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
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
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
integrate by parts
If = zero and
For initial value problem
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 ?
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.
Void optimality system
for exact solution
Definition of errors:
system for errors:
There exists a unique representation
In the form of non-linear operator equation,
we call - the Hessian operator.
are defined similarly.
Hessianoperator – is the only one to be
inverted in error analysis for any inverse
problem, therefore plays the key role in
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 .
Assume errors are normally distributed, unbiased and mutually uncorrelated, and
(tangent linear hypothesis
is a ‘local’ sufficient condition, not necessary!
Holds for any control or combination of controls!
1) Errors are normally distributed,
2) is moderately non-linear or are small, i.e.
1) errors have arbitrary pdf,
a) pre-compute (define) .
b) produce single errors implementation
using certain pdf generators.
2) ‘weak’ non-linear conditions
c) compute (no inversion involved)!
d) generate ensemble of
e) consider pdf of , which is not
f) use it, if you can.
1) Errors have arbitrary pdf,
All as above, but compute by
iterative procedure using as
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
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)
- 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 .
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
4. Start ensemble loop k=1,…,m.
Generate , which correspond to the given pdf. Put
Solve the non-linear optimization problem defined in 3).
5. End ensemble loop.
6. Compute statistic
NUMERICAL EXAMPLES: case 1
Model (non-linear convection-diffusion):
- driving bc
NUMERICAL EXAMPLES: case 1
Covariance matrix of the initial control problem for the diffusion equation.