intrusive schemes?

Stochastic collocation method （ SCM ） • Based on polynomial interpolations in random space      M is a set of nodes in -dimensional random space M N  N i 1 i Xiu and Hesthaven (2005) M         ( ) ( ) ( ) f f L i i  1 i    M         j ( ) , ( ) ,1 , L L i j M    i i j ij  j 1 i j  j i • Collocation points: Smolyak sparse grid algorithm    Xiu and Hesthaven (2005) 1 N          q i      i i ( , ) 1 ... q N U U 1 N   q i      1 q N i q is univariate interpolation U • Converge fast in case of smooth functions Chang & Zhang (2009) Lin & Tartakovsky (2009) 7

Choices ices of C f Colloca ocation tion Points ts • Tensor product of one-dimensional nodal sets Each dimension: knots m 3 2  N dimension: N M m 1 0 -1 • Smolyak sparse grid (level: k = q-N ) -2 -3 For N>1, preserving interpolation -3 -2 -1 0 1 2 3 property of N=1 with a small number 3 2 of knots 1 0 -1 • Tensor product vs. level-2 sparse grid -2 • N =2, 49 knots vs. 17 (shown right) -3 -3 -2 -1 0 1 2 3 • N =6, 117,649 knots vs. 97

MCS S vs. . PCM/S M/SCM CM 2 = 1.0 MC: 1000 realizations  = 4.0,  Y MCS: 7 • Random sampling of 6.8 ( realizations ) 6.6 • Equal weights for h j 6.4 6.2 ( realizations ) Head, h 6 5.8 2 nd order PCM: 28 representations,  = 4.0,  Y 2 = 1.0 5.6 7 5.4 6.8 5.2 6.6 5 0 1 2 3 4 5 6 7 8 9 10 x 6.4 6.2 PCM/SCM: Head, h 6 • Structured sampling 5.8 ( collocation points ) 5.6 • Non-equal weights for h j 5.4 5.2 ( representations ) 5 0 1 2 3 4 5 6 7 8 9 10 x

Stochastic collocation method • Inaccurate results − Non-physical realizations/Gibbs oscillation − Inaccurate statistical moments and probability density functions Lin & Tartakovsky (2009) Zhang et al. (2010) 10

Stochastic collocation method • Inaccurate results − When: advection dominated ( Pe = 100)  low regularity − Why: physical space  random space • Illustration − Unit mass instantaneously released at x = 0, t = 0 − Input parameter: conductivity k = exp(0.3 θ ), θ ~ N (0,1) − Output response: concentration c at x = 0.3, t = 1 11

Transformed stochastic collocation method • Stochastic collocation method (SCM) t  − Approximate s as a function of θ at fixed x and t ( , ; ) s x • Transformed stochastic collocation method (TSCM) s t  ( , ; ) x − Approximate x as a function of θ for a given s at fixed t s  − Approximate t as a function of θ for a given s at fixed x ( , ; ) t x Liao & Zhang (WRR, 2016) 12

1D example • Continuous injection − Input parameter: conductivity k = exp(0.3 θ ), θ ~ N (0,1), − Output response: concentration c at x = 0.3, t = 1 13

1D example • Forward solution approximation and posterior approximation true parameter θ = 0.2 true observation c = 0.842 14 observation error e ~ N (0, 0.01)

1D example • Convergence rate Marzouk & Xiu (2009) M     2               surrogate model: ( ) ( ) ( ) error: ( ) ( ) ( ) 0, s f L s s s s p d M M i i M 2 M L  1 i   ( )                 poster ior: ( ) ( | ) Kullback-Leibler diverg ence : || ( )log M 0, p s D d M   M M ( ) 15

2D example • Assume conductivity field is known • Top and bottom are no-flow boundaries , right head h 2 =0 • One instantaneous release location (circle), four observation wells (triangles) • Input parameters: release time t 0 ∈ [0,20], mass m 0 ∈ [1,2], left head h 1 ∈ [3,10] • Output responses: concentration c from t = 0 to 80, observation error e ~ N (0, 0.001) 16

2D example • Compare true concentration and approximated concentration 17

2D example • Surrogate approximation error • Adaptive transformed SCM (ATSCM) − Dimension-adaptive: automatically select important dimensions Klimke (2006) Liao et al. (JCP, 2016) − Further reduce the number of collocation points 18

2D example • Marginal PDF − MCMC with 10 5 model runs as a reference − ATSCM with 67 model runs is more accurate than SCM with 6017 model runs 19

2D example • Marginal PDF − Black: MCMC, red: SCM, blue: ATSCM 20

Inverse modeling • Maximizing the posterior PDF obs ( ) ( | ) p m p d m  obs ( | ) p m d obs ( ) p d • For Gaussian prior and error, minimizing an objective function 1 1         1 1 obs T obs pr T pr ( ) ( ( ) ) ( ( ) ) ( ) ( ). J m g m d C g m d m m C m m D M 2 2 • Ensemble based methods EnKF ES Iterative ES  Sequentially assimilate  Simultaneously assimilate  Simultaneously assimilate the data all the data all the data  One step method  One step method  Multi-step method  Moderate simulation  Small simulation  Large simulation effort (restart required ) effort (no restart) effort (no restart, iteration)  Suitable for highly non-linear problems 21