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Uncertainty Quantification of Large Complex Dynamical Systems

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Uncertainty Quantification of Large Complex Dynamical Systems Powered By Docstoc
					                     Uncertainty Quantification of
                  Large Complex Dynamical Systems


                                           Qingyun Duan
                                       Beijing Normal University
                                 The 2nd Summer School on Land Surface
                                Observing, Modeling and Data Assimilation
                                             Beijing, China
                                            July 13-16, 2010
                                                                            1

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               Structure of the presentation

            § Introduction to Uncertainty Quantification (UQ)

            § UQ methodologies

            § UQ applications to large complex dynamical systems

            § Concluding remarks




                                                                   2

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             Reality is uncertain!!!




                                                    “So far as the laws of
                                                    mathematics refer to reality,
                                                    they are not certain.”


                                                    “And so far as they are
                                                    certain, they do not refer to
                                                    reality.”



                                                                                    3

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           Why uncertainty quantification (UQ)?

                                        § Donald Rumsfeld, The Former U.S.
                                          Secretary of Defense:

                                            “There are known knowns; there are
                                            things we know we know.
                                            We also know there are known
                                            unknowns; that is to say we know
                                            there are some things we do not
                                            know.
                                            But there are also unknown
                                            unknowns - the ones we don't know
                                            we don't know”

                                                                                 4

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             A taxonomy of uncertainties

                § Aleatory (aleatoric) uncertainty :
                     • An inherent variation associated with the physical system or
                       the environment
                     − Also referred to as natural variability, and stochastic
                       uncertainty, random uncertainty


                • Examples:
                     −   The outcome of a flip of a coin
                     −   Variations in atmospheric conditions
                     −   Highway traffic flows
                     −   Measurement errors


                • Irreducible, usually with known PDFs
                                                                                      5

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            A taxonomy of uncertainties

              § Epistemic uncertainties:
                 • An uncertainty due to a lack of knowledge of quantities or
                   processes of the system or the environment
                    − Also referred to as subjective uncertainty, and model
                      uncertainty


                 • Examples:
                      − Missing physics due to “unknown unknowns”
                      − Lack of experimental data to characterize a new process
                      − Poor understanding of coupled physics phenomena


                 • Reducible, with unknown PDFs

                                                                                  6

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       Many uncertainties arise when mapping
       from reality to computer models
                                     Mathematical model/simulation code

              Control                                                      Model responses
              parameters/inputs

                      Uncertainties in:               Uncertainties in:
                      - Parameter values              - Physics sub-models
                      - Initial/boundary conditions     * imprecise physics
                                                                                         - Measurement
                      - Measurement errors              * data-driven empirical models
                                                                                           errors
                      - Surrounding environment       - Sub-model couplings
                                                                                         - Data scarcity
                      - Forcing inputs                - Missing physics
                                                      - Model implementation
                                                      - Roundoff errors
                                                      - Algorithmic errors (e.g. MC)
                                                      - Discretization errors

                                            Physical phenomenon

                 Experiments                                                 Observations


         ** need to account for ALL sources of uncertainties (a very difficult task)

                                                                                                           7

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      Model output variability as a result of
      parameter/input uncertainties
                                    Mathematical model/simulation code

              Control                                                    Model responses
              Parameters / inputs
         Uncertain parameter                                                             Output
         / input space                                                                   space




                                         Propagate distributions
                                         through computer model




                 Experiments                                              Observations

                                          Physical phenomenon

                                                                                                  8

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          Sensitivity analysis
                                    Mathematical model/simulation code

              Control                                                    Model responses
              Parameters/inputs


          Uncertain parameter                                                            Output
          / input space                                                                  space




                 Experiments                                              Observations

                                          Physical phenomenon

                                                                                                  9

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           Data assimilation
                                    Mathematical model/simulation code
             Control                                                     Model responses
             Parameters/Inputs

                                                                                         Output
          Uncertain                                                                      space
          input space




                Inferred input                                                    Known output
                uncertainties                                                     uncertainties


                  Experiments                                             Observations

                                          Physical phenomenon

                                                                                                  10

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           Model calibration
                                     Mathematical model/simulation code
             Control
                                                                          Model responses
             parameters/ Inputs

                                                                                          Output
          Uncertain                                                                       space
          Parameter space




                Inferred parameter                                                 Known output
                uncertainties                                                      uncertainties


                 Experiments                                               Observations

                                          Physical phenomenon

                                                                                                   11

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           Risk/realibility analysis –
           Assessing probability of failure
                                    Mathematical model/simulation code
              Control
                                                                     Model responses
              Parameters/Inputs

                                                                                  Failed region
          Uncertain
          parameter space




                                                                                        Output
                                                                                        space



                 Experiments                                             Observations

                                          Physical phenomenon

                                                                                                 12

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           What Does UQ Do?

