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Computing a posteriori covariance in variational DA I.Gejadze, F.-X. Le Dimet, V.Shutyaev. What is a posteriori covariance and why one needs to compute it ? - 1. Model of evolution process. can be defined in 2D-3D, can be a vector-function. Measure of event. - target state.

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Computing a posteriori covariance in variational DA I.Gejadze, F.-X. Le Dimet, V.Shutyaev

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Computing a posteriori covariance in variational da i gejadze f x le dimet v shutyaev

Computing a posteriori covariance in variational DA

I.Gejadze, F.-X. Le Dimet, V.Shutyaev


Computing a posteriori covariance in variational da i gejadze f x le dimet v shutyaev

What is a posteriori covariance and why one needs

to compute it ? - 1

Model of evolution process

can be defined in 2D-3D, can be a vector-function

Measure of event

- target state

Uncertainty in initial, boundary conditions,

distributed coefficients, model error ….

Uncertainty in the event measure

Data assimilation is used to reduce uncertainty

in controls / parameters, and eventually in the

event measure

Objective function (for the initial value control)


What is a posteriori covariance and why one needs to compute it 2 idea of adjoint sensitivities

What is a posteriori covariance and why one needs to compute it ? – 2(idea of adjoint sensitivities)

Trivial relationship between (small)

uncertainties in controls / parameters

and in the event measure

How to compute the gradient?

1. Direct method ( ) : for solve forward model with compute

2. Adjoint method:

form the Lagrangian

zero

Gateaux derivative

take variation

integrate by parts

If = zero and

For initial value problem

Generally


What is a posteriori covariance and why one needs to compute it 3

What is a posteriori covariance and why one needs to compute it ? - 3

- n-vector

scalar

sensitivity vector

Discussion is mainly focused on the question:

does the linearised error evolution model (adjoint sensitivity) approximate the non-linear model well?

However, another issue is equally (if not more) important:

do and how well we really now ?

Objective function

  • the covariance matrix of the background error or a priori covariance

  • (measure of uncertainty in before DA)

  • must be a posteriori covariance of the estimation error

  • (measure of uncertainty in after DA)


How to compute a posteriori covariance

How to compute a-posteriori covariance?


Hessian

Hessian

Definitions:

Function

Gradient

, where

Hessian

For large-scale problems, keeping Hessian or its inverse in the matrix form is not always

feasible. If the state vector dimension is , the Hessian contains elements.


Hessian operator function product form 1

Hessian – operator-function product form 1

Control problem:

Optimality system:

Void optimality system

for exact solution


Hessian operator function product form 2

Hessian – operator-function product form - 2

Definition of errors:

Non-linear optimality

system for errors:

There exists a unique representation

In the form of non-linear operator equation,

we call - the Hessian operator.

Hessian operator-function

product definition:

All operators

are defined similarly.


Hessian operator function product form 3

Hessian– operator-function product form - 3

Hessianoperator – is the only one to be

inverted in error analysis for any inverse

problem, therefore plays the key role in

defining !

Since we do not store full matrices,

we must find a way to define .

We do not know (and do not want to

know) , only .

Main Theorem

Assume errors are normally distributed, unbiased and mutually uncorrelated, and

, ;

a) if

b) if

(tangent linear hypothesis

is a ‘local’ sufficient condition, not necessary!


Optimal solution error covariance reminder

Optimal solution error covariance reminder

Holds for any control or combination of controls!


Optimal solution error covariance

Optimal solution error covariance

Case I:

1) Errors are normally distributed,

2) is moderately non-linear or are small, i.e.

Case II:

1) errors have arbitrary pdf,

a) pre-compute (define) .

b) produce single errors implementation

using certain pdf generators.

2) ‘weak’ non-linear conditions

hold

c) compute (no inversion involved)!

d) generate ensemble of

e) consider pdf of , which is not

normal!

f) use it, if you can.

Case III:

1) Errors have arbitrary pdf,

  • 2) is strongly non-linear (chaotic?)

  • or/and are very big

All as above, but compute by

iterative procedure using as

a pre-conditioner


Methods for computing the inverse hessian

Methods for computing the inverse Hessian

Direct methods:

1. Fisher information matrix:

Requires full matrix storage and algebraic inversion.

2. Sensitivity matrix method (A. Bennett), not suitable for large space-temporal data sets, i.e. good for 3DVar;

3. Finite difference method. Not suitable when constraints are partial differential

equations due to large truncation errors; requires full matrix storage and algebraic

matrix Inversion.

