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Approximate iterative methods for data assimilation

Approximate iterative methods for data assimilation. Amos Lawless 1 , Serge Gratton 2 and Nancy Nichols 1 1 Department of Mathematics, University of Reading 2 CERFACS, France. Outline. Data assimilation and the Gauss-Newton iteration Two common approximations

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Approximate iterative methods for data assimilation

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  1. Approximate iterative methods for data assimilation Amos Lawless1, Serge Gratton2 and Nancy Nichols1 1Department of Mathematics, University of Reading 2CERFACS, France

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

  3. 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.

  4. Nonlinear least squares problem subject to - Background state - Observations - Observation operator - Background error covariance matrix - Observation error covariance matrix

  5. Incremental 4D-Var Set (usually equal to background) Fork = 0, …, K Solve inner loop minimization problem with ; Update

  6. General nonlinear least squares 4D-Var cost function can be written in this form with

  7. Gauss-Newton iteration The Gauss-Newton iteration is

  8. 1D Shallow Water Model Nonlinear continuous equations with We discretize using a semi-implicit semi-Lagrangian scheme.

  9. 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.

  10. Truncation of inner loop

  11. 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

  12. Convergence of Truncated Gauss-Newton Theorem (GLN): (i) implies G-N converges. (ii) implies TG-N converges.

  13. Convergence of Truncated Gauss-Newton (2) Theorem (GLN): Conditions of D&S hold, then implies TG-N converges. The rate of convergence can also be established. Proof: Extension of D&S

  14. Approximation of linear model

  15. Approximation of linear model

  16. Perturbed Gauss-Newton The linear model is approximated. Equivalent to replacing by ExactG-N solves Perturbed G-N solves

  17. 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*). ~ ~ ~ ~

  18. 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.

  19. Convergence - Case 1 12 G-N iterations

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