            § UQ seeks to answer the following questions:
                  • What impact do parameter/model uncertainties have on model
                    outputs (uncertainty analysis)?

                  • Which parameters cause the most output uncertainties?
                    (sensitivity analysis)

                  • How do output uncertainties affect input uncertainties? (data
                    assimilation)

                  • How to use experimental data to find the best parameter values?
                    (calibration)

                  • In view of uncertainty, how do we quantify risk of failure?
                    (risk/reliability analysis)

                                                                                      13

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       UQ for Multi-physics/scales applications is
       especially challenging
           •   Difficult to prescribe parameter uncertainties (the priors)
           •   High-dimensionality of the uncertain parameters (10’s -100’s)
           •   High-dimensionality of the model outputs (can be millions)
           •   Models may be expensive to evaluate (many CPU-hours)
           •   Complex models show highly nonlinear (may be discontinuous) input-
               output relationships
           • Data scarcity for the full system (difficult to calibrate)
           • Models are often created by data far from operating conditions
                • extrapolation may be needed
           • Model-specific uncertainties are difficult/expensive to quantify
           • “Unknown unknowns” can greatly complicate the UQ process.

                                                                                    14

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       In order to perform UQ on a given
       application, we need
        § A UQ methodology
           • A well-thought process/plan with a well-defined objective
           • Consisting of a number of steps
           • Each step may require expert judgment or suitable UQ algorithms

        § Relevant UQ methods (forward propagation, SA, calibration)
           • Intrusive methods
           • Non-intrusive methods
           • Hybrid (intrusive+nonintrusive) methods

        § Adequate hardware/software infrastructure to perform UQ
           • Job management: scheduling, monitoring
           • Data processing
           • Analysis and visualization of results
                                                                               15

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         A typical UQ Analysis

                                                                            Expert judgment
                             Problem specification (model, variables)
                                                                            diligence

                                                                            Derive credible ranges
                        Characterize parameter/model uncertainties          Shapes and forms


                                 Parameter Screening: stage I
                                                                            For nParams >> 100
                                                                            Single effect analysis
                                                                            For nParams ~ 100
                                 Parameter Screening: stage II              e.g. use MOAT/GP/MARS
                                                                            (multi-algorithmic)
                                                                            For expensive models ~10
                                   Response surface analysis
                                                                            (use MARS,ANN,SVM,GP)


                                            Quantify uncertainty,       Design optimization/
               calibration
                                            Sensitivity, reliability        exploration
                                                                                                     16

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            A review of UQ methodologies

            § Uncertainty analysis and sampling

            § Dimension reduction (screening)

            § Response surface analysis

            § Global sensitivity analysis

            § Model calibration/design optimization




                                                      17

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         A classical UQ exercise –
         Monte Carlo Sampling
           •    Create N random sample points in the uncertain parameter space
           •    Run the points through the function and gather the Y’s
           •    Compute basic statistical quantities: mean, std. dev.
           •    Bin the Y’s and create an output histogram




                                                    f


               Sample points in parameter space         An example output distribution



                                                                                         18

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        Examples of experiment designs
        Or sampling methods
                            Monte Carlo (MC)           A quasi-random sequence (LPTAU)




                        Full factorial design (FACT)         Latin hypercube (LH)




                                                                                         19

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           What is full factorial design?


                                                    •   space-filling in all dimensions
                                                    •   sample size = s x m
                                                    •   s: number of levels
                                                    •   m: number of inputs
                                                    •   can be randomized by small
                                                        perturbations
                                                    • can resolve m-way interactions
                                                    • only suitable for small number of
                                                        inputs (expensive)




                                                                                          20

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           What is Latin Hypercube?