Iterative methods:

4. Lanczos type methods

Require only the product . The convergence could be erratic and slow if

eigenvalues are no well separated.

5. Quasi-Newton BFGS/LBFGS (generate inverse Hessian as a collateral result)

Require only the product . The convergence seems uniform. Required

storage can be controlled. Exact method (J. Greenstadt, J.Bard)


Preconditioned lbfgs

Preconditioned LBFGS

BFGS algorithm:

- pairs of vectors to be kept in

memory. LBFGS keeps only M latest

pairs and the diagonal.

Solve by LBFGS the following auxiliary control problem:

for chosen . Retrieve pairs which define projected inverse Hessian .

Then,

How to get ? For example one can use Cholesky factors .


Stupid yet 100 non linear ensemble method

Stupid, yet 100% non-linear ensemble method

1. Consider function as the exact solution to the problem

2. Assume that and .

3. The solution of the control problem

is

4. Start ensemble loop k=1,…,m.

Generate , which correspond to the given pdf. Put

Solve the non-linear optimization problem defined in 3).

Compute .

5. End ensemble loop.

6. Compute statistic


Computing a posteriori covariance in variational da i gejadze f x le dimet v shutyaev

NUMERICAL EXAMPLES: case 1

Model (non-linear convection-diffusion):

- driving bc


Computing a posteriori covariance in variational da i gejadze f x le dimet v shutyaev

NUMERICAL EXAMPLES: case 1

downstream

boundary

upstream

boundary


Computing a posteriori covariance in variational da i gejadze f x le dimet v shutyaev

NUMERICAL EXAMPLES: case 2


Computing a posteriori covariance in variational da i gejadze f x le dimet v shutyaev

NUMERICAL EXAMPLES: case 2


Computing a posteriori covariance in variational da i gejadze f x le dimet v shutyaev

NUMERICAL EXAMPLES: case 2


Computing a posteriori covariance in variational da i gejadze f x le dimet v shutyaev

NUMERICAL EXAMPLES: case 3


Computing a posteriori covariance in variational da i gejadze f x le dimet v shutyaev

NUMERICAL EXAMPLES: case 3


Computing a posteriori covariance in variational da i gejadze f x le dimet v shutyaev

NUMERICAL EXAMPLES: case 3


Summary

SUMMARY

  • Hessian plays the crucial role in the analysis of the

  • inverse problem solution errors as the only invertible

  • operator;

  • If errors are normally distributed and constraints are

  • linear the inverse Hessian is itself the covariance operator

  • (matrix) of the optimal solution error;

  • If the problem is moderately non-linear, the inverse Hessian could be a good

  • approximation of the optimal solution error covariance far beyond the validity of

  • the tangent linear hypothesis. Higher order terms could be considered in problem

  • canonical decomposition.

  • 4. Inverse Hessian can be well approximated by a sequence of quasi-Newton

  • updates (LBFGS) using the operator-vector product only. This sequence seems

  • to converge uniformly;

  • 5. Preconditioning dramatically accelerates the computing;

  • 6. The computational cost of computing inverse Hessian should not exceed the

  • cost of data assimilation procedure itself.

  • 7. Inverse Hessian is useful for uncertainty analysis, experiment design, adaptive

  • measuring techniques, etc.


Publications

PUBLICATIONS

  • I.Yu. Gejadze, V.P. Shutyaev, An optimal control problem of initial data

  • restoration, Comput. Math. & Math. Physics, 39/9 (1999), pp.1416-1425

  • F-X. Le-Dimet, V.P. Shutyaev, On deterministic error analysis in variational

  • data assimilation, Non-linear Processes in Geophysics 14(2005), pp1-10

  • F.-X. Le-Dimet, V.P. Shutyaev, I.Yu. Gejadze, On optimal solution error in

  • variational data assimilation: theoretical aspects. Russ. J. Numer. Analysis and

  • Math. Modelling (2006), v21/2, pp.139-152

  • 4. I. Gejadze, F-X. Le-Dimet and V. Shutyaev. On analysis error covariances

  • in variational data assimilation, SIAM J. Sci. Comp. (2008), v.30, no.4, 1847-74.

  • 5. I. Gejadze, F-X. Le-Dimet and V. Shutyaev. Optimal solution error

  • covariances in variational data assimilation problems, SIAM J. Sci. Comp.

  • (2008), to be published.

Covariance matrix of the initial control problem for the diffusion equation.

3-sensor configuration


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