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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

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)

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

- 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)

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.

Control problem:

Optimality system:

Void optimality system

for exact solution

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.

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!

Holds for any control or combination of controls!

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

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)

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 .

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

NUMERICAL EXAMPLES: case 1

Model (non-linear convection-diffusion):

- driving bc

NUMERICAL EXAMPLES: case 1

downstream

boundary

upstream

boundary

NUMERICAL EXAMPLES: case 2

NUMERICAL EXAMPLES: case 2

NUMERICAL EXAMPLES: case 2

NUMERICAL EXAMPLES: case 3

NUMERICAL EXAMPLES: case 3

NUMERICAL EXAMPLES: case 3

- 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.

- 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