                                                    •   space-filling in any one dimension
                                                    •   faster convergence than MC
                                                    •   esp. for monotonic functions
                                                    •   LHS(N, m, s) + noise
                                                    •   N: sample size (5 here)
                                                    •   m: number of parameters
                                                    •   s: number of symbols
                                                    •   r = N/s: number of replications
                Latin hypercube
                (stratified in each dimensiion)     •   How to choose sample size?
                                                    •   Sampling refinement



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         Does the sampling size matter?

                                 Distribution of the sample mean




              100 Monte Carlo runs (N=100)          100 Monte Carlo runs (N=1000)

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        Does the sampling method matter?

                              Distribution of the sample mean




                   100 Monte Carlo runs             100 Latin hypercube runs


                                                                               23

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            Other sampling designs

            •   LH with refinement :no need for a preselected sample size
            •   Quasi-Monte Carlo: LP-tau, Halton sequence, etc.
            •   Central composite designs (inscribed, circumscribed)
            •   OA-based Latin hypercube (more space-filling than LH)
            •   Plackett-Burman (screening design for linear problems)
            •   Box-Behnken (3 level, fit quadratic)
            •   Morris screening design (screening for nonlinear problems)
            •   Fourier Amplitude Sampling Test (FAST): quantitative SA
            •   Metis (space-filling, less restrictive than full factorial)




                                                                              24

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        Parameter Screening Methods

               • The Morris screening method
                    • based on sampling the gradients
               • The Delta test
                    • based on nearest-neighbor analysis
               • The Sum-of-trees method
                    • belongs to the class of tree-based methods
               • MARS-based importance analysis
                    • based on analysis from spline interpolation


                                                                    25

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        The Morris screening method
                                                            C    B”
         1. Start at a random point (A)
         2. Create the next point by
            perturbing one input (A’)
         3. Create the next point by
                                                    A
            perturbing another input (A’’)
              §    Repeat step 1-3 r times (B,C..)
              §    Form r gradients for each
                   input and compute modified C”        B   C’   B’
                   means and standard
                   deviations
              §    Plot mean vs standard dev.
                   for each input à screening               A”
                   diagram                         A’




                                                                      26

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         How does the Morris screening
         method work?
           Gradient of response w.r.t the j-th input

                    y(x1 ,x2 ,L,x j +∆x j ,Lxm )− y(x1 ,x2 ,L,x j ,Lxm )
             zj=
                                                    ∆x j
           Vector of gradients: with m input parameters

              Z r =( z1 , z 2 ,L, z m )
           Collection of gradient vectors (R paths or replications):

               Ω={Z 1 ,Z 2 ,L,Z R }
                                                                1 R
                                                           Z j = ∑ Z ij
                                                                R i =1
           Study the statistics (mean and standard deviation) of           Ω
                                                                               27

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       Interpretation: Screening diagram is a
       distillation of the Morris screening data

         Each point refers to one                                         Each point represents the average
         particular input parameter                                       “effect” of that particular input on
                                                                          the outputs
                            0.06


                            0.05


                            0.04
                                                                                                based on R points
                            0.03                                                                (R = # replicates)
                            0.02


                            0.01


                              0
                                   0    0.05         0.1         0.15         0.2      0.25
                                               2
               1 R                   
                                                   Morris Modified Mean
                               1 R
         σ j=    ∑      Z ji − ∑Z ji 
              R−1 i =1        R i =1 
                                                                        1 R
                                                                   Z j = ∑ Z ji
                                                                                               Note: mean is based
                                                                                               on absolute value of
                                                                        R i =1                 the output
         Large σ = non-linear relationship or inter-
         parameter interactions                                           Large mean = “sensitive” parameter
                                                                                                                     28

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        Response Surface Methods

            § Response surfaces are representations of the model
               output everywhere the parameter space
                                          ˆ
                               Y = F(X) ≈ F(X) in Ω
            § Other names
               • surrogate model
               • (stochastic or statistical) emulator
               • meta-model

            § Basic ingredients of a response surface analysis
               • a sample (input-output pairs, space-filling)
               • a response surface fitting method
                                                                   29

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         Response Surface Fitting Method

            § Parametric:
                 § linear regression, quadratic, cubic, quartic, etc.
                 § special polynomial: Legendre
                 § nonlinear regression functions
            § Nonparametric:
                 • multivariate adaptive splines (MARS) + bootstrap aggr
                 • artificial neural network
                 • Gaussian process (GP, kriging, treed-GP)
                 • Support vector machines
                 • Sum-of-trees
                 • Many others: e.g. wavelet, …
            § Selection depends on knowledge of the function and
              sample size (e.g. GP is very expensive)
                                                                           30

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         How to create response surfaces?


