Approximate iterative methods for data assimilation

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Approximate iterative methods for data
assimilation

• Amos Lawless1, Serge Gratton2
• and
• Nancy Nichols1
• 1Department of Mathematics,
• University of Reading
• 2CERFACS, France

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Outline

• Data assimilation and the Gauss-Newton iteration
• Two common approximations
• Truncation of inner minimization
• Approximation of linear model
• Conclusions

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Data Assimilation
Aim Find the best estimate (analysis) of the
true state of the atmosphere, consistent with
both observations distributed in time and system
dynamics.
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Nonlinear least squares problem
subject to
- Background state - Observations - Observation
operator - Background error covariance matrix -
Observation error covariance matrix
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Incremental 4D-Var

• Set (usually equal to background)
• For k 0, , K
• Solve inner loop minimization problem
• with
• Update

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General nonlinear least squares
4D-Var cost function can be written in this form
with
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Gauss-Newton iteration
The Gauss-Newton iteration is
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1D Shallow Water Model
Nonlinear continuous equations with
We discretize using a semi-implicit
semi-Lagrangian scheme.
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Assimilation experiments

• Observations are generated from a model run with
  the true initial state
• First guess estimate is truth with phase error.
• No background term included.
• Inner problem solved using minimization by CONMIN
  algorithm, with stopping criterion on relative
  change in objective function.
• Assimilation window is 100 time steps.

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Truncation of inner loop
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Truncated Gauss-Newton

• Linear quadratic problem is not solved exactly on
  each iteration. The residual error is given by
  rk.
• Solve such that on each iteration

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Convergence of Truncated Gauss-Newton
Theorem (GLN) (i) implies G-N
converges. (ii) implies TG-N converges.
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Convergence of Truncated Gauss-Newton (2)
Theorem (GLN) Conditions of DS hold,
then implies TG-N converges. The rate of
convergence can also be established. Proof
Extension of DS
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Approximation of linear model
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Approximation of linear model
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Perturbed Gauss-Newton

• The linear model is approximated. Equivalent to
  replacing by
• Exact G-N solves
• Perturbed G-N solves

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Convergence of Perturbed Gauss-Newton
Theorem (GLN) implies convergence of PG-N to
x . Theorem (GLN) Distance between fixed
points depends on distance between
pseudo-inverses (JTJ)-1JT and (JTJ)-1JT
calculated at x and on residual f(x).




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Conclusions

• Incremental 4D-Var without approximations is
  equivalent to a Gauss-Newton iteration.
• In operational implementation we usually
  approximate the solution procedure.
• Truncation of inner minimization may improve
  overall convergence.
• Good solution obtained even with approximate
  linear model (PFM).
• Theoretical results obtained by reference to
  Gauss-Newton method.

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Convergence - Case 1
12 G-N iterations