Analysis of the impact of model nonlinearities in inverse problem - - PDF document

1

Analysis of the impact of model nonlinearities in inverse problem solving

Tomislava Vukicevic

University of Colorado Collaboration

Derek Posselt

University of Michigan

Inverse problem formulation Diagnostic numerical analysis

– Impact of

• model nonlinearity
• modeling errors
• observations
• prior information

Examples of parameter, initial condition and state estimation

Conclusions and Discussion

2

Inverse Problem Formulation

after Mosegaard and Tarantola (2002)

(International Handbook of Earthquake & Engineering Seismology, Academic Press)

• Conjunction of information in a joint space of observations and

parameters (includes state)

• Information is represented by probability density functions

Parameters are quantities which we wish to estimate, hereafter denoted

m

[ ]

I i m m m M

K i i

, 1 ), ,...., ( :

1

= ∈

Observations are in space

[ ]

L l y y y D

N l l

, 1 ), ,...., ( :

1

= ∈

Mapping or forward model is

) (

*

m f y =

Conjunction of pdfs

Joint space

M D×

) , (

1

y m p ) , ( ) , ( ) , ( 1 ) , (

2 1

y m y m p y m p y m p ν γ =

∫ ×

=

M D

y m y m p y m p ) , ( ) , ( ) , (

2 1

ν γ

, Joint pdf given information from model only

) , (

2

y m p

Joint pdf given observations and prior parameters (no model involved)

) , ( y m ν

Joint homogenous pdf on space pdf of unit volume in the space; constant if space is linear

M D× ) , ( y m p

Joint posterior pdf

3

Breakout of contributing pdfs

) ( ) / ( ) , (

1 1

m m y p y m p ν =

Model only Marginal pdf for parameters is homogenous Information about existence only Prior and Observations

) ( ) ( ) , (

2 2 2

y p m p y m p =

Observations and prior assumed independent because before particular observations are used there is no dependence Prior Observations

) ( ) ( ) , ( y m y m ν ν ν =

Homogenous Independent for the same reason

Inverse problem solution

∫

=

D m

dy y m p m p ) , ( ) (

Marginal of joint posterior pdf

[ ]

dy m y p y p m p m p

D m

∫

= ) / ( ) ( ) ( 1 ) (

1 2 2 *

γ

homogenous pdf folded into the constant

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Familiar case

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − − =

−

)) ( ) ( 2 1 exp det ) 2 ( 1 ) (

1 2 1 2

y y C y y C y p

y T y

π ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − − =

−

)) ( ( )) ( ( 2 1 exp det ) 2 ( 1 ) / (

1 2 1 1

m f y C m f y C m y p

s T s

π

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − − =

−

) ) ( ( ) ) ( ( 2 1 exp ) ( ) (

1 2

y m f C y m f m kp m p

D T m s y D

C C C + =

, , ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − − =

−

)) ( ) ( 2 1 exp det ) 2 ( 1 ) (

1 2 1 2 prior m T prior m

m m C m m C m p π

( ) ( ) ( ) (

) ( )

[ ]

prior m T prior D T m

m m C m m y m f C y m f m J m J const m p − − + − − = − =

− − 1 1

) ( ) ( 2 1 ) ( ) ( exp ) ( Observations Model Convolving two Gaussians Prior CONJUCTED

Linear familiar case

Hm m f ≡ ) (

( )

1 1 1 1

) ( ) (

− − − −

+ = − + + =

m D T prior D T m T m prior

C H C H C Hm y C H HC H C m m

First moment or mean is also minimum of cost function Posterior marginal pdf is Gaussian with first and second moments, respectively

( ) ( ) (

) ( )

[ ]

prior m T prior D T

m m C m m y m f C y m f m J − − + − − =

− − 1 1

) ( ) ( 2 1 ) (

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Nonlinear case

Explicit numerical evaluation of posterior joint and marginal pdfs for a nonlinear model given observations and priors When this is possible could evaluate Impact of

• model nonlinearity
• model uncertainties
• Gaussian prior
• observation uncertainties

Combination of function mapping on a discrete multidimensional grid in phase space and Monte Carlo sampling of known parametric pdfs Not a data assimilation algorithm !

Discretization

[ ]

dy m y p y p m p m p

D m

∫

= ) / ( ) ( ) ( 1 ) (

1 2 2 *

γ

6 Example 1

• Model error is assumed Gaussian at each solution y=f(m), for discrete set of m

values within interval of permissible values;

• Each m unit volume has the same probability (homogenous pdf for marginal in m)

2 2

= + + ωχ τ χ α τ χ d d d d

τ η λ τ η λ ) ( 2 ) ( 1

) (

− − + −

Α + Α = e e m f

Damped harmonic oscillations

m is initial condition m is natural frequency y y m m

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − − =

−

)) ( ( )) ( ( 2 1 exp det ) 2 ( 1 ) / (

1 2 1

m f y C m f y C m y p

s T s

π

Example 1 Example 1, continued

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − − =

−

)) ( ) ( 2 1 exp det ) 2 ( 1 ) (

1 2 1

y y C y y C y p

y T y

π

Observation: Gaussian with mean from true reference at an arbitrary time point

y

pdf

Prior: uniform

7 Result of conjunction

) (m pm ) (m pm

initial condition, linear model Natural frequency, exponential model

Resulting posterior is Gaussian Resulting posterior is approximately Log-Normal

) (m pm

) (m pm

) , ( y m p ) , ( y m p

Uniform prior

Parameterized dry convection model (Lorenz, 1963)

Example 2

bZ XY d dZ XZ Y rX d dY Y X a d dX − = − − = − − = τ τ τ ) (

Only X component is

• bserved 4 times

8 Inverse problems with Lorenz model Coefficients Initial conditions State estimation

pdfs at individual observation instances for coefficient a

time solution

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Cumulative influence of observations

still inverse problem for coefficient a

Final marginal for the given observation set is asymmetrical, but unimodal Posterior after first

• bservation

time; prior for the second Posterior after second

• bservation

time sequential

pdfs at individual observation instances for initial condition X

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Multidimensional pdfs

3 coefficients or initial conditions

State estimation - 3 state components Experiments

• Varying amplitude of model error
• Gaussian prior update at each observation input

3 coefficients / negligible model error

True pdf Final posterior pdf using all 4 observation times Intermediate times

Well constrained problem with 4 observatios

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Impact of model error and Gaussian prior

estimation of coefficients

Small model error Large model error Gaussian prior update

Impact of model error and Gaussian prior

initial condition problem

Small model error Large model error Gaussian prior update

12 State estimation Full pdf or Gaussian prior update

time Gaussian “true pdf”

X is observed

Example of joint pdf in state

full Gaussian

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Conclusions and discussion

Nonmonotonic forward model gives rise to the potential for a multimodal posterior pdf, the realization of which depends on the information content of the observations, and on

• bservation and model uncertainties

The presence of model error greatly increases the possibility

• f capturing multiple modes in the posterior pdf

Cumulative effect of observations, over time, space or both, could render unimodal final posterior pdf even with the nonmonotonic forward model A greater number of independent observations are needed to constrain the solution in the case of a nonmonotonic nonlinear model than for a monotonic model for same number of degrees of freedom in control parameter space Gaussian prior update has a similar effect to an increase in model error, which indicates there is the potential for inaccurate estimate (relative to truth) even when

• bservations and model are unbiased.