         1. Choose a sampling method (LP-tau, Metis, LH, etc.)
         2.       Run the simulator with the sample
         3.       Use response surface check to see goodness of fit
              •     examine training errors
              •     examine cross validation errors
         4.       If errors are not acceptable, add more points
         5.       Create a FF IV design to sample some corners
              •     to test the robustness against extrapolation
         6.       Use ‘rstest’ to examine extrapolation errors
         7.       If good, add FF design and create new response surface


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           Illustration of a statistical emulator
                  From O’Hagan, 2006




                                                    32

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           Rosenbrock Function Example




                    Monte Carlo (>100000 samples)   MARS (100 samples)


                                                                         33

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           Rosenbrock Function Example




                    Monte Carlo (>100000 samples)   Quartic (100 samples)


                                                                            34

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          A word about response surfaces

          § Conclusions from the analysis are valid only if the
             response model approximates the output response well
             (some smoothness assumptions)
          §
          §Need response surface validation
               • response surface design: adequate resolution
               • response surface design: true space filling
               • response surface design: avoid extrapolation
               • validation via training set and test set
               • cross validation (e.g. bootstrap, jackknife), k-fold CV
               • R-square in regression



                                                                           35

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           What is sensitivity analysis (SA)?


            § Sensitivity Analysis is the study of how the variation in
              the output of a model can be apportioned, qualitatively
              or quantitatively, to different sources of variation.
            § It is thus the natural next step after the output
              uncertainties have been quantified.
            § It can be classified into 3 groups:
                 • Local sensitivity analysis
                 • Screening (qualitative SA, covered previously)
                 • Global sensitivity analysis


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        Global Sensitivity Analysis is more
        suitable for multi-physics applications

            § Local Sensitivity Analysis
               • Computing partial derivatives of output w.r.t input
                 parameters over a small range

            § Global Sensitivity Analysis
              • Including the influence of scale and shape
                 (nonlinearities, wide range)
              • Evaluating the individual effect while all other factors
                 are varying (complex interactions)




                                                                           37

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           Variance decomposition:
           The Sobol’ property

             § Any function can be decomposed into terms of increasing
               dimensionality, i.e. (such that the mean of each term is 0.)

           F ( x1, ..., x k ) = ∑ i =1 Fi ( xi ) + ∑ i =1 ∑ j > i Fij (x i , x j ) + ... + F1... k ( x1 ,..., x k )
                                      k                       k       k



             § Then,   the total variance is the sum of the variances of the
                individual terms.

                       ∑          V i + ∑ i =1 ∑
                           k              k        k
                V =        i =1                    j >i
                                                          V ij + ... + V1 ... k

             § This  holds true only for functions with uncorrelated inputs
                (the joint probability distribution function is 0)



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         Sensitivity measures


             § Sensitivity index for input I (main effect or 1st order)
                      Si =     Vi
                               V

             § Sensitivity index for input i and j (second order)

                      Sij =
                               Vij
                               V

             § Total sensitivity              S Ti = ∑ all V ' s involving i
                index for input i



                                                                               39

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           Another useful property from statistics


            § Variance decomposition based on conditioning input i

                          V = V [ E (Y | X i )] + E[V (Y | X i )]

                          Variance of conditional expectation   Remaining variability due to
                          (conditioned on input i)              other inputs


            §Sensitivity index for input i

                                         V [ E (Y | X i )]
                           Si =   Vi
                                  V
                                       =
                                                V
                                                                                               40

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           A pictorial view of variance decomposition

              § Given a scatter plot of output with respect to input i




               V [ E (Y | X i )]                        E[V (Y | X )]
           Variance of the means (the red line)     Each column shows the distribution
           The variance of the trend shows the      of Y given a fixed X. Calculate the
           importance of X.                         variances and take the mean of all X’s


                                                                                             41

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          Similarly, we can derive interaction
          and total sensitivity indices

