Minimisation Techniques 1 Assimilation Algorithms: Minimisation Techniques Yannick Trémolet ECMWF Data Assimilation Training Course March 2006.

Présentation au sujet: "Minimisation Techniques 1 Assimilation Algorithms: Minimisation Techniques Yannick Trémolet ECMWF Data Assimilation Training Course March 2006."— Transcription de la présentation:

1 Minimisation Techniques 1 Assimilation Algorithms: Minimisation Techniques Yannick Trémolet ECMWF Data Assimilation Training Course March 2006

2 Minimisation Techniques 2 4D Variational Data Assimilation

3 Minimisation Techniques 3 Incremental 4D-Var

4 Minimisation Techniques 4 Incremental 4D-Var

5 Minimisation Techniques 5 Incremental 4D-Var

6 Minimisation Techniques 6 The Outer Iterations

7 Minimisation Techniques 7 The Inner Iterations

8 Minimisation Techniques 8 Minimisation: Newton method

9 Minimisation Techniques 9 Minimisation: Newton method

10 Minimisation Techniques 10 Minimisation: Quasi-Newton method

11 Minimisation Techniques 11 Minimisation: Quasi-Newton method

12 Minimisation Techniques 12 Limited Memory Quasi-Newton

13 Minimisation Techniques 13 Minimisation: Steepest Descent

14 Minimisation Techniques 14 Minimisation: Steepest Descent The first step fully minimizes the function in the descent direction, but this is undone by subsequent steps. We want to avoid this.

15 Minimisation Techniques 15 Minimisation: Conjugate Gradient

16 Minimisation Techniques 16 Conjugate Gradient Convergence

17 Minimisation Techniques 17 Preconditioning

18 Minimisation Techniques 18 4D-Var Preconditioning

19 Minimisation Techniques 19 A case of poor convergence

20 Minimisation Techniques 20 Theoretical example

21 Minimisation Techniques 21 Theoretical example

22 Minimisation Techniques 22 A case of poor convergence

23 Minimisation Techniques 23 Hessian Preconditioning

24 Minimisation Techniques 24 Hessian Eigenvectors Preconditioning

25 Minimisation Techniques 25 4D-Var Eigenvalues Eigenvalue N =1 1 =3105.4 26 =492.75 Preconditioning reduces the condition number k= 1 / N from 3105.4 to 492.75

26 Minimisation Techniques 26 Conjugate Gradient and Lanczos Algorithm

27 Minimisation Techniques 27 Lanczos Algorithm

28 Minimisation Techniques 28 Lanczos Algorithm

29 Minimisation Techniques 29 Conjugate Gradient and Lanczos Algorithm

30 Minimisation Techniques 30 Superlinear Convergence

31 Minimisation Techniques 31 Rounding Error

32 Minimisation Techniques 32 CG Cost function reduction Quasi-Newton with inexact line searches Quasi-Newton with exact line searches Conjugate Gradient without orthogonalisation Conjugate Gradient with orthogonalisation

33 Minimisation Techniques 33 CG Gradient norm reduction Quasi-Newton with inexact line searches Quasi-Newton with exact line searches Conjugate Gradient without orthogonalisation Conjugate Gradient with orthogonalisation

34 Minimisation Techniques 34 4D-Var Cost function reduction Variational Quality Control

35 Minimisation Techniques 35 4D-Var gradient norm reduction Convergence is roughly twice as fast with Hessian preconditioning.

36 Minimisation Techniques 36 CG reduction of norm of gradient 0.05

37 Minimisation Techniques 37 Spectrum of preconditioned Hessian

38 Minimisation Techniques 38 RMS of T analysis increments

39 Minimisation Techniques 39 4D-Var Convergence

40 Minimisation Techniques 40 4D-Var Convergence

41 Minimisation Techniques 41 Summary

42 Minimisation Techniques 42