            § interaction study (need different sampling methods)
                 • use replicated orthogonal array design

                    V = V [ E (Y | X i , X j )] + E[V (Y | X i , X j )]

            § total sensitivity indices
                 • with correlated inputs, these are better measures
                 • can use Fourier Amplitude Sampling Test (FAST) design

                    V = V [ E (Y | X −i )] + E[V (Y | X −i )]
                    STi = E[V (Y | X −i )] / V (Y )


                                                                           42

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          Variance Decomposition: An Example

            § Given:
                      Y = F ( x1 , x2 ) = x1 + 2 x2 + 3 x1 x2

                     where      x1, x2 ∈[0,1]        and uniformly distributed

            §Basic statistics:

                      mean              Y = 2 . 25

                      variance          σ   2
                                                = 1 . 604


                                                                                 43

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          Apply Sobol’ decomposition to the example

           1. Rewrite: (so that the mean of each term is 0.)

             Y = 5 ( x1 − 1 ) + 7 ( x2 − 1 ) + 3( x1 − 1 )( x2 − 1 ) + 9
                 2        2     2        2             2         2     4

           2. Calculate the variance of each term:
                                                                    mean

              V = V1 + V2 + V12 =               25
                                                48   + 49 + 16
                                                       48
                                                             1


           3. hence,
                   Main effect of x1 = 25/48
                   Main effect of x2 = 49/48
                   Interaction (1,2) = 1/16
                   Total sensitivity of x1 = 25/48 + 1/16 = 7/12

                                                                           44

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           Model calibration
                 Measured                           Measured
                  Inputs                            Outputs

                               Real World
                                                                    Yt
                                                               +
                                                               -
                                MODEL (θ)           Computed                 t
                                                     Outputs




                                       θ
                    Prior
                    Info                                   Optimization
                                                            Procedure



              “Calibration: constraining the model to be consistent with observations”

                                                                                         45

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            Why numerical optimization?

                § Optimal design – finding the best configuration




                                                    X




                                                                    46

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           Model calibration – 2 approaches
             § Based on deterministic optimization
                  § formulate an objective function (e.g. least-squares)
                  § define independent variables and bounds
                  § define any inequality constraints
                  § run optimization algorithms
             §Stochastic optimization (e.g. Bayesian)
                  § given data and standard deviation (assume normal)
                  § define a likelihood function
                  § define independent variables and distributions
                  § run Markov Chain Monte Carlo algorithm
             § For   efficiency reason, response surface is preferred.

                                                                           47

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          The Shuffled Complex Evolution
          (SCE-UA)Algorithm
                                      The SCE-UA Algorithm   …




                            Duan, Gupta, and Sorooshian, 1992, WRR

                                                                     48

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                         A Case Study in Multi-Species
                           Reactive Transport Model




                                                         49

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           PSUADE is a toolkit for UQ of
           large scale models




          A Problem Solving environment for Uncertainty
                  Analysis and Design Exploration

                                                          50

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       Problem Solving environment for Uncertainty
       Analysis and Design Exploration (PSUADE)

                    Parameters                           Simulation                        Emulation
                                         Sampling
                                      method selection
                                                                          Output global
              pdf




                                                                          sensitivities
                                       PSUADE.in

                                 x1
              pdf




                                                             PSUADE
                                 x2
                                          Sample points
              pdf




                                 x3           Model
                                            generation
              pdf




                                 x4                                   Evaluate objective
                                           Run model                       function


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           Searching for Best Strategies for
           Contaminant Removal




             Biodegradation sequence: TCE à DCE à VC à ETH
             Computational model is used to compute concentration at different locations
             Wells are dug at various locations and contaminants are removed
             Question: what is the most cost effective removal strategy (rates/cost)?
                                                                                           52

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         Conceptual Model and Uncertain
         Parameters
                                 1.    Transport in a 2D heterogeneous aquifer
                                 2.    A TCE source with a constant concentration at
                                       (0,0)
                                 3.    TCEàDCEàVCàETH, all reactions are assumed
                                       to be first order, but reaction rates are unknown.
                                 4.    Adsorption coefficients of 4 species are unknown.
                                 5.    Available data are obtained from species
                                       concentrations simulated using mean values of
                                       those unknown parameters and normally
                                       distributed noise.
                         Constant P1




                                                                                            Constant P2
                                                            V




                                                                                                          53

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          Reactive transport system

              Sequential Reactions                            Reactive Transport

               
                           '
                            k1                                ∂c1
                A1 + C1 ⇒ y1C2                               ∂t = L(c1 ) − k1c1
                           k2'
                                                              ∂c
                A2 + C2 ⇒ y 2C3                              2 = L(c2 ) − k 2c2 + y1k1c1
                        ...'                                 ∂t
                                                            ...
                          ki
                Ai + Ci ⇒ yiCi +1                            ∂c
                                                             i = L(ci ) − ki ci + yi −1ki −1ci −1
                        ...'                                 ∂t
                         kn
                                                             ...
                An + Cn ⇒ y nCn +1                           ∂c n
                                                              ∂t = L(cn ) − k n cn + yn−1k n −1cn−1
                                                             
                   where
                   Ai ,i=1,…,n is a reactant participating in reaction i,
                   Ci is the product of reaction i-1,
                   yi is the yield coefficient of reaction i
                   ki’ is the reaction rate constant in reaction i.



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             Experiment Setup
           Field-test data are from Major et al. (2002):

           1.Kinetics identification fails when t < 28 days.
           2.Data after 42 days follow the first-order chain
           reactions.

          1. Sample size:1000 samples
          2. Sampling method: Latin hypercube
                                                   nt              
                                                   ∑ ci − ci  ˆj
                                                         j
                                            n                       
                             Min    f =   ∑i =1
                                                   j =1
                                                   max( c i j )
                                                                    
                                                                    
                                                                   
                                                                   

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             Data in time (no transport)

                                                TCEàDCEàVCàETH
                                                                          k2= 0.03167 1/d



                                           k1= 0.06125 1/d

                                          k1




                              k3= 0.05943 1/d                k4= 1.852E-8 1/d




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           Response surfaces




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           Response surface




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           Data in space (with transport)
         1. Transport in a 2D heterogeneous aquifer
         2. A TCE plume from a unknown location
         3. TCEàDCEàVCàETH, all reactions are assumed to be first order, but reaction rates
            are unknown
         4. Available data are species concentrations on the central line at time zero and one
            year




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         Response Surface Analysis -
         Objective Function 1:

                                                       1 4  nx j       j 2
                                                    f = ∑ ∑ (ci − ci ) 
                                                                      ˆ
                                                       nx i =1  j =1      


                                                               Central line only
                                                               t=365.0 days




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         Response Surface Analysis -
         Objective Function 2:

                                                       1     4  nx  c j − c j  2 
                                                                            ˆi
                                                    f = ∑ ∑          max(c )  
                                                                         i
                                                                                
                                                       nx i =1  j =1       i  
                                                                                   


                                                                   Central line only
                                                                   t=365.0 days




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         Response Surface Analysis

                    1        nx  c j − c j  2 
                            4
                                         ˆi           Central line only
                 f = ∑ ∑          max(c )  
                                      i
                                             
                                                      t=365.0 days

                    nx i =1  j =1       i  
                                                
                         Effect of Sampling Number:

                                     N=100                                N=1000




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         Effect of Sampling Size: 3d view
                                    N=100           N=1000




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         Sensitivity of Adsorption Coefficients
         Based on Response Surface MCMC

                                                   1       nt   4   nx  c j − c j  2 
                                                                                ˆi
                                             f =         ∑∑ ∑  max(c )  
                                                 nx ⋅ nt k =1 i =1  j =1 
                                                                            i
                                                                                    
                                                                                 i  
                                                                                      

                                                                           ρ B kd
                                                                      Kd =
                                                                             φ




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         Concentration envelopes




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            Concluding Remarks

            § UQ for large-scale multi-physics applications is challenging due to
              nonlinearity, interactions, correlations, expensive evaluation, high-
              dimensionality…

            § Advances in computer architectures have made it feasible to
              perform a large number of calculations

            § Still need advances in general methodology and rigorous and
              efficient mathematical methods (as physics complexity increases)

            § Current and future UQ method development will benefit from
              intelligent sampling and analysis

            § UQ will be a major driving force for more computing power, and it
              will require innovations in hardware and software infrastructure
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